research article numerical optimal control for problems
TRANSCRIPT
Research ArticleNumerical Optimal Control for Problems withRandom Forced SPDE Constraints
R Naseri and A Malek
Department of Applied Mathematics Faculty of Mathematical Sciences Tarbiat Modares University PO Box 14115-134 Tehran Iran
Correspondence should be addressed to A Malek malamodaresacir
Received 22 September 2013 Accepted 10 December 2013 Published 20 February 2014
Academic Editors H Homeier and F Zirilli
Copyright copy 2014 R Naseri and A Malek This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
A numerical algorithm for solving optimization problems with stochastic diffusion equation as a constraint is proposed Firstseparation of randomanddeterministic variables is done viaKarhunen-Loeve expansionThen the problem is discretized in spatialpart using the finite element method and the polynomial chaos expansion in the stochastic part of the problemThis process leadsto the optimal control problem with a large scale system in its constraint To overcome these difficulties the adjoint technique forderivative computation to implementation of the optimal control issue in preconditioned Newtonrsquos conjugate gradient method isused By some numerical simulation it is shown that this hybrid approach is efficient and simple to implement
1 Introduction
Physical problems in many cases can be formulated asoptimization problems These problems are utilized to gaina more widely understanding of physical systems Typicallythese problems depend on some models which in manycases are deterministic Uncertainty might plague every-thing from modeling assumptions to experimental dataAs such in order to accommodate for these uncertaintiesmany practitioners have developed stochastic models Inorder to make sense of and solve these models in additionto randomness of the models some additional theory isrequired to the resulting optimization problems This paperproposes an adjoint based approach for solving optimizationproblems governed by stochastic diffusion equations Inorder to deal with the stochastic partial differential equations(SPDE) as a constraint in an optimization problem one mayfirst solve the SPDE Since the provided systems by thisapproach are random and nonlinear such kind of problemsare difficult to handle and very challenging One of the mostchallenging examples in this area is the control of stochasticdiffusion equations with random forcing [1] Polynomialchaos expansion (PCE) [2] provides a good direction forsolution of nonlinear SPDEs numerically In the presence ofthe random forcing PCE seems to be more accurate and
efficient numerical method than Monte Carlo simulation Infact the PCE can be interpreted as the Fourier expansionin the probability space Particularly the aim of this work isthe numerical solution for the distributed control problemsinvolving the stochastic diffusion equations in the form
minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω (1)
119906 (x 120596) = 0 on 120597119863 times Ω (2)
where 119863 is the spatial domain 120597119863 is the boundary of thespatial domain Ω is probability space x isin 119863 120596 isin Ω119906 is the solution of the SPDE 119911 is the source functionand 119886(x 120596) is the permeability field of the problem As animportant assumption for the stochastic diffusion equation(1) it is assumed that the random coefficient 119886(x 120596) satisfiesthe elliptic condition [3] that is there exists a constant 119886minsuch that
0 lt 119886min le 119886 (x 120596) forall (x 120596) isin 119863 times Ω (3)
Karhunen-Loeve expansion (KLE) of correlated randomfunctions is used to separate the random and deterministicparts in random coefficient [4 5] Therefore a finite dimen-sional approximation 119886
119872(x 120596) is performed by truncating the
Hindawi Publishing CorporationISRN Applied MathematicsVolume 2014 Article ID 974305 11 pageshttpdxdoiorg1011552014974305
2 ISRN Applied Mathematics
Karhunen-Loeve expansion of the permeability field 119886(x 120596)Then (1) is approximated by
nabla (119886
119872 (x 120596) nabla119906) = 119911 (4)
For approximation of the function 119906 with 119873randomvariables and the Gaussian random variables of the highestdegree119870 the number of PCE coefficients are(119873+119870)119873119870[6] Now weak formulation of (4) and then Galerkinfinite element method are used to solve the SPDE problemapproximately Generally in nature optimization with SPDE-constraint is infinite dimensional large and complex Thereare two numerical approaches for solving this problem [2 7]
(i) discretize then optimize (DO) for problems that canbe trivially discretized first
(ii) optimize then discretize (OD) for problems that aredifferentiable
Our intention is to work with discretizethen optimizationalgorithms for smooth functions Thus we follow the DOapproach Galerkin finite element method incorporated withpolynomial chaos expansion is used for discretizing theSPDE By computing gradient and Hessian of the Lagrangianaugmented function preconditionedNewtonrsquos conjugate gra-dient method is used to compute the best right hand side vec-tor (control values at grid points) where the correspondingsolution of SPDE (state values) according to this vector staysas near as to the desired value approximation and has lowestcost in the objective function The organization of this paperis as follows in Section 2 problem formulation in additionto KLE PCE and stochastic Galerkin method is presentedIn Section 3 distributed control of random forced diffusionequation with some numerical examples is proposed
2 Problem Formulation
We consider optimal control problems of the form
min119906isin119880 119911isinZ
119869 (119906 119911)
st 119890 (119906 119911) = 0
(5)
where 119869 119867
1
0(119863)⨂119871
2(Ω) rarr R is the objective function
119890 119880 = 119867
1
0(119863)⨂119871
2(Ω) rarr Z = 119871
2(119863) is an operator which
stands as the constraint 119863 is the spatial domain Ω is theprobability space and⨂ is the tensor product
To provide a concrete setting for our discussion thefollowing problem is considered as the constraint operator
minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω
119906 (x 120596) = 0 on 120597119863 times Ω
(6)
where 119886(x 120596) is the correlated random fields Hence thesolutions 119906(x 120596) of the SPDE (6) are also random fields It is
assumed that the control space 119911(x) isin 119871
2(119863) The weak form
of the constraint function (6) can be defined for all V isin 119880 as
⟨119890 (119906 119911) V⟩119880lowast 119880
= int
Ω
120588 (120596)int
119863
(119886 (x 120596) nabla119906 (x 120596)
sdot nablaV (x 120596) 119889x119889120596 minus119911 (x) V (x 120596)) 119889x119889120596
(7)
It is assumed that the objective or cost functional is as
119869 (119906 119911) = 119864 [
1
2
119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
(8)
where 119864(sdot) denotes the expected value is a given desiredfunction and 120572 is the regularization parameter Obviouslycontrolling the solution of this problem leads to controllingthe statistics of the solution for example the expected valueof the solution given by
119864[119906 (x 120596) minus (x)]21198712(119863)
le 119864 [119906 (x 120596) minus (x)21198712(119863)
]
le 119906 (x 120596) minus (x)21198712(119863)otimes119871
2(Ω)
(9)
So the optimal control problem can be interpretedas follows given the random field 119886(x 120596) minimize theobjective function (8) over all 119911 isin Z subject to 119906 isin 119880 suchthat for the given random field 119886(x 120596) satisfies in the weakformulation (7) [1]
21 Basic Definitions In (5) 119869 and 119890 are assumed to becontinuously F-differentiable and that for each 119911 isin Z thestate equation 119890(119906 119911) = 0 possesses a unique correspondingsolution 119906(119911) isin 119880 Similar to the deterministic PDEsexistence and uniqueness of the solution for these kind ofSPDEs can be established using the Lax-Milgram theoremThus there is a solution operator 119911 isin Z rarr 119906(119911) isin 119880In addition it is assumed that 119890
119906(119906(119911) 119911) isin L(119880Z) is
continuously invertible Then existence and uniqueness ofthe solution for the above optimal control problems as well asthe continuously differentiability of 119906(119911) are ensured by theimplicit function theorem [8]
22 Karhunen-Loeve Expansion (KLE) Consider a randomfield 119886(x 120596) x isin 119863 with finite second order moment
int
Ω
119864 [119886
2(x 120596)] 119889119909 lt infin (10)
Assume that 119864[119886] = 119886(x) It is possible to expand 119886(x 120596)for a given orthonormal basis 120595
119896 in 119871
2(119863) as a generalized
Fourier series
119886 (x 120596) = 119886 (x) +infin
sum
119896=1
119886
119896 (120596) 120595119896 (
x) (11)
ISRN Applied Mathematics 3
where
119886
119896 (120596) = int
Ω
119886 (x 120596) 120595119896 (x) 119889x 119896 = 1 2 (12)
are random variables with zero means It is importantnow to find a special basis 120601
119896 that makes corresponding
119886
119896uncorrelated 119864[119886
119894119886
119895] = 0 for all 119894 = 119895 Denoting the
covariance function of 119886(x 120596) by 119877(x y) = 119864[119886(x 120596)119886(y 120596)]the basis functions 120601
119896 should satisfy
119864 [119886
119894119886
119895] = int
119863
120601
119894 (x) 119889xint
119863
119877 (x y) 120601119895(y) 119889y = 0 119894 = 119895
(13)
Completion and orthonormality of 120601119896 in 119871
2(119863) follow
that 120601119896(x) are eigenfunctions of 119877(x y)
int
119863
119877 (x y) 120601119895(y) 119889119910 = 120582
119895120601
119895 (x) 119895 = 1 2 (14)
where 120582
119895= 119864[119886
2
119895] gt 0 Indeed by choosing basis
functions 120601119896(119909) as the solutions of the eigenproblem (14) the
randomvariables 119886119896(120596)will be uncorrelated In (14) denoting
120579
119896= 119886
119896radic120582
119896 yields the following expansion
119886 (x 120596) = 119886 (x) +infin
sum
119896=1
radic
120582
119896120579
119896(120596) 120601119896 (
x) (15)
where 120579
119896satisfy 119864[120579
119896] = 0 and 119864[120579
119894120579
119895] = 120575
119894119895 In the case
that 119886(x 120596) is considered as a Gaussian process 119886119896(120596) 119896 =
1 2 will be independent Gaussian random variables Theexpansion (15) is known as the Karhunen-Loeve expansion(KLE) of the stochastic process 119886(x 120596)
Using the KLE (15) the stochastic process can be rep-resented as a series of uncorrelated random variables Sincethe basis functions 120601
119896(x) are deterministic the spatial depen-
dence of the random process can be resolved by them Theconvergence property of the KLE to the random process119886(x 120596) can be represented in the mean square sense
lim119873rarrinfin
int
119863
119864
1003816
1003816
1003816
1003816
119886 (x 120596) minus 119886
119873 (x 120596)
1003816
1003816
1003816
1003816
2119889x = 0 (16)
where
119886
119873= 119886 (x) +
119873
sum
119896=1
radic
120582
119896120579
119896120601
119896(17)
is a finite term KLE [4 5]
23 Polynomial Chaos Expansion (PCE) There are problemsas the solution of a PDE with random inputs that thecovariance function of a random process 119906(x 120596) x isin 119863 isnot knownThe solution of such problems can be representedusing a polynomial chaos expansion (PCE) given by
119906 (x 120596) =
infin
sum
119896=1
119906
119896 (x) Ψ119896 (120585) (18)
where the functions 119906
119896(x) are deterministic coefficients 120585
is a vector of orthonormal random variables and Ψ
119896(120585) are
multidimensional orthogonal polynomials that satisfy in thefollowing properties
⟨Ψ
1⟩ equiv 119864 [Ψ
1] = 1 ⟨Ψ
119896⟩ = 0 119896 gt 1
⟨Ψ
119894Ψ
119895⟩ = ℎ
119894120575
119894119895
(19)
The convergence property of PCE for a random quantityin 119871
2 is ensured by the Cameron-Martin theorem [6 9] thatis
⟨119906 (x 120596) minus
infin
sum
119896=1
119906
119896 (x) Ψ119896 (120585)⟩
1198712
997888rarr 0 (20)
Hence this convergence justifies a truncation of PCE to afinite number of terms
119906 (x 120596) =
119875
sum
119896=1
119906
119896 (x) Ψ119896 (120585) (21)
where the value of 119875 is determined by the highest degree ofpolynomial 119870 used to represent 119906 and the number 119873 ofrandom variablesmdashthe length of 120585mdashwith the formula 119875+1 =
(119873 + 119870)119873119870 Generally the value of 119873 is the same asthe number of uncorrelated random variables in the systemor equivalently the truncation length of the truncated KLETypically the value of119870 is chosen by some heuristic methodIndeed in the case of119870 = 1 and119873 randomvariables the KLEis a special case of the PCE [9 10]
24 Stochastic Galerkin Suppose 119882
119894sub 119871
2
120588119894
(Ω
119894) with dimen-
sion 120588
119894for 119894 = 1 119872 and 119881 sub 119867
1
0(119863) with dimension
119873 In addition let 120595
119894
119899
119901119894
119899=1for 119894 = 1 119872 be basis of
119882
119894and 120601
119894
119873
119894=1a basis of 119881 The finite dimensional tensor
product space 119882
1⨂sdot sdot sdot⨂119882
119872⨂119881 can be defined as the
space spanned by the functions 120595
1
1198991
120595
119872
119899119872
120601
119894 for all 119899
1isin
1 119901
1 119899
119872isin 1 119901
119872 and 119894 isin 1 119873
For simplification let n denote a multi-index whose 119896thcomponent 119899
119896isin 1 119901
119896 and I denote the set of such
multi-indices then the basis functions for the tensor productspace have the form
Vn119894 (x 120596) = 120601
119894(x) Ψ119899 (120596) (22)
where
Ψ
119899=
119872
prod
119896=1
120595
119896
119899119896
(120596
119896) (23)
Then the Galerkin method is looking for a solution isin 119882
1⨂sdot sdot sdot⨂119882
119872⨂119881 such that for x = (119909 119910) all n isinI
and 119894 = 1 119873
int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla119906
ℎ119875 (x 120596) sdot nablaVn119894 (x 120596) 119889x119889120596
= int
Ω
120588 (120596) int
119863
119911 (x 120596) Vn119894 (x 120596) 119889x119889120596
(24)
4 ISRN Applied Mathematics
where 120588(120596) is the integration weight Equation (24) is equiv-alent to the following equation which results by plugging inthe explicit formula for the basis functions
int
Ω
120588 (120596)Ψn (120596) int
119863
119886 (x 120596) nabla119906
ℎ119875 (x 120596) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψn (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(25)
for all n isinI and 119894 = 1 119873Now considering the approximate solution as
119906
ℎ119875 (x 120596) =
119873
sum
119895=1
sum
misinI119906
119895m120601
119895 (x) Ψm (120596) (26)
and substituting (26) into (25) yields119873
sum
119895=1
sum
misinI119906
119895m int
Ω
120588 (120596)Ψn (120596)Ψm (120596)
times int
119863
119886 (x 120596) nabla120601
119895 (x) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψn (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(27)
for allm isinI and 119894 = 1 119873 Let 119875 = |I| = prod
119872
119896=1119901
119896 then
a natural bijection between 1 119875 and Ican be definedThus (27) can be written as
119875
sum
119898=1
119873
sum
119895=1
119906
119898119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596)
times int
119863
119886 (x 120596) nabla120601
119895 (x) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψ119899 (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(28)
Now if 119886 and 119911 have the following KLE
119886 (x 120596) = 119886
0 (x) +
119872
sum
119894=1
120596
119894119886
119894 (x)
119911 (x 120596) = 119911
0 (x) +
119872
sum
119894=1
120596
119894119911
119894 (x)
(29)
then we can rewrite the Galerkin projection (28) as119875
sum
119898=1
119873
sum
119895=1
119906
119898119895((119870
0)
119894119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596
+
119872
sum
119896=1
(119870
119896)
119894119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 120596119896
119889120596)
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(30)
where
(119870
119896)
119894119895= int
119863
119886
119896 (x) nabla120601
119894 (x) sdot nabla120601
119895 (x) 119889x
cong sum
119903
119908
119903119886
119896(x119903) nabla120601
119894(x119903) sdot nabla120601
119895(x119903)
(119911
119896)
119894= int
119863
119911
119894 (x) 120601119894 (x) 119889x cong sum
119903
119908
119903119911
119894(x119903) 120601
119894(x119903)
(31)
for 119896 = 1 119872 [3] Equation (31) is approximated byquadrature rule in spatial domain Now assume that thereexist functions Ψ
119899 119899 = 1 119875 such that
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596 = 120575
119898119899
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 120596119896
119889120596 = 119862
119896119899120575
119899119898
(32)
where for 119898 = 119899 the above integral is approximated usingquadrature rule in random domain as follows
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119899 (
120596) 120596119896119889120596 cong sum
119904
119908
1015840
119904120588 (120596
119904) Ψ
2
119899(120596
119904) 120596
119896 (33)
Then (30) becomes
119875
sum
119898=1
119873
sum
119895=1
119906
119898119895((119870
0)
119894119895+
119872
sum
119896=1
119862
119896119899(119870
119896)
119894119895)120575
119899119898
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(34)
or equivalently
119873
sum
119895=1
119906m119895 ((119870
0)
119894119895+
119872
sum
119896=1
119862
119896119899(119870
119896)
119894119895)
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(35)
In (34) and (35) we have
int
Ω
120588 (120596)Ψ119899 (120596) 119889120596 = 0 for 119899 = 0
int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596 = int
Ω
120588 (120596)Ψ0 (120596)Ψ119899 (
120596) 120596119896119889120596
= 119862
119896119899120575
0119899= 119862
1198960
(36)
ISRN Applied Mathematics 5
Finally (35) can be considered as the following blockdiagonal system
(
(
(
119870
0+
119872
sum
119896=1
119862
1198961119870
119896sdot sdot sdot 0
d
0 sdot sdot sdot 119870
0+
119872
sum
119896=1
119862
119896119901119870
119896
)
)
)
(
1
119875
)
= (
119911
1
119911
119901
)
(37)
or briefly 119870 =
119911
3 Distributed Optimal Control Problem
Using the stochastic Galerkin method for general objectivefunctions computing the derivatives becomes very tediousand rather messy Consider the following optimal controlproblem
min119906isin119880 119911isinz
119869 (119906 119911) =
1
2
119864 [119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
st minus nabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω
119906 (x 120596) = 0 on 120597119863 times Ω
(38)
The solutions to linear elliptic SPDEs live in the space119880 =
119867
1
0(119863)⨂119871
2(Ω) and the control space Z = 119871
2(119863) Denoting
the constraint equation by 119890(119906 119911) = 0 usually it is possibleto invoke the implicit function theorem to find a solution119906(119911) to the constraint equation This leads to an implicitlydefined objective function
119869(119911) = 119869(119906(119911) 119911) Now we focuson computing derivatives of 119869 [11] For any direction 119904 isin Z
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= ⟨119869
119906 (119906 (119911) 119911) 119906119911 (
119911) 119904⟩
119880lowast119880
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(39)
where119880lowast andZlowast are the dual space of119880 andZ respectivelyNow computing the derivative of the constraint 119890with respectto 119911 in the direction 119904 and applying it to the implicit solutionto 119890(119906(119911) 119911) = 0 yield
119890
119906 (119906 (119911) 119911) 119906119911 (
119911) 119904 + 119890
119911 (119906 (119911) 119911) 119904 = 0 (40)
So 119906
119911(119911) is
119906
119911 (119911) 119904 = minus119890
119906(119906 (119911) 119911)
minus1119890
119911 (119906 (119911) 119911) 119904
(41)
Hence
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= minus ⟨119869
119906 (119906 (119911) 119911) 119890119906(
119906 (119911) 119911)
minus1
times 119890
119911 (119906 (119911) 119911) 119904⟩
119880lowast119880
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(42)
Applying the adjoint to the 119906
119911(119911) term
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= minus ⟨119890
119911(119906 (119911) 119911)
lowast119890
119906(119906 (119911) 119911)
minuslowast
times 119869
119906 (119906 (119911) 119911) 119904⟩
Zlowast Z
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
= minus ⟨119890
119911(119906 (119911) 119911)
lowast119890
119906(119906 (119911) 119911)
minuslowast119869
119906(119906 (119911) 119911)
+119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(43)
Considering the adjoint state 120583 = 120583(119911) isin Zlowast as the solutionto
119890
119906(119906 (119911) 119911)
lowast120583 = minus119869
119906 (119906 (119911) 119911) (44)
the derivative of 1198691015840 becomes
119869
1015840(119911) = 119890
119911(119906 (119911) 119911)
lowast120583 (119911) + 119869
119911 (119906 (119911) 119911)
(45)
Theweak formof the constraint function is defined as followsfor all V isin 119880
⟨119890 (119906 119911) V⟩119880lowast 119880 = int
Ω
120588 (120596)int
119863
(119886 (x 120596) nabla119906 (x 120596) sdot nablaV (x 120596)
minus119911 (x 120596) V (x 120596)) 119889x119889120596
(46)
Thus the constraint function 119890 119880 times Z rarr 119880
lowast and 119884 = 119880
lowastSince the constraint function 119890 acts linearly with respect to 119906
and 119911 thus the corresponding derivative of 119890 with respect to119911 in the direction 120575119911 isin Z for all V isin 119880 is given by
⟨119890
119911 (119906 119911) 120575119911 V⟩
119880lowast119880
= minusint
Ω
120588 (120596)int
119863
120575119911 (x 120596) V (x 120596) 119889x119889120596
(47)
Similarly the derivative of 119890 with respect to 119906 for anydirection 120575119906 isin 119880 and for all V isin 119880 is given by
⟨119890
119906 (119906 119911) 120575119906 V⟩
119880lowast119880
= int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla120575119906 (x 120596)
sdot nablaV (x 120596) 119889x119889120596
(48)
Both of these derivatives are symmetric so 119890
119906(119906 119911)
lowast=
119890
119906(119906 119911) and 119890
119911(119906 119911)
lowast= 119890
119911(119906 119911) From here the adjoint can
be computed as
119890
119906 (119906 (119911) 119911) 120583 = minus119869
119906 (119906 (119911) 119911) (49)
Using this to any direction V isin 119880
⟨119890
119906 (119906 (119911) 119911) 120583 V⟩
119880lowast119880
= int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla120583 (x 120596)
sdot nablaV (x 120596) 119889x119889120596
= minus119869
119906 (119906 (119911) 119911) V
(50)
6 ISRN Applied Mathematics
By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is
⟨119869
119906 (119906 119911) 120575119906⟩
119880lowast119880
= 119864 [119906 (x 120596) minus (x)21198712(119863)
times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)
]
(51)
Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin
Z direction is
⟨119869
119911 (119906 119911) 120575119911⟩
Zlowast Z= 120572⟨119911 120575119911⟩Z (52)
With these computations one can compute the firstderivative of
119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883
ℎsub 119867
1
0(119863) is a finite dimensional
subspace of dimension 119873 and 119884
ℎsub 119871
2(Ω) is a finite
dimensional subspace of dimension 119875 then 119883
ℎ⨂119884
ℎis a
finite dimensional subspace of the state space 119880 Similarlylet Z
ℎsub Z be a finite dimensional subspace of the control
space (with dimension) For the stochastic Galerkin methodit is assumed that 119884
ℎ= P119875minus1 the space of polynomials with
highest degree 119875minus1 and119883
ℎis any finite element space here
119883
ℎis the space of linear functions built on a given meshT
ℎ
We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884
ℎ that is 119884
ℎ= spanΨ
1(120596) Ψ
119875(120596)
where
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596 = 120575
119898119899 (53)
The discretized optimization problem in the stochasticGalerkin framework is
min119911ℎisinZℎ
119869
ℎ119875(119911
ℎ) =
1
2
119864
[
[
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
119875
sum
119896=1
119873
sum
119895=1
(
119896(119911
ℎ))
119895Ψ
119896 (120596) 120601119895 (
x)
minus (x)1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
2
1198712(119863)
]
]
+
120572
2
1003817
1003817
1003817
1003817
119911
ℎ
1003817
1003817
1003817
1003817
2
1198712(119863)
(54)
where
119896(119911
ℎ) = (
1(119911
ℎ)
119879
119875(119911
ℎ)
119879)
119879
=
(55)
is the stochastic Galerkin solution to the state equation
(
119870
11sdot sdot sdot 119870
1119875
d
119870
1198751sdot sdot sdot 119870
119875119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119870
(
1
119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=
= (
119864 [Ψ
1]119872
119911
ℎ
119864 [Ψ
119875]119872
119911
ℎ
)
= (
120575
11119872
119911
ℎ
120575
1119875119872
119911
ℎ
)
(56)
since 119864[Ψ
119896] = 119864[1Ψ
119896] = 119864[Ψ
1Ψ
119896] = 120575
1119896 The blocks of the
above 119870matrix have the form
(119870
119896119897)
119894119895= int
Ω
120588 (120596)Ψ
119896(120596)Ψ119897 (
120596)
times int
119863
119886 (x 120596) nabla120601
119894 (x) sdot nabla120601
119895 (x) 119889x119889120596
(57)
First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations
minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))
in 119863 times Ω
120583 (x 120596) = 0 on 120597119863 times Ω
(58)
Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865
119879
1 119865
119879
119901)
119879 and 119865
119896is defined
as
(
119865
119896)
119894= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596
= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895120601
119895 (x) Ψ119897 (120596) 120601119894 (
x)
minus (x) 120601119894 (x))119889x119889120596
= minus
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
(x) 120601119894 (x) 119889x
=
119887
119894120575
1119896minus
119875
sum
119897=1
119873
sum
119895=1
119872
119894119895(
119897)
119895120575
119896119897
=
119887
119894120575
1119896minus
119873
sum
119895=1
119872
119894119895(
119897)
119895
(59)
in which
119887
119894= int 120601
119894 and119872
119894119895= int 120601
119894120601
119895
Thus
(
119865
119896)
119894=
119887 minus 119872119906
1if 119896 = 1
minus119872119906 if 119896 = 2 119875
(60)
where b= [
119887
1
119887
119873] and the matrix 119872 = (119872
119894119895) is of order
119873 times 119873
ISRN Applied Mathematics 7
Now the derivative of
119869(119911) in the direction 120601
119895(x) for all
119895 = 1 119873 with the computed adjoint state is
⟨
119869
1015840(119911) 120601119895 (
x)⟩Zlowast Z
= ⟨119890
119911(119906 (119911 sdot sdot) 119911)
lowast120583
+ 119869
119911 (119906 (119911 sdot sdot) 119911) 120601119895 (
x) ⟩Zlowast Z
= minusint
Ω
120588 (120596)int
119863
120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596
+ 120572int
119863
119911 (119909) 120601119895 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894int
Ω
120588 (120596)Ψ119896 (120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ 120572
119873
sum
119894=1
119911
119894int
119863
120601
119895 (x) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894119872
119894119895120575
1119896+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
= minus
119873
sum
119894=1
(
1)
119894119872
119894119895+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
(61)
Therefore the gradient can be computed as follows
119869
1015840(119911) = 119872(120572
119911
ℎminus
1)
(62)
Now for any vector V isin Z it is possible to write V as
V =
119873
sum
119896=1
V119896120601
119896 (x) (63)
In order to compute the Hessian times a vector that ismultiply the Hessian of the
119869 with a some vector V isin Z theequation 119890
119906(119906(119911) 119911)119908 = 119890
119911(119906(119911) 119911)V for 119908 must be solved
Similar to the adjoint computation119908 solves the linear system119870 = 119866 where
119866 = (
119866
119879
1
119866
119879
119875) and
119866
119896for 119896 = 1 119875 is
(
119866
119896)
119894= int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
119873
sum
119895=1
V119895120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895V119895120575
1119896
(64)
for all 119894 = 1 119873 Thus
119866
119896=
119872V if 119896 = 1
0 if 119896 = 2 119875
(65)
Now solving 119890
119906(119906(119911) 119911)119902 = 119869
119906119906(119906(119911) 119911)119908 for 119902 requires
the solution to the linear system 119870 119902 = 119867 where
(
119867
119896)
119894=
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895(
119896)
119895
(66)
or equivalently
119867
119896= 119872
119896for all 119896 = 1 119875 Hence in the
direction 120601
119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by
⟨
119869
10158401015840(119911) V 120601119894⟩Zlowast Z
asymp ⟨
119869
10158401015840
ℎ119901(119911
ℎ) V 120601
119894⟩
Zlowast Z
= minusint
Ω
120588 (120596)int
119863
119875
sum
119896=1
119873
sum
119895=1
( 119902
119896)
119895120601
119895 (x) Ψ119896 (120596)
times 120601
119894 (x) 119889x119889120596
+ 120572int
119863
V (119909) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119895=1
119872
119894119895( 119902
119896)
119895120575
1119896+ 120572
119873
sum
119895=1
119872
119894119895V119895
=
119873
sum
119895=1
119872
119894119895(120572V
119895minus ( 119902
1)
119895)
(67)
Therefore
119869
10158401015840(119911) V asymp 119872(120572V minus 119902
1)
(68)
Having (62) and (68) we can use the iterative solvers tothe Newton equation
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) (69)
Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572
119896and using this step size to generate
the iterate 119911
119896+1= 119911
119896+ 120572
119896119878
119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is
119869 (119911
119896+ 120572
119896119878
119896) le 119869 (119911
119896) + 119888120572
119896119869
1015840(119911
119896) 119878
119896 (70)
where 119888 isin (0 1) and typically is quite small for example 119888 =
10
minus4 [12 13]
8 ISRN Applied Mathematics
(1) Given 119911
0 and tol gt 0 set 119896 = 0
(2) Compute 119870119872 119860 and
119887
(3) Compute (
119865
119896)
119894=
119887 minus 119872119906 if 119896 = 1
minus119872119906 if 119896 = 2 119875
(4) Solve 119870 120583 = 119865
(5) Compute
119869
1015840(119911
119896) = 119872(120572119911
119896minus
1)
(6) Compute
119866
1= 119872V and
119866
119905= 0 for 119905 = 2 119875
(7) Solve 119870 = 119866 = (
119866
119879
1
119866
119879
119875)
119879
(8) Compute
119867
119896= 119872
119896for 119896 = 1 119875
(9) Solve 119870 119902 = 119867 = (
119867
119879
1
119867
119879
119875)
119879
(10) Compute
119869
10158401015840(119911)V asymp 119872(120572V minus 119902
1)
(11) If 1003817100381710038171003817
1003817
nabla
119869 (119911
119896)
1003817
1003817
1003817
1003817
1003817
lt tol stop(12) Solve
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) using PCG method with preconditioner
119875 = 119866⨂119870
0
(i) Set 1199090= 0
(ii) Set 1199030larr
119869
10158401015840(119911
119896) 119909
0+
119869
1015840(119911
119896)
(iii) Solve
119875119910
0= 119903
0for 119910
0
(iv) Set 1199010= minus119903
0 119905 larr 0
(v) While 1003817
1003817
1003817
1003817
119903
119905
1003817
1003817
1003817
1003817
gt tol
120572
119896larr
119903
119879
119905119910
119905
119901
119879
119905119860119901
119905
119909
119905+1larr 119909
119905+ 120572
119905119901
119905
119903
119905+1larr 119903
119905+ 120572
119905119860119901
119905
119875119910
119905+1larr 119903
119905+1
120573
119905+1larr
119903
119879
119905+1119910
119905+1
119903
119879
119905119910
119905
119901
119905+1larr minus119910
119905+1+ 120573
119905+1119901
119905
119905 larr 119905 + 1
End (while)(vi) 119904119896 = 119909
119905+1
(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911
119896+ 120572
119896119878
119896)
(ii) While 119869 (119911
119896+ 120572
119896119878
119896) gt 119869 (119911
119896) + 10
minus4120572
119896119878
119896nabla
119869 (119911
119896) do
(i) Set 120572119896 = 120572
1198962 and evaluate 119869 (119911
119896+ 120572
119896119878
119896)
(14) Set 119911119896+1 = 119911
119896+ 120572
119896119878
119896 119896 larr 119896 + 1 Go to (2)
Algorithm 1
31 Kronecker Product Preconditioners Let us to rewrite (37)in the form
119860 =
119911 where
119860 = (
119872
sum
119896=0
119866
119896⨂119870
119896) (71)
The strategy here is to introduce
119875 = 119866⨂119870
0as a
preconditioner such that
119866 = argmin 119867 isin R119875times119875
1003817
1003817
1003817
1003817
1003817
119860 minus 119867⨂119870
0
1003817
1003817
1003817
1003817
1003817119865 (72)
where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot
119865denotes the Frobe-
nius norm The closed form of the solution can be written asfollow [14]
119866 = 119868 + sum
119875
tr (119870119879
119896119870
0)
tr (119870119879
0119870
0)
119866
119896
(73)
Here tr(119870119879
119896119870
0) = sum
119873119902
119894=1[119870
119896]
119894119894[119870
0]
119894119894and hence the coeffi-
cients in the above equality can be computed straightforwardIn addition since
119860 and 119870
0are symmetric and positive
definite thus 119866 and
119875 = 119866⨂119870
0have also these properties
[15 16]
Example 1 Consider the following distributed optimal con-trol problem
min119906119911
119869 (119906 119911) = 119864 [
1
2
119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω
(74)
where 119863 = [minus1 1]
2 Ω sim 119880[minus1 1] the boundary conditionsare given as
119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0
(75)
ISRN Applied Mathematics 9
005
1
005
10
05
1y d
xy
(a) Desired values
005
1
005
10
05
1
uM
ean
xy
times10minus3
(b) Initial state
005
1
005
10
05
1
u opt
xy
(c) Approximated solution (state)
005
1
005
10
200
400
z opt
xy
(d) Control
005
1
0
05
10
05
1
times 10minus3
xy
120583
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
05
1
15
2
25
times 10minus3
(f) Expected value
20 40
10
20
30
40
50
05
1
15
2
25
times 10minus3
(g) Standard dev
0 5 10times 104
0
5
10
times 104 0
(h) Sparsity of global matrix119899119911 = 1815980
Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))
2+ (119910 minus (12))
2)]
119873 = 3 119870 = 5 and 120572 = 000005
and the desired function is given by
(119909 119910) = exp [minus64 ((119909 minus
1
2
)
2
+ (119910 minus
1
2
)
2
)] (76)
The random field 119886(x 120596) is characterized by its mean andcovariance function
119864 [119886] = 119886 = 10
119877 (119909
1 119909
2) = 119890
minus|1199091minus1199092| 119909
1 119909
2isin [minus1 1]
(77)
Following Section 22 the truncated KLE can be representedas
119886 (119909 119910 120596) = 10 +
119873
sum
119895=0
radic120582
119895120579
119895120601
119895 (119909) (78)
The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867
1
0(119863)⨂119871
2(Ω) and all 119911isin 119871
2(119863)
119869 (119906opt 119911opt) le 119869 (119906 119911) (79)
The eigenpairs 120582119895 120579
119895 in truncated KLE solve the integral
equation
int
1
minus1
119890
minus|1199091minus1199092|120601
119895(119909
2) 119889119909
2= 120582
119895120601
119895(119909
1)
(80)
For this special case of the covariance function we haveexplicit expressions for 120601
119895and 120582
119895 [6] Let120596
119895evenand120596
119895oddsolve
the equations
1 minus 120596
119895eventan (120596
119895even) = 0
120596
119895odd+ tan (120596
119895odd) = 0
(81)
Then the even and odd indexed eigenfunctions are given by
120601
119895even(119909) =
cos (120596119895even
119909)
radic1 + (sin (2120596
119895even) 2120596
119895even)
120601
119895odd(119909)
=
sin (120596
119895odd119909)
radic1 minus (sin (2120596
119895odd) 2120596
119895odd)
(82)
10 ISRN Applied Mathematics
005
1
005
10
05
1y ha
nd
xy
(a) Desired values
005
1
005
1minus05
05
0
u Mea
n
xy
(b) Mean values
005
1
005
1
0
Num
sol
xy
0
1
(c) Approximated solution (State)
005
1
005
10
05
1
xy
(d) Control
005
1
005
10
05
1
Lend
a
xy
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
0
005
01
015
02
(f) Expected value-SGS
20 40
10
20
30
40
50
0
005
01
015
02
(g) Standard dev-SGS
0 2 4 6 8times 104
0
2
4
6
8
times 104
(h) Sparsity of global matrix119899119911 = 3046160
Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001
and corresponding eigenvalues
120582
119895even(119909) =
2
120596
2
119895even+ 1
120582
119895odd(119909) =
2
120596
2
119895odd+ 1
(83)
We choose 120579 = (120579
1 120579
119873)
119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]
First of all we find numerical approximations of120596119895even
and120596
119895oddwith bisection method and then with the eigenpairs
evaluated by this 120596119895even
and 120596
119895odd by choosing dimension 119873 =
3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1
In Figure 1(a) the desired function (119909 119910) is plottedfor 119909
119894= 119910
119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial
state is computed and plotted in Figure 1(b) by solving (37)where
119911 = 0 Approximated 119906opt corresponding to the
optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial
119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911
119900119901119905must approximate the
initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane
In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use
Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary
ISRN Applied Mathematics 11
conditions is replaced by (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2
4 Conclusion
The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996
[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987
[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005
[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947
[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946
[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991
[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007
[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009
[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938
[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993
[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008
[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999
[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006
[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010
[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010
[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007
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Stochastic AnalysisInternational Journal of
2 ISRN Applied Mathematics
Karhunen-Loeve expansion of the permeability field 119886(x 120596)Then (1) is approximated by
nabla (119886
119872 (x 120596) nabla119906) = 119911 (4)
For approximation of the function 119906 with 119873randomvariables and the Gaussian random variables of the highestdegree119870 the number of PCE coefficients are(119873+119870)119873119870[6] Now weak formulation of (4) and then Galerkinfinite element method are used to solve the SPDE problemapproximately Generally in nature optimization with SPDE-constraint is infinite dimensional large and complex Thereare two numerical approaches for solving this problem [2 7]
(i) discretize then optimize (DO) for problems that canbe trivially discretized first
(ii) optimize then discretize (OD) for problems that aredifferentiable
Our intention is to work with discretizethen optimizationalgorithms for smooth functions Thus we follow the DOapproach Galerkin finite element method incorporated withpolynomial chaos expansion is used for discretizing theSPDE By computing gradient and Hessian of the Lagrangianaugmented function preconditionedNewtonrsquos conjugate gra-dient method is used to compute the best right hand side vec-tor (control values at grid points) where the correspondingsolution of SPDE (state values) according to this vector staysas near as to the desired value approximation and has lowestcost in the objective function The organization of this paperis as follows in Section 2 problem formulation in additionto KLE PCE and stochastic Galerkin method is presentedIn Section 3 distributed control of random forced diffusionequation with some numerical examples is proposed
2 Problem Formulation
We consider optimal control problems of the form
min119906isin119880 119911isinZ
119869 (119906 119911)
st 119890 (119906 119911) = 0
(5)
where 119869 119867
1
0(119863)⨂119871
2(Ω) rarr R is the objective function
119890 119880 = 119867
1
0(119863)⨂119871
2(Ω) rarr Z = 119871
2(119863) is an operator which
stands as the constraint 119863 is the spatial domain Ω is theprobability space and⨂ is the tensor product
To provide a concrete setting for our discussion thefollowing problem is considered as the constraint operator
minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω
119906 (x 120596) = 0 on 120597119863 times Ω
(6)
where 119886(x 120596) is the correlated random fields Hence thesolutions 119906(x 120596) of the SPDE (6) are also random fields It is
assumed that the control space 119911(x) isin 119871
2(119863) The weak form
of the constraint function (6) can be defined for all V isin 119880 as
⟨119890 (119906 119911) V⟩119880lowast 119880
= int
Ω
120588 (120596)int
119863
(119886 (x 120596) nabla119906 (x 120596)
sdot nablaV (x 120596) 119889x119889120596 minus119911 (x) V (x 120596)) 119889x119889120596
(7)
It is assumed that the objective or cost functional is as
119869 (119906 119911) = 119864 [
1
2
119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
(8)
where 119864(sdot) denotes the expected value is a given desiredfunction and 120572 is the regularization parameter Obviouslycontrolling the solution of this problem leads to controllingthe statistics of the solution for example the expected valueof the solution given by
119864[119906 (x 120596) minus (x)]21198712(119863)
le 119864 [119906 (x 120596) minus (x)21198712(119863)
]
le 119906 (x 120596) minus (x)21198712(119863)otimes119871
2(Ω)
(9)
So the optimal control problem can be interpretedas follows given the random field 119886(x 120596) minimize theobjective function (8) over all 119911 isin Z subject to 119906 isin 119880 suchthat for the given random field 119886(x 120596) satisfies in the weakformulation (7) [1]
21 Basic Definitions In (5) 119869 and 119890 are assumed to becontinuously F-differentiable and that for each 119911 isin Z thestate equation 119890(119906 119911) = 0 possesses a unique correspondingsolution 119906(119911) isin 119880 Similar to the deterministic PDEsexistence and uniqueness of the solution for these kind ofSPDEs can be established using the Lax-Milgram theoremThus there is a solution operator 119911 isin Z rarr 119906(119911) isin 119880In addition it is assumed that 119890
119906(119906(119911) 119911) isin L(119880Z) is
continuously invertible Then existence and uniqueness ofthe solution for the above optimal control problems as well asthe continuously differentiability of 119906(119911) are ensured by theimplicit function theorem [8]
22 Karhunen-Loeve Expansion (KLE) Consider a randomfield 119886(x 120596) x isin 119863 with finite second order moment
int
Ω
119864 [119886
2(x 120596)] 119889119909 lt infin (10)
Assume that 119864[119886] = 119886(x) It is possible to expand 119886(x 120596)for a given orthonormal basis 120595
119896 in 119871
2(119863) as a generalized
Fourier series
119886 (x 120596) = 119886 (x) +infin
sum
119896=1
119886
119896 (120596) 120595119896 (
x) (11)
ISRN Applied Mathematics 3
where
119886
119896 (120596) = int
Ω
119886 (x 120596) 120595119896 (x) 119889x 119896 = 1 2 (12)
are random variables with zero means It is importantnow to find a special basis 120601
119896 that makes corresponding
119886
119896uncorrelated 119864[119886
119894119886
119895] = 0 for all 119894 = 119895 Denoting the
covariance function of 119886(x 120596) by 119877(x y) = 119864[119886(x 120596)119886(y 120596)]the basis functions 120601
119896 should satisfy
119864 [119886
119894119886
119895] = int
119863
120601
119894 (x) 119889xint
119863
119877 (x y) 120601119895(y) 119889y = 0 119894 = 119895
(13)
Completion and orthonormality of 120601119896 in 119871
2(119863) follow
that 120601119896(x) are eigenfunctions of 119877(x y)
int
119863
119877 (x y) 120601119895(y) 119889119910 = 120582
119895120601
119895 (x) 119895 = 1 2 (14)
where 120582
119895= 119864[119886
2
119895] gt 0 Indeed by choosing basis
functions 120601119896(119909) as the solutions of the eigenproblem (14) the
randomvariables 119886119896(120596)will be uncorrelated In (14) denoting
120579
119896= 119886
119896radic120582
119896 yields the following expansion
119886 (x 120596) = 119886 (x) +infin
sum
119896=1
radic
120582
119896120579
119896(120596) 120601119896 (
x) (15)
where 120579
119896satisfy 119864[120579
119896] = 0 and 119864[120579
119894120579
119895] = 120575
119894119895 In the case
that 119886(x 120596) is considered as a Gaussian process 119886119896(120596) 119896 =
1 2 will be independent Gaussian random variables Theexpansion (15) is known as the Karhunen-Loeve expansion(KLE) of the stochastic process 119886(x 120596)
Using the KLE (15) the stochastic process can be rep-resented as a series of uncorrelated random variables Sincethe basis functions 120601
119896(x) are deterministic the spatial depen-
dence of the random process can be resolved by them Theconvergence property of the KLE to the random process119886(x 120596) can be represented in the mean square sense
lim119873rarrinfin
int
119863
119864
1003816
1003816
1003816
1003816
119886 (x 120596) minus 119886
119873 (x 120596)
1003816
1003816
1003816
1003816
2119889x = 0 (16)
where
119886
119873= 119886 (x) +
119873
sum
119896=1
radic
120582
119896120579
119896120601
119896(17)
is a finite term KLE [4 5]
23 Polynomial Chaos Expansion (PCE) There are problemsas the solution of a PDE with random inputs that thecovariance function of a random process 119906(x 120596) x isin 119863 isnot knownThe solution of such problems can be representedusing a polynomial chaos expansion (PCE) given by
119906 (x 120596) =
infin
sum
119896=1
119906
119896 (x) Ψ119896 (120585) (18)
where the functions 119906
119896(x) are deterministic coefficients 120585
is a vector of orthonormal random variables and Ψ
119896(120585) are
multidimensional orthogonal polynomials that satisfy in thefollowing properties
⟨Ψ
1⟩ equiv 119864 [Ψ
1] = 1 ⟨Ψ
119896⟩ = 0 119896 gt 1
⟨Ψ
119894Ψ
119895⟩ = ℎ
119894120575
119894119895
(19)
The convergence property of PCE for a random quantityin 119871
2 is ensured by the Cameron-Martin theorem [6 9] thatis
⟨119906 (x 120596) minus
infin
sum
119896=1
119906
119896 (x) Ψ119896 (120585)⟩
1198712
997888rarr 0 (20)
Hence this convergence justifies a truncation of PCE to afinite number of terms
119906 (x 120596) =
119875
sum
119896=1
119906
119896 (x) Ψ119896 (120585) (21)
where the value of 119875 is determined by the highest degree ofpolynomial 119870 used to represent 119906 and the number 119873 ofrandom variablesmdashthe length of 120585mdashwith the formula 119875+1 =
(119873 + 119870)119873119870 Generally the value of 119873 is the same asthe number of uncorrelated random variables in the systemor equivalently the truncation length of the truncated KLETypically the value of119870 is chosen by some heuristic methodIndeed in the case of119870 = 1 and119873 randomvariables the KLEis a special case of the PCE [9 10]
24 Stochastic Galerkin Suppose 119882
119894sub 119871
2
120588119894
(Ω
119894) with dimen-
sion 120588
119894for 119894 = 1 119872 and 119881 sub 119867
1
0(119863) with dimension
119873 In addition let 120595
119894
119899
119901119894
119899=1for 119894 = 1 119872 be basis of
119882
119894and 120601
119894
119873
119894=1a basis of 119881 The finite dimensional tensor
product space 119882
1⨂sdot sdot sdot⨂119882
119872⨂119881 can be defined as the
space spanned by the functions 120595
1
1198991
120595
119872
119899119872
120601
119894 for all 119899
1isin
1 119901
1 119899
119872isin 1 119901
119872 and 119894 isin 1 119873
For simplification let n denote a multi-index whose 119896thcomponent 119899
119896isin 1 119901
119896 and I denote the set of such
multi-indices then the basis functions for the tensor productspace have the form
Vn119894 (x 120596) = 120601
119894(x) Ψ119899 (120596) (22)
where
Ψ
119899=
119872
prod
119896=1
120595
119896
119899119896
(120596
119896) (23)
Then the Galerkin method is looking for a solution isin 119882
1⨂sdot sdot sdot⨂119882
119872⨂119881 such that for x = (119909 119910) all n isinI
and 119894 = 1 119873
int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla119906
ℎ119875 (x 120596) sdot nablaVn119894 (x 120596) 119889x119889120596
= int
Ω
120588 (120596) int
119863
119911 (x 120596) Vn119894 (x 120596) 119889x119889120596
(24)
4 ISRN Applied Mathematics
where 120588(120596) is the integration weight Equation (24) is equiv-alent to the following equation which results by plugging inthe explicit formula for the basis functions
int
Ω
120588 (120596)Ψn (120596) int
119863
119886 (x 120596) nabla119906
ℎ119875 (x 120596) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψn (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(25)
for all n isinI and 119894 = 1 119873Now considering the approximate solution as
119906
ℎ119875 (x 120596) =
119873
sum
119895=1
sum
misinI119906
119895m120601
119895 (x) Ψm (120596) (26)
and substituting (26) into (25) yields119873
sum
119895=1
sum
misinI119906
119895m int
Ω
120588 (120596)Ψn (120596)Ψm (120596)
times int
119863
119886 (x 120596) nabla120601
119895 (x) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψn (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(27)
for allm isinI and 119894 = 1 119873 Let 119875 = |I| = prod
119872
119896=1119901
119896 then
a natural bijection between 1 119875 and Ican be definedThus (27) can be written as
119875
sum
119898=1
119873
sum
119895=1
119906
119898119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596)
times int
119863
119886 (x 120596) nabla120601
119895 (x) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψ119899 (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(28)
Now if 119886 and 119911 have the following KLE
119886 (x 120596) = 119886
0 (x) +
119872
sum
119894=1
120596
119894119886
119894 (x)
119911 (x 120596) = 119911
0 (x) +
119872
sum
119894=1
120596
119894119911
119894 (x)
(29)
then we can rewrite the Galerkin projection (28) as119875
sum
119898=1
119873
sum
119895=1
119906
119898119895((119870
0)
119894119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596
+
119872
sum
119896=1
(119870
119896)
119894119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 120596119896
119889120596)
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(30)
where
(119870
119896)
119894119895= int
119863
119886
119896 (x) nabla120601
119894 (x) sdot nabla120601
119895 (x) 119889x
cong sum
119903
119908
119903119886
119896(x119903) nabla120601
119894(x119903) sdot nabla120601
119895(x119903)
(119911
119896)
119894= int
119863
119911
119894 (x) 120601119894 (x) 119889x cong sum
119903
119908
119903119911
119894(x119903) 120601
119894(x119903)
(31)
for 119896 = 1 119872 [3] Equation (31) is approximated byquadrature rule in spatial domain Now assume that thereexist functions Ψ
119899 119899 = 1 119875 such that
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596 = 120575
119898119899
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 120596119896
119889120596 = 119862
119896119899120575
119899119898
(32)
where for 119898 = 119899 the above integral is approximated usingquadrature rule in random domain as follows
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119899 (
120596) 120596119896119889120596 cong sum
119904
119908
1015840
119904120588 (120596
119904) Ψ
2
119899(120596
119904) 120596
119896 (33)
Then (30) becomes
119875
sum
119898=1
119873
sum
119895=1
119906
119898119895((119870
0)
119894119895+
119872
sum
119896=1
119862
119896119899(119870
119896)
119894119895)120575
119899119898
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(34)
or equivalently
119873
sum
119895=1
119906m119895 ((119870
0)
119894119895+
119872
sum
119896=1
119862
119896119899(119870
119896)
119894119895)
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(35)
In (34) and (35) we have
int
Ω
120588 (120596)Ψ119899 (120596) 119889120596 = 0 for 119899 = 0
int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596 = int
Ω
120588 (120596)Ψ0 (120596)Ψ119899 (
120596) 120596119896119889120596
= 119862
119896119899120575
0119899= 119862
1198960
(36)
ISRN Applied Mathematics 5
Finally (35) can be considered as the following blockdiagonal system
(
(
(
119870
0+
119872
sum
119896=1
119862
1198961119870
119896sdot sdot sdot 0
d
0 sdot sdot sdot 119870
0+
119872
sum
119896=1
119862
119896119901119870
119896
)
)
)
(
1
119875
)
= (
119911
1
119911
119901
)
(37)
or briefly 119870 =
119911
3 Distributed Optimal Control Problem
Using the stochastic Galerkin method for general objectivefunctions computing the derivatives becomes very tediousand rather messy Consider the following optimal controlproblem
min119906isin119880 119911isinz
119869 (119906 119911) =
1
2
119864 [119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
st minus nabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω
119906 (x 120596) = 0 on 120597119863 times Ω
(38)
The solutions to linear elliptic SPDEs live in the space119880 =
119867
1
0(119863)⨂119871
2(Ω) and the control space Z = 119871
2(119863) Denoting
the constraint equation by 119890(119906 119911) = 0 usually it is possibleto invoke the implicit function theorem to find a solution119906(119911) to the constraint equation This leads to an implicitlydefined objective function
119869(119911) = 119869(119906(119911) 119911) Now we focuson computing derivatives of 119869 [11] For any direction 119904 isin Z
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= ⟨119869
119906 (119906 (119911) 119911) 119906119911 (
119911) 119904⟩
119880lowast119880
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(39)
where119880lowast andZlowast are the dual space of119880 andZ respectivelyNow computing the derivative of the constraint 119890with respectto 119911 in the direction 119904 and applying it to the implicit solutionto 119890(119906(119911) 119911) = 0 yield
119890
119906 (119906 (119911) 119911) 119906119911 (
119911) 119904 + 119890
119911 (119906 (119911) 119911) 119904 = 0 (40)
So 119906
119911(119911) is
119906
119911 (119911) 119904 = minus119890
119906(119906 (119911) 119911)
minus1119890
119911 (119906 (119911) 119911) 119904
(41)
Hence
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= minus ⟨119869
119906 (119906 (119911) 119911) 119890119906(
119906 (119911) 119911)
minus1
times 119890
119911 (119906 (119911) 119911) 119904⟩
119880lowast119880
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(42)
Applying the adjoint to the 119906
119911(119911) term
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= minus ⟨119890
119911(119906 (119911) 119911)
lowast119890
119906(119906 (119911) 119911)
minuslowast
times 119869
119906 (119906 (119911) 119911) 119904⟩
Zlowast Z
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
= minus ⟨119890
119911(119906 (119911) 119911)
lowast119890
119906(119906 (119911) 119911)
minuslowast119869
119906(119906 (119911) 119911)
+119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(43)
Considering the adjoint state 120583 = 120583(119911) isin Zlowast as the solutionto
119890
119906(119906 (119911) 119911)
lowast120583 = minus119869
119906 (119906 (119911) 119911) (44)
the derivative of 1198691015840 becomes
119869
1015840(119911) = 119890
119911(119906 (119911) 119911)
lowast120583 (119911) + 119869
119911 (119906 (119911) 119911)
(45)
Theweak formof the constraint function is defined as followsfor all V isin 119880
⟨119890 (119906 119911) V⟩119880lowast 119880 = int
Ω
120588 (120596)int
119863
(119886 (x 120596) nabla119906 (x 120596) sdot nablaV (x 120596)
minus119911 (x 120596) V (x 120596)) 119889x119889120596
(46)
Thus the constraint function 119890 119880 times Z rarr 119880
lowast and 119884 = 119880
lowastSince the constraint function 119890 acts linearly with respect to 119906
and 119911 thus the corresponding derivative of 119890 with respect to119911 in the direction 120575119911 isin Z for all V isin 119880 is given by
⟨119890
119911 (119906 119911) 120575119911 V⟩
119880lowast119880
= minusint
Ω
120588 (120596)int
119863
120575119911 (x 120596) V (x 120596) 119889x119889120596
(47)
Similarly the derivative of 119890 with respect to 119906 for anydirection 120575119906 isin 119880 and for all V isin 119880 is given by
⟨119890
119906 (119906 119911) 120575119906 V⟩
119880lowast119880
= int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla120575119906 (x 120596)
sdot nablaV (x 120596) 119889x119889120596
(48)
Both of these derivatives are symmetric so 119890
119906(119906 119911)
lowast=
119890
119906(119906 119911) and 119890
119911(119906 119911)
lowast= 119890
119911(119906 119911) From here the adjoint can
be computed as
119890
119906 (119906 (119911) 119911) 120583 = minus119869
119906 (119906 (119911) 119911) (49)
Using this to any direction V isin 119880
⟨119890
119906 (119906 (119911) 119911) 120583 V⟩
119880lowast119880
= int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla120583 (x 120596)
sdot nablaV (x 120596) 119889x119889120596
= minus119869
119906 (119906 (119911) 119911) V
(50)
6 ISRN Applied Mathematics
By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is
⟨119869
119906 (119906 119911) 120575119906⟩
119880lowast119880
= 119864 [119906 (x 120596) minus (x)21198712(119863)
times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)
]
(51)
Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin
Z direction is
⟨119869
119911 (119906 119911) 120575119911⟩
Zlowast Z= 120572⟨119911 120575119911⟩Z (52)
With these computations one can compute the firstderivative of
119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883
ℎsub 119867
1
0(119863) is a finite dimensional
subspace of dimension 119873 and 119884
ℎsub 119871
2(Ω) is a finite
dimensional subspace of dimension 119875 then 119883
ℎ⨂119884
ℎis a
finite dimensional subspace of the state space 119880 Similarlylet Z
ℎsub Z be a finite dimensional subspace of the control
space (with dimension) For the stochastic Galerkin methodit is assumed that 119884
ℎ= P119875minus1 the space of polynomials with
highest degree 119875minus1 and119883
ℎis any finite element space here
119883
ℎis the space of linear functions built on a given meshT
ℎ
We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884
ℎ that is 119884
ℎ= spanΨ
1(120596) Ψ
119875(120596)
where
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596 = 120575
119898119899 (53)
The discretized optimization problem in the stochasticGalerkin framework is
min119911ℎisinZℎ
119869
ℎ119875(119911
ℎ) =
1
2
119864
[
[
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
119875
sum
119896=1
119873
sum
119895=1
(
119896(119911
ℎ))
119895Ψ
119896 (120596) 120601119895 (
x)
minus (x)1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
2
1198712(119863)
]
]
+
120572
2
1003817
1003817
1003817
1003817
119911
ℎ
1003817
1003817
1003817
1003817
2
1198712(119863)
(54)
where
119896(119911
ℎ) = (
1(119911
ℎ)
119879
119875(119911
ℎ)
119879)
119879
=
(55)
is the stochastic Galerkin solution to the state equation
(
119870
11sdot sdot sdot 119870
1119875
d
119870
1198751sdot sdot sdot 119870
119875119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119870
(
1
119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=
= (
119864 [Ψ
1]119872
119911
ℎ
119864 [Ψ
119875]119872
119911
ℎ
)
= (
120575
11119872
119911
ℎ
120575
1119875119872
119911
ℎ
)
(56)
since 119864[Ψ
119896] = 119864[1Ψ
119896] = 119864[Ψ
1Ψ
119896] = 120575
1119896 The blocks of the
above 119870matrix have the form
(119870
119896119897)
119894119895= int
Ω
120588 (120596)Ψ
119896(120596)Ψ119897 (
120596)
times int
119863
119886 (x 120596) nabla120601
119894 (x) sdot nabla120601
119895 (x) 119889x119889120596
(57)
First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations
minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))
in 119863 times Ω
120583 (x 120596) = 0 on 120597119863 times Ω
(58)
Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865
119879
1 119865
119879
119901)
119879 and 119865
119896is defined
as
(
119865
119896)
119894= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596
= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895120601
119895 (x) Ψ119897 (120596) 120601119894 (
x)
minus (x) 120601119894 (x))119889x119889120596
= minus
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
(x) 120601119894 (x) 119889x
=
119887
119894120575
1119896minus
119875
sum
119897=1
119873
sum
119895=1
119872
119894119895(
119897)
119895120575
119896119897
=
119887
119894120575
1119896minus
119873
sum
119895=1
119872
119894119895(
119897)
119895
(59)
in which
119887
119894= int 120601
119894 and119872
119894119895= int 120601
119894120601
119895
Thus
(
119865
119896)
119894=
119887 minus 119872119906
1if 119896 = 1
minus119872119906 if 119896 = 2 119875
(60)
where b= [
119887
1
119887
119873] and the matrix 119872 = (119872
119894119895) is of order
119873 times 119873
ISRN Applied Mathematics 7
Now the derivative of
119869(119911) in the direction 120601
119895(x) for all
119895 = 1 119873 with the computed adjoint state is
⟨
119869
1015840(119911) 120601119895 (
x)⟩Zlowast Z
= ⟨119890
119911(119906 (119911 sdot sdot) 119911)
lowast120583
+ 119869
119911 (119906 (119911 sdot sdot) 119911) 120601119895 (
x) ⟩Zlowast Z
= minusint
Ω
120588 (120596)int
119863
120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596
+ 120572int
119863
119911 (119909) 120601119895 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894int
Ω
120588 (120596)Ψ119896 (120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ 120572
119873
sum
119894=1
119911
119894int
119863
120601
119895 (x) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894119872
119894119895120575
1119896+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
= minus
119873
sum
119894=1
(
1)
119894119872
119894119895+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
(61)
Therefore the gradient can be computed as follows
119869
1015840(119911) = 119872(120572
119911
ℎminus
1)
(62)
Now for any vector V isin Z it is possible to write V as
V =
119873
sum
119896=1
V119896120601
119896 (x) (63)
In order to compute the Hessian times a vector that ismultiply the Hessian of the
119869 with a some vector V isin Z theequation 119890
119906(119906(119911) 119911)119908 = 119890
119911(119906(119911) 119911)V for 119908 must be solved
Similar to the adjoint computation119908 solves the linear system119870 = 119866 where
119866 = (
119866
119879
1
119866
119879
119875) and
119866
119896for 119896 = 1 119875 is
(
119866
119896)
119894= int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
119873
sum
119895=1
V119895120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895V119895120575
1119896
(64)
for all 119894 = 1 119873 Thus
119866
119896=
119872V if 119896 = 1
0 if 119896 = 2 119875
(65)
Now solving 119890
119906(119906(119911) 119911)119902 = 119869
119906119906(119906(119911) 119911)119908 for 119902 requires
the solution to the linear system 119870 119902 = 119867 where
(
119867
119896)
119894=
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895(
119896)
119895
(66)
or equivalently
119867
119896= 119872
119896for all 119896 = 1 119875 Hence in the
direction 120601
119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by
⟨
119869
10158401015840(119911) V 120601119894⟩Zlowast Z
asymp ⟨
119869
10158401015840
ℎ119901(119911
ℎ) V 120601
119894⟩
Zlowast Z
= minusint
Ω
120588 (120596)int
119863
119875
sum
119896=1
119873
sum
119895=1
( 119902
119896)
119895120601
119895 (x) Ψ119896 (120596)
times 120601
119894 (x) 119889x119889120596
+ 120572int
119863
V (119909) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119895=1
119872
119894119895( 119902
119896)
119895120575
1119896+ 120572
119873
sum
119895=1
119872
119894119895V119895
=
119873
sum
119895=1
119872
119894119895(120572V
119895minus ( 119902
1)
119895)
(67)
Therefore
119869
10158401015840(119911) V asymp 119872(120572V minus 119902
1)
(68)
Having (62) and (68) we can use the iterative solvers tothe Newton equation
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) (69)
Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572
119896and using this step size to generate
the iterate 119911
119896+1= 119911
119896+ 120572
119896119878
119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is
119869 (119911
119896+ 120572
119896119878
119896) le 119869 (119911
119896) + 119888120572
119896119869
1015840(119911
119896) 119878
119896 (70)
where 119888 isin (0 1) and typically is quite small for example 119888 =
10
minus4 [12 13]
8 ISRN Applied Mathematics
(1) Given 119911
0 and tol gt 0 set 119896 = 0
(2) Compute 119870119872 119860 and
119887
(3) Compute (
119865
119896)
119894=
119887 minus 119872119906 if 119896 = 1
minus119872119906 if 119896 = 2 119875
(4) Solve 119870 120583 = 119865
(5) Compute
119869
1015840(119911
119896) = 119872(120572119911
119896minus
1)
(6) Compute
119866
1= 119872V and
119866
119905= 0 for 119905 = 2 119875
(7) Solve 119870 = 119866 = (
119866
119879
1
119866
119879
119875)
119879
(8) Compute
119867
119896= 119872
119896for 119896 = 1 119875
(9) Solve 119870 119902 = 119867 = (
119867
119879
1
119867
119879
119875)
119879
(10) Compute
119869
10158401015840(119911)V asymp 119872(120572V minus 119902
1)
(11) If 1003817100381710038171003817
1003817
nabla
119869 (119911
119896)
1003817
1003817
1003817
1003817
1003817
lt tol stop(12) Solve
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) using PCG method with preconditioner
119875 = 119866⨂119870
0
(i) Set 1199090= 0
(ii) Set 1199030larr
119869
10158401015840(119911
119896) 119909
0+
119869
1015840(119911
119896)
(iii) Solve
119875119910
0= 119903
0for 119910
0
(iv) Set 1199010= minus119903
0 119905 larr 0
(v) While 1003817
1003817
1003817
1003817
119903
119905
1003817
1003817
1003817
1003817
gt tol
120572
119896larr
119903
119879
119905119910
119905
119901
119879
119905119860119901
119905
119909
119905+1larr 119909
119905+ 120572
119905119901
119905
119903
119905+1larr 119903
119905+ 120572
119905119860119901
119905
119875119910
119905+1larr 119903
119905+1
120573
119905+1larr
119903
119879
119905+1119910
119905+1
119903
119879
119905119910
119905
119901
119905+1larr minus119910
119905+1+ 120573
119905+1119901
119905
119905 larr 119905 + 1
End (while)(vi) 119904119896 = 119909
119905+1
(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911
119896+ 120572
119896119878
119896)
(ii) While 119869 (119911
119896+ 120572
119896119878
119896) gt 119869 (119911
119896) + 10
minus4120572
119896119878
119896nabla
119869 (119911
119896) do
(i) Set 120572119896 = 120572
1198962 and evaluate 119869 (119911
119896+ 120572
119896119878
119896)
(14) Set 119911119896+1 = 119911
119896+ 120572
119896119878
119896 119896 larr 119896 + 1 Go to (2)
Algorithm 1
31 Kronecker Product Preconditioners Let us to rewrite (37)in the form
119860 =
119911 where
119860 = (
119872
sum
119896=0
119866
119896⨂119870
119896) (71)
The strategy here is to introduce
119875 = 119866⨂119870
0as a
preconditioner such that
119866 = argmin 119867 isin R119875times119875
1003817
1003817
1003817
1003817
1003817
119860 minus 119867⨂119870
0
1003817
1003817
1003817
1003817
1003817119865 (72)
where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot
119865denotes the Frobe-
nius norm The closed form of the solution can be written asfollow [14]
119866 = 119868 + sum
119875
tr (119870119879
119896119870
0)
tr (119870119879
0119870
0)
119866
119896
(73)
Here tr(119870119879
119896119870
0) = sum
119873119902
119894=1[119870
119896]
119894119894[119870
0]
119894119894and hence the coeffi-
cients in the above equality can be computed straightforwardIn addition since
119860 and 119870
0are symmetric and positive
definite thus 119866 and
119875 = 119866⨂119870
0have also these properties
[15 16]
Example 1 Consider the following distributed optimal con-trol problem
min119906119911
119869 (119906 119911) = 119864 [
1
2
119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω
(74)
where 119863 = [minus1 1]
2 Ω sim 119880[minus1 1] the boundary conditionsare given as
119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0
(75)
ISRN Applied Mathematics 9
005
1
005
10
05
1y d
xy
(a) Desired values
005
1
005
10
05
1
uM
ean
xy
times10minus3
(b) Initial state
005
1
005
10
05
1
u opt
xy
(c) Approximated solution (state)
005
1
005
10
200
400
z opt
xy
(d) Control
005
1
0
05
10
05
1
times 10minus3
xy
120583
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
05
1
15
2
25
times 10minus3
(f) Expected value
20 40
10
20
30
40
50
05
1
15
2
25
times 10minus3
(g) Standard dev
0 5 10times 104
0
5
10
times 104 0
(h) Sparsity of global matrix119899119911 = 1815980
Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))
2+ (119910 minus (12))
2)]
119873 = 3 119870 = 5 and 120572 = 000005
and the desired function is given by
(119909 119910) = exp [minus64 ((119909 minus
1
2
)
2
+ (119910 minus
1
2
)
2
)] (76)
The random field 119886(x 120596) is characterized by its mean andcovariance function
119864 [119886] = 119886 = 10
119877 (119909
1 119909
2) = 119890
minus|1199091minus1199092| 119909
1 119909
2isin [minus1 1]
(77)
Following Section 22 the truncated KLE can be representedas
119886 (119909 119910 120596) = 10 +
119873
sum
119895=0
radic120582
119895120579
119895120601
119895 (119909) (78)
The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867
1
0(119863)⨂119871
2(Ω) and all 119911isin 119871
2(119863)
119869 (119906opt 119911opt) le 119869 (119906 119911) (79)
The eigenpairs 120582119895 120579
119895 in truncated KLE solve the integral
equation
int
1
minus1
119890
minus|1199091minus1199092|120601
119895(119909
2) 119889119909
2= 120582
119895120601
119895(119909
1)
(80)
For this special case of the covariance function we haveexplicit expressions for 120601
119895and 120582
119895 [6] Let120596
119895evenand120596
119895oddsolve
the equations
1 minus 120596
119895eventan (120596
119895even) = 0
120596
119895odd+ tan (120596
119895odd) = 0
(81)
Then the even and odd indexed eigenfunctions are given by
120601
119895even(119909) =
cos (120596119895even
119909)
radic1 + (sin (2120596
119895even) 2120596
119895even)
120601
119895odd(119909)
=
sin (120596
119895odd119909)
radic1 minus (sin (2120596
119895odd) 2120596
119895odd)
(82)
10 ISRN Applied Mathematics
005
1
005
10
05
1y ha
nd
xy
(a) Desired values
005
1
005
1minus05
05
0
u Mea
n
xy
(b) Mean values
005
1
005
1
0
Num
sol
xy
0
1
(c) Approximated solution (State)
005
1
005
10
05
1
xy
(d) Control
005
1
005
10
05
1
Lend
a
xy
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
0
005
01
015
02
(f) Expected value-SGS
20 40
10
20
30
40
50
0
005
01
015
02
(g) Standard dev-SGS
0 2 4 6 8times 104
0
2
4
6
8
times 104
(h) Sparsity of global matrix119899119911 = 3046160
Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001
and corresponding eigenvalues
120582
119895even(119909) =
2
120596
2
119895even+ 1
120582
119895odd(119909) =
2
120596
2
119895odd+ 1
(83)
We choose 120579 = (120579
1 120579
119873)
119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]
First of all we find numerical approximations of120596119895even
and120596
119895oddwith bisection method and then with the eigenpairs
evaluated by this 120596119895even
and 120596
119895odd by choosing dimension 119873 =
3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1
In Figure 1(a) the desired function (119909 119910) is plottedfor 119909
119894= 119910
119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial
state is computed and plotted in Figure 1(b) by solving (37)where
119911 = 0 Approximated 119906opt corresponding to the
optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial
119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911
119900119901119905must approximate the
initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane
In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use
Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary
ISRN Applied Mathematics 11
conditions is replaced by (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2
4 Conclusion
The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996
[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987
[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005
[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947
[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946
[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991
[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007
[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009
[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938
[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993
[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008
[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999
[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006
[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010
[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010
[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007
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Stochastic AnalysisInternational Journal of
ISRN Applied Mathematics 3
where
119886
119896 (120596) = int
Ω
119886 (x 120596) 120595119896 (x) 119889x 119896 = 1 2 (12)
are random variables with zero means It is importantnow to find a special basis 120601
119896 that makes corresponding
119886
119896uncorrelated 119864[119886
119894119886
119895] = 0 for all 119894 = 119895 Denoting the
covariance function of 119886(x 120596) by 119877(x y) = 119864[119886(x 120596)119886(y 120596)]the basis functions 120601
119896 should satisfy
119864 [119886
119894119886
119895] = int
119863
120601
119894 (x) 119889xint
119863
119877 (x y) 120601119895(y) 119889y = 0 119894 = 119895
(13)
Completion and orthonormality of 120601119896 in 119871
2(119863) follow
that 120601119896(x) are eigenfunctions of 119877(x y)
int
119863
119877 (x y) 120601119895(y) 119889119910 = 120582
119895120601
119895 (x) 119895 = 1 2 (14)
where 120582
119895= 119864[119886
2
119895] gt 0 Indeed by choosing basis
functions 120601119896(119909) as the solutions of the eigenproblem (14) the
randomvariables 119886119896(120596)will be uncorrelated In (14) denoting
120579
119896= 119886
119896radic120582
119896 yields the following expansion
119886 (x 120596) = 119886 (x) +infin
sum
119896=1
radic
120582
119896120579
119896(120596) 120601119896 (
x) (15)
where 120579
119896satisfy 119864[120579
119896] = 0 and 119864[120579
119894120579
119895] = 120575
119894119895 In the case
that 119886(x 120596) is considered as a Gaussian process 119886119896(120596) 119896 =
1 2 will be independent Gaussian random variables Theexpansion (15) is known as the Karhunen-Loeve expansion(KLE) of the stochastic process 119886(x 120596)
Using the KLE (15) the stochastic process can be rep-resented as a series of uncorrelated random variables Sincethe basis functions 120601
119896(x) are deterministic the spatial depen-
dence of the random process can be resolved by them Theconvergence property of the KLE to the random process119886(x 120596) can be represented in the mean square sense
lim119873rarrinfin
int
119863
119864
1003816
1003816
1003816
1003816
119886 (x 120596) minus 119886
119873 (x 120596)
1003816
1003816
1003816
1003816
2119889x = 0 (16)
where
119886
119873= 119886 (x) +
119873
sum
119896=1
radic
120582
119896120579
119896120601
119896(17)
is a finite term KLE [4 5]
23 Polynomial Chaos Expansion (PCE) There are problemsas the solution of a PDE with random inputs that thecovariance function of a random process 119906(x 120596) x isin 119863 isnot knownThe solution of such problems can be representedusing a polynomial chaos expansion (PCE) given by
119906 (x 120596) =
infin
sum
119896=1
119906
119896 (x) Ψ119896 (120585) (18)
where the functions 119906
119896(x) are deterministic coefficients 120585
is a vector of orthonormal random variables and Ψ
119896(120585) are
multidimensional orthogonal polynomials that satisfy in thefollowing properties
⟨Ψ
1⟩ equiv 119864 [Ψ
1] = 1 ⟨Ψ
119896⟩ = 0 119896 gt 1
⟨Ψ
119894Ψ
119895⟩ = ℎ
119894120575
119894119895
(19)
The convergence property of PCE for a random quantityin 119871
2 is ensured by the Cameron-Martin theorem [6 9] thatis
⟨119906 (x 120596) minus
infin
sum
119896=1
119906
119896 (x) Ψ119896 (120585)⟩
1198712
997888rarr 0 (20)
Hence this convergence justifies a truncation of PCE to afinite number of terms
119906 (x 120596) =
119875
sum
119896=1
119906
119896 (x) Ψ119896 (120585) (21)
where the value of 119875 is determined by the highest degree ofpolynomial 119870 used to represent 119906 and the number 119873 ofrandom variablesmdashthe length of 120585mdashwith the formula 119875+1 =
(119873 + 119870)119873119870 Generally the value of 119873 is the same asthe number of uncorrelated random variables in the systemor equivalently the truncation length of the truncated KLETypically the value of119870 is chosen by some heuristic methodIndeed in the case of119870 = 1 and119873 randomvariables the KLEis a special case of the PCE [9 10]
24 Stochastic Galerkin Suppose 119882
119894sub 119871
2
120588119894
(Ω
119894) with dimen-
sion 120588
119894for 119894 = 1 119872 and 119881 sub 119867
1
0(119863) with dimension
119873 In addition let 120595
119894
119899
119901119894
119899=1for 119894 = 1 119872 be basis of
119882
119894and 120601
119894
119873
119894=1a basis of 119881 The finite dimensional tensor
product space 119882
1⨂sdot sdot sdot⨂119882
119872⨂119881 can be defined as the
space spanned by the functions 120595
1
1198991
120595
119872
119899119872
120601
119894 for all 119899
1isin
1 119901
1 119899
119872isin 1 119901
119872 and 119894 isin 1 119873
For simplification let n denote a multi-index whose 119896thcomponent 119899
119896isin 1 119901
119896 and I denote the set of such
multi-indices then the basis functions for the tensor productspace have the form
Vn119894 (x 120596) = 120601
119894(x) Ψ119899 (120596) (22)
where
Ψ
119899=
119872
prod
119896=1
120595
119896
119899119896
(120596
119896) (23)
Then the Galerkin method is looking for a solution isin 119882
1⨂sdot sdot sdot⨂119882
119872⨂119881 such that for x = (119909 119910) all n isinI
and 119894 = 1 119873
int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla119906
ℎ119875 (x 120596) sdot nablaVn119894 (x 120596) 119889x119889120596
= int
Ω
120588 (120596) int
119863
119911 (x 120596) Vn119894 (x 120596) 119889x119889120596
(24)
4 ISRN Applied Mathematics
where 120588(120596) is the integration weight Equation (24) is equiv-alent to the following equation which results by plugging inthe explicit formula for the basis functions
int
Ω
120588 (120596)Ψn (120596) int
119863
119886 (x 120596) nabla119906
ℎ119875 (x 120596) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψn (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(25)
for all n isinI and 119894 = 1 119873Now considering the approximate solution as
119906
ℎ119875 (x 120596) =
119873
sum
119895=1
sum
misinI119906
119895m120601
119895 (x) Ψm (120596) (26)
and substituting (26) into (25) yields119873
sum
119895=1
sum
misinI119906
119895m int
Ω
120588 (120596)Ψn (120596)Ψm (120596)
times int
119863
119886 (x 120596) nabla120601
119895 (x) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψn (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(27)
for allm isinI and 119894 = 1 119873 Let 119875 = |I| = prod
119872
119896=1119901
119896 then
a natural bijection between 1 119875 and Ican be definedThus (27) can be written as
119875
sum
119898=1
119873
sum
119895=1
119906
119898119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596)
times int
119863
119886 (x 120596) nabla120601
119895 (x) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψ119899 (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(28)
Now if 119886 and 119911 have the following KLE
119886 (x 120596) = 119886
0 (x) +
119872
sum
119894=1
120596
119894119886
119894 (x)
119911 (x 120596) = 119911
0 (x) +
119872
sum
119894=1
120596
119894119911
119894 (x)
(29)
then we can rewrite the Galerkin projection (28) as119875
sum
119898=1
119873
sum
119895=1
119906
119898119895((119870
0)
119894119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596
+
119872
sum
119896=1
(119870
119896)
119894119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 120596119896
119889120596)
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(30)
where
(119870
119896)
119894119895= int
119863
119886
119896 (x) nabla120601
119894 (x) sdot nabla120601
119895 (x) 119889x
cong sum
119903
119908
119903119886
119896(x119903) nabla120601
119894(x119903) sdot nabla120601
119895(x119903)
(119911
119896)
119894= int
119863
119911
119894 (x) 120601119894 (x) 119889x cong sum
119903
119908
119903119911
119894(x119903) 120601
119894(x119903)
(31)
for 119896 = 1 119872 [3] Equation (31) is approximated byquadrature rule in spatial domain Now assume that thereexist functions Ψ
119899 119899 = 1 119875 such that
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596 = 120575
119898119899
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 120596119896
119889120596 = 119862
119896119899120575
119899119898
(32)
where for 119898 = 119899 the above integral is approximated usingquadrature rule in random domain as follows
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119899 (
120596) 120596119896119889120596 cong sum
119904
119908
1015840
119904120588 (120596
119904) Ψ
2
119899(120596
119904) 120596
119896 (33)
Then (30) becomes
119875
sum
119898=1
119873
sum
119895=1
119906
119898119895((119870
0)
119894119895+
119872
sum
119896=1
119862
119896119899(119870
119896)
119894119895)120575
119899119898
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(34)
or equivalently
119873
sum
119895=1
119906m119895 ((119870
0)
119894119895+
119872
sum
119896=1
119862
119896119899(119870
119896)
119894119895)
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(35)
In (34) and (35) we have
int
Ω
120588 (120596)Ψ119899 (120596) 119889120596 = 0 for 119899 = 0
int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596 = int
Ω
120588 (120596)Ψ0 (120596)Ψ119899 (
120596) 120596119896119889120596
= 119862
119896119899120575
0119899= 119862
1198960
(36)
ISRN Applied Mathematics 5
Finally (35) can be considered as the following blockdiagonal system
(
(
(
119870
0+
119872
sum
119896=1
119862
1198961119870
119896sdot sdot sdot 0
d
0 sdot sdot sdot 119870
0+
119872
sum
119896=1
119862
119896119901119870
119896
)
)
)
(
1
119875
)
= (
119911
1
119911
119901
)
(37)
or briefly 119870 =
119911
3 Distributed Optimal Control Problem
Using the stochastic Galerkin method for general objectivefunctions computing the derivatives becomes very tediousand rather messy Consider the following optimal controlproblem
min119906isin119880 119911isinz
119869 (119906 119911) =
1
2
119864 [119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
st minus nabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω
119906 (x 120596) = 0 on 120597119863 times Ω
(38)
The solutions to linear elliptic SPDEs live in the space119880 =
119867
1
0(119863)⨂119871
2(Ω) and the control space Z = 119871
2(119863) Denoting
the constraint equation by 119890(119906 119911) = 0 usually it is possibleto invoke the implicit function theorem to find a solution119906(119911) to the constraint equation This leads to an implicitlydefined objective function
119869(119911) = 119869(119906(119911) 119911) Now we focuson computing derivatives of 119869 [11] For any direction 119904 isin Z
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= ⟨119869
119906 (119906 (119911) 119911) 119906119911 (
119911) 119904⟩
119880lowast119880
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(39)
where119880lowast andZlowast are the dual space of119880 andZ respectivelyNow computing the derivative of the constraint 119890with respectto 119911 in the direction 119904 and applying it to the implicit solutionto 119890(119906(119911) 119911) = 0 yield
119890
119906 (119906 (119911) 119911) 119906119911 (
119911) 119904 + 119890
119911 (119906 (119911) 119911) 119904 = 0 (40)
So 119906
119911(119911) is
119906
119911 (119911) 119904 = minus119890
119906(119906 (119911) 119911)
minus1119890
119911 (119906 (119911) 119911) 119904
(41)
Hence
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= minus ⟨119869
119906 (119906 (119911) 119911) 119890119906(
119906 (119911) 119911)
minus1
times 119890
119911 (119906 (119911) 119911) 119904⟩
119880lowast119880
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(42)
Applying the adjoint to the 119906
119911(119911) term
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= minus ⟨119890
119911(119906 (119911) 119911)
lowast119890
119906(119906 (119911) 119911)
minuslowast
times 119869
119906 (119906 (119911) 119911) 119904⟩
Zlowast Z
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
= minus ⟨119890
119911(119906 (119911) 119911)
lowast119890
119906(119906 (119911) 119911)
minuslowast119869
119906(119906 (119911) 119911)
+119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(43)
Considering the adjoint state 120583 = 120583(119911) isin Zlowast as the solutionto
119890
119906(119906 (119911) 119911)
lowast120583 = minus119869
119906 (119906 (119911) 119911) (44)
the derivative of 1198691015840 becomes
119869
1015840(119911) = 119890
119911(119906 (119911) 119911)
lowast120583 (119911) + 119869
119911 (119906 (119911) 119911)
(45)
Theweak formof the constraint function is defined as followsfor all V isin 119880
⟨119890 (119906 119911) V⟩119880lowast 119880 = int
Ω
120588 (120596)int
119863
(119886 (x 120596) nabla119906 (x 120596) sdot nablaV (x 120596)
minus119911 (x 120596) V (x 120596)) 119889x119889120596
(46)
Thus the constraint function 119890 119880 times Z rarr 119880
lowast and 119884 = 119880
lowastSince the constraint function 119890 acts linearly with respect to 119906
and 119911 thus the corresponding derivative of 119890 with respect to119911 in the direction 120575119911 isin Z for all V isin 119880 is given by
⟨119890
119911 (119906 119911) 120575119911 V⟩
119880lowast119880
= minusint
Ω
120588 (120596)int
119863
120575119911 (x 120596) V (x 120596) 119889x119889120596
(47)
Similarly the derivative of 119890 with respect to 119906 for anydirection 120575119906 isin 119880 and for all V isin 119880 is given by
⟨119890
119906 (119906 119911) 120575119906 V⟩
119880lowast119880
= int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla120575119906 (x 120596)
sdot nablaV (x 120596) 119889x119889120596
(48)
Both of these derivatives are symmetric so 119890
119906(119906 119911)
lowast=
119890
119906(119906 119911) and 119890
119911(119906 119911)
lowast= 119890
119911(119906 119911) From here the adjoint can
be computed as
119890
119906 (119906 (119911) 119911) 120583 = minus119869
119906 (119906 (119911) 119911) (49)
Using this to any direction V isin 119880
⟨119890
119906 (119906 (119911) 119911) 120583 V⟩
119880lowast119880
= int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla120583 (x 120596)
sdot nablaV (x 120596) 119889x119889120596
= minus119869
119906 (119906 (119911) 119911) V
(50)
6 ISRN Applied Mathematics
By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is
⟨119869
119906 (119906 119911) 120575119906⟩
119880lowast119880
= 119864 [119906 (x 120596) minus (x)21198712(119863)
times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)
]
(51)
Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin
Z direction is
⟨119869
119911 (119906 119911) 120575119911⟩
Zlowast Z= 120572⟨119911 120575119911⟩Z (52)
With these computations one can compute the firstderivative of
119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883
ℎsub 119867
1
0(119863) is a finite dimensional
subspace of dimension 119873 and 119884
ℎsub 119871
2(Ω) is a finite
dimensional subspace of dimension 119875 then 119883
ℎ⨂119884
ℎis a
finite dimensional subspace of the state space 119880 Similarlylet Z
ℎsub Z be a finite dimensional subspace of the control
space (with dimension) For the stochastic Galerkin methodit is assumed that 119884
ℎ= P119875minus1 the space of polynomials with
highest degree 119875minus1 and119883
ℎis any finite element space here
119883
ℎis the space of linear functions built on a given meshT
ℎ
We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884
ℎ that is 119884
ℎ= spanΨ
1(120596) Ψ
119875(120596)
where
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596 = 120575
119898119899 (53)
The discretized optimization problem in the stochasticGalerkin framework is
min119911ℎisinZℎ
119869
ℎ119875(119911
ℎ) =
1
2
119864
[
[
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
119875
sum
119896=1
119873
sum
119895=1
(
119896(119911
ℎ))
119895Ψ
119896 (120596) 120601119895 (
x)
minus (x)1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
2
1198712(119863)
]
]
+
120572
2
1003817
1003817
1003817
1003817
119911
ℎ
1003817
1003817
1003817
1003817
2
1198712(119863)
(54)
where
119896(119911
ℎ) = (
1(119911
ℎ)
119879
119875(119911
ℎ)
119879)
119879
=
(55)
is the stochastic Galerkin solution to the state equation
(
119870
11sdot sdot sdot 119870
1119875
d
119870
1198751sdot sdot sdot 119870
119875119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119870
(
1
119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=
= (
119864 [Ψ
1]119872
119911
ℎ
119864 [Ψ
119875]119872
119911
ℎ
)
= (
120575
11119872
119911
ℎ
120575
1119875119872
119911
ℎ
)
(56)
since 119864[Ψ
119896] = 119864[1Ψ
119896] = 119864[Ψ
1Ψ
119896] = 120575
1119896 The blocks of the
above 119870matrix have the form
(119870
119896119897)
119894119895= int
Ω
120588 (120596)Ψ
119896(120596)Ψ119897 (
120596)
times int
119863
119886 (x 120596) nabla120601
119894 (x) sdot nabla120601
119895 (x) 119889x119889120596
(57)
First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations
minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))
in 119863 times Ω
120583 (x 120596) = 0 on 120597119863 times Ω
(58)
Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865
119879
1 119865
119879
119901)
119879 and 119865
119896is defined
as
(
119865
119896)
119894= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596
= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895120601
119895 (x) Ψ119897 (120596) 120601119894 (
x)
minus (x) 120601119894 (x))119889x119889120596
= minus
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
(x) 120601119894 (x) 119889x
=
119887
119894120575
1119896minus
119875
sum
119897=1
119873
sum
119895=1
119872
119894119895(
119897)
119895120575
119896119897
=
119887
119894120575
1119896minus
119873
sum
119895=1
119872
119894119895(
119897)
119895
(59)
in which
119887
119894= int 120601
119894 and119872
119894119895= int 120601
119894120601
119895
Thus
(
119865
119896)
119894=
119887 minus 119872119906
1if 119896 = 1
minus119872119906 if 119896 = 2 119875
(60)
where b= [
119887
1
119887
119873] and the matrix 119872 = (119872
119894119895) is of order
119873 times 119873
ISRN Applied Mathematics 7
Now the derivative of
119869(119911) in the direction 120601
119895(x) for all
119895 = 1 119873 with the computed adjoint state is
⟨
119869
1015840(119911) 120601119895 (
x)⟩Zlowast Z
= ⟨119890
119911(119906 (119911 sdot sdot) 119911)
lowast120583
+ 119869
119911 (119906 (119911 sdot sdot) 119911) 120601119895 (
x) ⟩Zlowast Z
= minusint
Ω
120588 (120596)int
119863
120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596
+ 120572int
119863
119911 (119909) 120601119895 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894int
Ω
120588 (120596)Ψ119896 (120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ 120572
119873
sum
119894=1
119911
119894int
119863
120601
119895 (x) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894119872
119894119895120575
1119896+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
= minus
119873
sum
119894=1
(
1)
119894119872
119894119895+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
(61)
Therefore the gradient can be computed as follows
119869
1015840(119911) = 119872(120572
119911
ℎminus
1)
(62)
Now for any vector V isin Z it is possible to write V as
V =
119873
sum
119896=1
V119896120601
119896 (x) (63)
In order to compute the Hessian times a vector that ismultiply the Hessian of the
119869 with a some vector V isin Z theequation 119890
119906(119906(119911) 119911)119908 = 119890
119911(119906(119911) 119911)V for 119908 must be solved
Similar to the adjoint computation119908 solves the linear system119870 = 119866 where
119866 = (
119866
119879
1
119866
119879
119875) and
119866
119896for 119896 = 1 119875 is
(
119866
119896)
119894= int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
119873
sum
119895=1
V119895120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895V119895120575
1119896
(64)
for all 119894 = 1 119873 Thus
119866
119896=
119872V if 119896 = 1
0 if 119896 = 2 119875
(65)
Now solving 119890
119906(119906(119911) 119911)119902 = 119869
119906119906(119906(119911) 119911)119908 for 119902 requires
the solution to the linear system 119870 119902 = 119867 where
(
119867
119896)
119894=
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895(
119896)
119895
(66)
or equivalently
119867
119896= 119872
119896for all 119896 = 1 119875 Hence in the
direction 120601
119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by
⟨
119869
10158401015840(119911) V 120601119894⟩Zlowast Z
asymp ⟨
119869
10158401015840
ℎ119901(119911
ℎ) V 120601
119894⟩
Zlowast Z
= minusint
Ω
120588 (120596)int
119863
119875
sum
119896=1
119873
sum
119895=1
( 119902
119896)
119895120601
119895 (x) Ψ119896 (120596)
times 120601
119894 (x) 119889x119889120596
+ 120572int
119863
V (119909) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119895=1
119872
119894119895( 119902
119896)
119895120575
1119896+ 120572
119873
sum
119895=1
119872
119894119895V119895
=
119873
sum
119895=1
119872
119894119895(120572V
119895minus ( 119902
1)
119895)
(67)
Therefore
119869
10158401015840(119911) V asymp 119872(120572V minus 119902
1)
(68)
Having (62) and (68) we can use the iterative solvers tothe Newton equation
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) (69)
Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572
119896and using this step size to generate
the iterate 119911
119896+1= 119911
119896+ 120572
119896119878
119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is
119869 (119911
119896+ 120572
119896119878
119896) le 119869 (119911
119896) + 119888120572
119896119869
1015840(119911
119896) 119878
119896 (70)
where 119888 isin (0 1) and typically is quite small for example 119888 =
10
minus4 [12 13]
8 ISRN Applied Mathematics
(1) Given 119911
0 and tol gt 0 set 119896 = 0
(2) Compute 119870119872 119860 and
119887
(3) Compute (
119865
119896)
119894=
119887 minus 119872119906 if 119896 = 1
minus119872119906 if 119896 = 2 119875
(4) Solve 119870 120583 = 119865
(5) Compute
119869
1015840(119911
119896) = 119872(120572119911
119896minus
1)
(6) Compute
119866
1= 119872V and
119866
119905= 0 for 119905 = 2 119875
(7) Solve 119870 = 119866 = (
119866
119879
1
119866
119879
119875)
119879
(8) Compute
119867
119896= 119872
119896for 119896 = 1 119875
(9) Solve 119870 119902 = 119867 = (
119867
119879
1
119867
119879
119875)
119879
(10) Compute
119869
10158401015840(119911)V asymp 119872(120572V minus 119902
1)
(11) If 1003817100381710038171003817
1003817
nabla
119869 (119911
119896)
1003817
1003817
1003817
1003817
1003817
lt tol stop(12) Solve
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) using PCG method with preconditioner
119875 = 119866⨂119870
0
(i) Set 1199090= 0
(ii) Set 1199030larr
119869
10158401015840(119911
119896) 119909
0+
119869
1015840(119911
119896)
(iii) Solve
119875119910
0= 119903
0for 119910
0
(iv) Set 1199010= minus119903
0 119905 larr 0
(v) While 1003817
1003817
1003817
1003817
119903
119905
1003817
1003817
1003817
1003817
gt tol
120572
119896larr
119903
119879
119905119910
119905
119901
119879
119905119860119901
119905
119909
119905+1larr 119909
119905+ 120572
119905119901
119905
119903
119905+1larr 119903
119905+ 120572
119905119860119901
119905
119875119910
119905+1larr 119903
119905+1
120573
119905+1larr
119903
119879
119905+1119910
119905+1
119903
119879
119905119910
119905
119901
119905+1larr minus119910
119905+1+ 120573
119905+1119901
119905
119905 larr 119905 + 1
End (while)(vi) 119904119896 = 119909
119905+1
(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911
119896+ 120572
119896119878
119896)
(ii) While 119869 (119911
119896+ 120572
119896119878
119896) gt 119869 (119911
119896) + 10
minus4120572
119896119878
119896nabla
119869 (119911
119896) do
(i) Set 120572119896 = 120572
1198962 and evaluate 119869 (119911
119896+ 120572
119896119878
119896)
(14) Set 119911119896+1 = 119911
119896+ 120572
119896119878
119896 119896 larr 119896 + 1 Go to (2)
Algorithm 1
31 Kronecker Product Preconditioners Let us to rewrite (37)in the form
119860 =
119911 where
119860 = (
119872
sum
119896=0
119866
119896⨂119870
119896) (71)
The strategy here is to introduce
119875 = 119866⨂119870
0as a
preconditioner such that
119866 = argmin 119867 isin R119875times119875
1003817
1003817
1003817
1003817
1003817
119860 minus 119867⨂119870
0
1003817
1003817
1003817
1003817
1003817119865 (72)
where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot
119865denotes the Frobe-
nius norm The closed form of the solution can be written asfollow [14]
119866 = 119868 + sum
119875
tr (119870119879
119896119870
0)
tr (119870119879
0119870
0)
119866
119896
(73)
Here tr(119870119879
119896119870
0) = sum
119873119902
119894=1[119870
119896]
119894119894[119870
0]
119894119894and hence the coeffi-
cients in the above equality can be computed straightforwardIn addition since
119860 and 119870
0are symmetric and positive
definite thus 119866 and
119875 = 119866⨂119870
0have also these properties
[15 16]
Example 1 Consider the following distributed optimal con-trol problem
min119906119911
119869 (119906 119911) = 119864 [
1
2
119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω
(74)
where 119863 = [minus1 1]
2 Ω sim 119880[minus1 1] the boundary conditionsare given as
119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0
(75)
ISRN Applied Mathematics 9
005
1
005
10
05
1y d
xy
(a) Desired values
005
1
005
10
05
1
uM
ean
xy
times10minus3
(b) Initial state
005
1
005
10
05
1
u opt
xy
(c) Approximated solution (state)
005
1
005
10
200
400
z opt
xy
(d) Control
005
1
0
05
10
05
1
times 10minus3
xy
120583
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
05
1
15
2
25
times 10minus3
(f) Expected value
20 40
10
20
30
40
50
05
1
15
2
25
times 10minus3
(g) Standard dev
0 5 10times 104
0
5
10
times 104 0
(h) Sparsity of global matrix119899119911 = 1815980
Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))
2+ (119910 minus (12))
2)]
119873 = 3 119870 = 5 and 120572 = 000005
and the desired function is given by
(119909 119910) = exp [minus64 ((119909 minus
1
2
)
2
+ (119910 minus
1
2
)
2
)] (76)
The random field 119886(x 120596) is characterized by its mean andcovariance function
119864 [119886] = 119886 = 10
119877 (119909
1 119909
2) = 119890
minus|1199091minus1199092| 119909
1 119909
2isin [minus1 1]
(77)
Following Section 22 the truncated KLE can be representedas
119886 (119909 119910 120596) = 10 +
119873
sum
119895=0
radic120582
119895120579
119895120601
119895 (119909) (78)
The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867
1
0(119863)⨂119871
2(Ω) and all 119911isin 119871
2(119863)
119869 (119906opt 119911opt) le 119869 (119906 119911) (79)
The eigenpairs 120582119895 120579
119895 in truncated KLE solve the integral
equation
int
1
minus1
119890
minus|1199091minus1199092|120601
119895(119909
2) 119889119909
2= 120582
119895120601
119895(119909
1)
(80)
For this special case of the covariance function we haveexplicit expressions for 120601
119895and 120582
119895 [6] Let120596
119895evenand120596
119895oddsolve
the equations
1 minus 120596
119895eventan (120596
119895even) = 0
120596
119895odd+ tan (120596
119895odd) = 0
(81)
Then the even and odd indexed eigenfunctions are given by
120601
119895even(119909) =
cos (120596119895even
119909)
radic1 + (sin (2120596
119895even) 2120596
119895even)
120601
119895odd(119909)
=
sin (120596
119895odd119909)
radic1 minus (sin (2120596
119895odd) 2120596
119895odd)
(82)
10 ISRN Applied Mathematics
005
1
005
10
05
1y ha
nd
xy
(a) Desired values
005
1
005
1minus05
05
0
u Mea
n
xy
(b) Mean values
005
1
005
1
0
Num
sol
xy
0
1
(c) Approximated solution (State)
005
1
005
10
05
1
xy
(d) Control
005
1
005
10
05
1
Lend
a
xy
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
0
005
01
015
02
(f) Expected value-SGS
20 40
10
20
30
40
50
0
005
01
015
02
(g) Standard dev-SGS
0 2 4 6 8times 104
0
2
4
6
8
times 104
(h) Sparsity of global matrix119899119911 = 3046160
Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001
and corresponding eigenvalues
120582
119895even(119909) =
2
120596
2
119895even+ 1
120582
119895odd(119909) =
2
120596
2
119895odd+ 1
(83)
We choose 120579 = (120579
1 120579
119873)
119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]
First of all we find numerical approximations of120596119895even
and120596
119895oddwith bisection method and then with the eigenpairs
evaluated by this 120596119895even
and 120596
119895odd by choosing dimension 119873 =
3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1
In Figure 1(a) the desired function (119909 119910) is plottedfor 119909
119894= 119910
119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial
state is computed and plotted in Figure 1(b) by solving (37)where
119911 = 0 Approximated 119906opt corresponding to the
optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial
119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911
119900119901119905must approximate the
initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane
In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use
Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary
ISRN Applied Mathematics 11
conditions is replaced by (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2
4 Conclusion
The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996
[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987
[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005
[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947
[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946
[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991
[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007
[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009
[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938
[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993
[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008
[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999
[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006
[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010
[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010
[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
4 ISRN Applied Mathematics
where 120588(120596) is the integration weight Equation (24) is equiv-alent to the following equation which results by plugging inthe explicit formula for the basis functions
int
Ω
120588 (120596)Ψn (120596) int
119863
119886 (x 120596) nabla119906
ℎ119875 (x 120596) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψn (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(25)
for all n isinI and 119894 = 1 119873Now considering the approximate solution as
119906
ℎ119875 (x 120596) =
119873
sum
119895=1
sum
misinI119906
119895m120601
119895 (x) Ψm (120596) (26)
and substituting (26) into (25) yields119873
sum
119895=1
sum
misinI119906
119895m int
Ω
120588 (120596)Ψn (120596)Ψm (120596)
times int
119863
119886 (x 120596) nabla120601
119895 (x) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψn (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(27)
for allm isinI and 119894 = 1 119873 Let 119875 = |I| = prod
119872
119896=1119901
119896 then
a natural bijection between 1 119875 and Ican be definedThus (27) can be written as
119875
sum
119898=1
119873
sum
119895=1
119906
119898119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596)
times int
119863
119886 (x 120596) nabla120601
119895 (x) sdot nabla120601
119894 (x) 119889x119889120596
= int
Ω
120588 (120596)Ψ119899 (120596) int
119863
119911 (x 120596) 120601119894 (x) 119889x119889120596
(28)
Now if 119886 and 119911 have the following KLE
119886 (x 120596) = 119886
0 (x) +
119872
sum
119894=1
120596
119894119886
119894 (x)
119911 (x 120596) = 119911
0 (x) +
119872
sum
119894=1
120596
119894119911
119894 (x)
(29)
then we can rewrite the Galerkin projection (28) as119875
sum
119898=1
119873
sum
119895=1
119906
119898119895((119870
0)
119894119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596
+
119872
sum
119896=1
(119870
119896)
119894119895int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 120596119896
119889120596)
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(30)
where
(119870
119896)
119894119895= int
119863
119886
119896 (x) nabla120601
119894 (x) sdot nabla120601
119895 (x) 119889x
cong sum
119903
119908
119903119886
119896(x119903) nabla120601
119894(x119903) sdot nabla120601
119895(x119903)
(119911
119896)
119894= int
119863
119911
119894 (x) 120601119894 (x) 119889x cong sum
119903
119908
119903119911
119894(x119903) 120601
119894(x119903)
(31)
for 119896 = 1 119872 [3] Equation (31) is approximated byquadrature rule in spatial domain Now assume that thereexist functions Ψ
119899 119899 = 1 119875 such that
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596 = 120575
119898119899
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 120596119896
119889120596 = 119862
119896119899120575
119899119898
(32)
where for 119898 = 119899 the above integral is approximated usingquadrature rule in random domain as follows
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119899 (
120596) 120596119896119889120596 cong sum
119904
119908
1015840
119904120588 (120596
119904) Ψ
2
119899(120596
119904) 120596
119896 (33)
Then (30) becomes
119875
sum
119898=1
119873
sum
119895=1
119906
119898119895((119870
0)
119894119895+
119872
sum
119896=1
119862
119896119899(119870
119896)
119894119895)120575
119899119898
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(34)
or equivalently
119873
sum
119895=1
119906m119895 ((119870
0)
119894119895+
119872
sum
119896=1
119862
119896119899(119870
119896)
119894119895)
= (119911
0)
119894int
Ω
120588 (120596)Ψ119899 (120596) 119889120596
+
119872
sum
119896=1
(119911
119896)
119894int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596
(35)
In (34) and (35) we have
int
Ω
120588 (120596)Ψ119899 (120596) 119889120596 = 0 for 119899 = 0
int
Ω
120588 (120596)Ψ119899 (120596) 120596119896
119889120596 = int
Ω
120588 (120596)Ψ0 (120596)Ψ119899 (
120596) 120596119896119889120596
= 119862
119896119899120575
0119899= 119862
1198960
(36)
ISRN Applied Mathematics 5
Finally (35) can be considered as the following blockdiagonal system
(
(
(
119870
0+
119872
sum
119896=1
119862
1198961119870
119896sdot sdot sdot 0
d
0 sdot sdot sdot 119870
0+
119872
sum
119896=1
119862
119896119901119870
119896
)
)
)
(
1
119875
)
= (
119911
1
119911
119901
)
(37)
or briefly 119870 =
119911
3 Distributed Optimal Control Problem
Using the stochastic Galerkin method for general objectivefunctions computing the derivatives becomes very tediousand rather messy Consider the following optimal controlproblem
min119906isin119880 119911isinz
119869 (119906 119911) =
1
2
119864 [119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
st minus nabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω
119906 (x 120596) = 0 on 120597119863 times Ω
(38)
The solutions to linear elliptic SPDEs live in the space119880 =
119867
1
0(119863)⨂119871
2(Ω) and the control space Z = 119871
2(119863) Denoting
the constraint equation by 119890(119906 119911) = 0 usually it is possibleto invoke the implicit function theorem to find a solution119906(119911) to the constraint equation This leads to an implicitlydefined objective function
119869(119911) = 119869(119906(119911) 119911) Now we focuson computing derivatives of 119869 [11] For any direction 119904 isin Z
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= ⟨119869
119906 (119906 (119911) 119911) 119906119911 (
119911) 119904⟩
119880lowast119880
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(39)
where119880lowast andZlowast are the dual space of119880 andZ respectivelyNow computing the derivative of the constraint 119890with respectto 119911 in the direction 119904 and applying it to the implicit solutionto 119890(119906(119911) 119911) = 0 yield
119890
119906 (119906 (119911) 119911) 119906119911 (
119911) 119904 + 119890
119911 (119906 (119911) 119911) 119904 = 0 (40)
So 119906
119911(119911) is
119906
119911 (119911) 119904 = minus119890
119906(119906 (119911) 119911)
minus1119890
119911 (119906 (119911) 119911) 119904
(41)
Hence
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= minus ⟨119869
119906 (119906 (119911) 119911) 119890119906(
119906 (119911) 119911)
minus1
times 119890
119911 (119906 (119911) 119911) 119904⟩
119880lowast119880
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(42)
Applying the adjoint to the 119906
119911(119911) term
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= minus ⟨119890
119911(119906 (119911) 119911)
lowast119890
119906(119906 (119911) 119911)
minuslowast
times 119869
119906 (119906 (119911) 119911) 119904⟩
Zlowast Z
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
= minus ⟨119890
119911(119906 (119911) 119911)
lowast119890
119906(119906 (119911) 119911)
minuslowast119869
119906(119906 (119911) 119911)
+119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(43)
Considering the adjoint state 120583 = 120583(119911) isin Zlowast as the solutionto
119890
119906(119906 (119911) 119911)
lowast120583 = minus119869
119906 (119906 (119911) 119911) (44)
the derivative of 1198691015840 becomes
119869
1015840(119911) = 119890
119911(119906 (119911) 119911)
lowast120583 (119911) + 119869
119911 (119906 (119911) 119911)
(45)
Theweak formof the constraint function is defined as followsfor all V isin 119880
⟨119890 (119906 119911) V⟩119880lowast 119880 = int
Ω
120588 (120596)int
119863
(119886 (x 120596) nabla119906 (x 120596) sdot nablaV (x 120596)
minus119911 (x 120596) V (x 120596)) 119889x119889120596
(46)
Thus the constraint function 119890 119880 times Z rarr 119880
lowast and 119884 = 119880
lowastSince the constraint function 119890 acts linearly with respect to 119906
and 119911 thus the corresponding derivative of 119890 with respect to119911 in the direction 120575119911 isin Z for all V isin 119880 is given by
⟨119890
119911 (119906 119911) 120575119911 V⟩
119880lowast119880
= minusint
Ω
120588 (120596)int
119863
120575119911 (x 120596) V (x 120596) 119889x119889120596
(47)
Similarly the derivative of 119890 with respect to 119906 for anydirection 120575119906 isin 119880 and for all V isin 119880 is given by
⟨119890
119906 (119906 119911) 120575119906 V⟩
119880lowast119880
= int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla120575119906 (x 120596)
sdot nablaV (x 120596) 119889x119889120596
(48)
Both of these derivatives are symmetric so 119890
119906(119906 119911)
lowast=
119890
119906(119906 119911) and 119890
119911(119906 119911)
lowast= 119890
119911(119906 119911) From here the adjoint can
be computed as
119890
119906 (119906 (119911) 119911) 120583 = minus119869
119906 (119906 (119911) 119911) (49)
Using this to any direction V isin 119880
⟨119890
119906 (119906 (119911) 119911) 120583 V⟩
119880lowast119880
= int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla120583 (x 120596)
sdot nablaV (x 120596) 119889x119889120596
= minus119869
119906 (119906 (119911) 119911) V
(50)
6 ISRN Applied Mathematics
By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is
⟨119869
119906 (119906 119911) 120575119906⟩
119880lowast119880
= 119864 [119906 (x 120596) minus (x)21198712(119863)
times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)
]
(51)
Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin
Z direction is
⟨119869
119911 (119906 119911) 120575119911⟩
Zlowast Z= 120572⟨119911 120575119911⟩Z (52)
With these computations one can compute the firstderivative of
119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883
ℎsub 119867
1
0(119863) is a finite dimensional
subspace of dimension 119873 and 119884
ℎsub 119871
2(Ω) is a finite
dimensional subspace of dimension 119875 then 119883
ℎ⨂119884
ℎis a
finite dimensional subspace of the state space 119880 Similarlylet Z
ℎsub Z be a finite dimensional subspace of the control
space (with dimension) For the stochastic Galerkin methodit is assumed that 119884
ℎ= P119875minus1 the space of polynomials with
highest degree 119875minus1 and119883
ℎis any finite element space here
119883
ℎis the space of linear functions built on a given meshT
ℎ
We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884
ℎ that is 119884
ℎ= spanΨ
1(120596) Ψ
119875(120596)
where
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596 = 120575
119898119899 (53)
The discretized optimization problem in the stochasticGalerkin framework is
min119911ℎisinZℎ
119869
ℎ119875(119911
ℎ) =
1
2
119864
[
[
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
119875
sum
119896=1
119873
sum
119895=1
(
119896(119911
ℎ))
119895Ψ
119896 (120596) 120601119895 (
x)
minus (x)1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
2
1198712(119863)
]
]
+
120572
2
1003817
1003817
1003817
1003817
119911
ℎ
1003817
1003817
1003817
1003817
2
1198712(119863)
(54)
where
119896(119911
ℎ) = (
1(119911
ℎ)
119879
119875(119911
ℎ)
119879)
119879
=
(55)
is the stochastic Galerkin solution to the state equation
(
119870
11sdot sdot sdot 119870
1119875
d
119870
1198751sdot sdot sdot 119870
119875119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119870
(
1
119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=
= (
119864 [Ψ
1]119872
119911
ℎ
119864 [Ψ
119875]119872
119911
ℎ
)
= (
120575
11119872
119911
ℎ
120575
1119875119872
119911
ℎ
)
(56)
since 119864[Ψ
119896] = 119864[1Ψ
119896] = 119864[Ψ
1Ψ
119896] = 120575
1119896 The blocks of the
above 119870matrix have the form
(119870
119896119897)
119894119895= int
Ω
120588 (120596)Ψ
119896(120596)Ψ119897 (
120596)
times int
119863
119886 (x 120596) nabla120601
119894 (x) sdot nabla120601
119895 (x) 119889x119889120596
(57)
First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations
minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))
in 119863 times Ω
120583 (x 120596) = 0 on 120597119863 times Ω
(58)
Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865
119879
1 119865
119879
119901)
119879 and 119865
119896is defined
as
(
119865
119896)
119894= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596
= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895120601
119895 (x) Ψ119897 (120596) 120601119894 (
x)
minus (x) 120601119894 (x))119889x119889120596
= minus
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
(x) 120601119894 (x) 119889x
=
119887
119894120575
1119896minus
119875
sum
119897=1
119873
sum
119895=1
119872
119894119895(
119897)
119895120575
119896119897
=
119887
119894120575
1119896minus
119873
sum
119895=1
119872
119894119895(
119897)
119895
(59)
in which
119887
119894= int 120601
119894 and119872
119894119895= int 120601
119894120601
119895
Thus
(
119865
119896)
119894=
119887 minus 119872119906
1if 119896 = 1
minus119872119906 if 119896 = 2 119875
(60)
where b= [
119887
1
119887
119873] and the matrix 119872 = (119872
119894119895) is of order
119873 times 119873
ISRN Applied Mathematics 7
Now the derivative of
119869(119911) in the direction 120601
119895(x) for all
119895 = 1 119873 with the computed adjoint state is
⟨
119869
1015840(119911) 120601119895 (
x)⟩Zlowast Z
= ⟨119890
119911(119906 (119911 sdot sdot) 119911)
lowast120583
+ 119869
119911 (119906 (119911 sdot sdot) 119911) 120601119895 (
x) ⟩Zlowast Z
= minusint
Ω
120588 (120596)int
119863
120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596
+ 120572int
119863
119911 (119909) 120601119895 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894int
Ω
120588 (120596)Ψ119896 (120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ 120572
119873
sum
119894=1
119911
119894int
119863
120601
119895 (x) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894119872
119894119895120575
1119896+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
= minus
119873
sum
119894=1
(
1)
119894119872
119894119895+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
(61)
Therefore the gradient can be computed as follows
119869
1015840(119911) = 119872(120572
119911
ℎminus
1)
(62)
Now for any vector V isin Z it is possible to write V as
V =
119873
sum
119896=1
V119896120601
119896 (x) (63)
In order to compute the Hessian times a vector that ismultiply the Hessian of the
119869 with a some vector V isin Z theequation 119890
119906(119906(119911) 119911)119908 = 119890
119911(119906(119911) 119911)V for 119908 must be solved
Similar to the adjoint computation119908 solves the linear system119870 = 119866 where
119866 = (
119866
119879
1
119866
119879
119875) and
119866
119896for 119896 = 1 119875 is
(
119866
119896)
119894= int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
119873
sum
119895=1
V119895120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895V119895120575
1119896
(64)
for all 119894 = 1 119873 Thus
119866
119896=
119872V if 119896 = 1
0 if 119896 = 2 119875
(65)
Now solving 119890
119906(119906(119911) 119911)119902 = 119869
119906119906(119906(119911) 119911)119908 for 119902 requires
the solution to the linear system 119870 119902 = 119867 where
(
119867
119896)
119894=
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895(
119896)
119895
(66)
or equivalently
119867
119896= 119872
119896for all 119896 = 1 119875 Hence in the
direction 120601
119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by
⟨
119869
10158401015840(119911) V 120601119894⟩Zlowast Z
asymp ⟨
119869
10158401015840
ℎ119901(119911
ℎ) V 120601
119894⟩
Zlowast Z
= minusint
Ω
120588 (120596)int
119863
119875
sum
119896=1
119873
sum
119895=1
( 119902
119896)
119895120601
119895 (x) Ψ119896 (120596)
times 120601
119894 (x) 119889x119889120596
+ 120572int
119863
V (119909) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119895=1
119872
119894119895( 119902
119896)
119895120575
1119896+ 120572
119873
sum
119895=1
119872
119894119895V119895
=
119873
sum
119895=1
119872
119894119895(120572V
119895minus ( 119902
1)
119895)
(67)
Therefore
119869
10158401015840(119911) V asymp 119872(120572V minus 119902
1)
(68)
Having (62) and (68) we can use the iterative solvers tothe Newton equation
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) (69)
Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572
119896and using this step size to generate
the iterate 119911
119896+1= 119911
119896+ 120572
119896119878
119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is
119869 (119911
119896+ 120572
119896119878
119896) le 119869 (119911
119896) + 119888120572
119896119869
1015840(119911
119896) 119878
119896 (70)
where 119888 isin (0 1) and typically is quite small for example 119888 =
10
minus4 [12 13]
8 ISRN Applied Mathematics
(1) Given 119911
0 and tol gt 0 set 119896 = 0
(2) Compute 119870119872 119860 and
119887
(3) Compute (
119865
119896)
119894=
119887 minus 119872119906 if 119896 = 1
minus119872119906 if 119896 = 2 119875
(4) Solve 119870 120583 = 119865
(5) Compute
119869
1015840(119911
119896) = 119872(120572119911
119896minus
1)
(6) Compute
119866
1= 119872V and
119866
119905= 0 for 119905 = 2 119875
(7) Solve 119870 = 119866 = (
119866
119879
1
119866
119879
119875)
119879
(8) Compute
119867
119896= 119872
119896for 119896 = 1 119875
(9) Solve 119870 119902 = 119867 = (
119867
119879
1
119867
119879
119875)
119879
(10) Compute
119869
10158401015840(119911)V asymp 119872(120572V minus 119902
1)
(11) If 1003817100381710038171003817
1003817
nabla
119869 (119911
119896)
1003817
1003817
1003817
1003817
1003817
lt tol stop(12) Solve
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) using PCG method with preconditioner
119875 = 119866⨂119870
0
(i) Set 1199090= 0
(ii) Set 1199030larr
119869
10158401015840(119911
119896) 119909
0+
119869
1015840(119911
119896)
(iii) Solve
119875119910
0= 119903
0for 119910
0
(iv) Set 1199010= minus119903
0 119905 larr 0
(v) While 1003817
1003817
1003817
1003817
119903
119905
1003817
1003817
1003817
1003817
gt tol
120572
119896larr
119903
119879
119905119910
119905
119901
119879
119905119860119901
119905
119909
119905+1larr 119909
119905+ 120572
119905119901
119905
119903
119905+1larr 119903
119905+ 120572
119905119860119901
119905
119875119910
119905+1larr 119903
119905+1
120573
119905+1larr
119903
119879
119905+1119910
119905+1
119903
119879
119905119910
119905
119901
119905+1larr minus119910
119905+1+ 120573
119905+1119901
119905
119905 larr 119905 + 1
End (while)(vi) 119904119896 = 119909
119905+1
(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911
119896+ 120572
119896119878
119896)
(ii) While 119869 (119911
119896+ 120572
119896119878
119896) gt 119869 (119911
119896) + 10
minus4120572
119896119878
119896nabla
119869 (119911
119896) do
(i) Set 120572119896 = 120572
1198962 and evaluate 119869 (119911
119896+ 120572
119896119878
119896)
(14) Set 119911119896+1 = 119911
119896+ 120572
119896119878
119896 119896 larr 119896 + 1 Go to (2)
Algorithm 1
31 Kronecker Product Preconditioners Let us to rewrite (37)in the form
119860 =
119911 where
119860 = (
119872
sum
119896=0
119866
119896⨂119870
119896) (71)
The strategy here is to introduce
119875 = 119866⨂119870
0as a
preconditioner such that
119866 = argmin 119867 isin R119875times119875
1003817
1003817
1003817
1003817
1003817
119860 minus 119867⨂119870
0
1003817
1003817
1003817
1003817
1003817119865 (72)
where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot
119865denotes the Frobe-
nius norm The closed form of the solution can be written asfollow [14]
119866 = 119868 + sum
119875
tr (119870119879
119896119870
0)
tr (119870119879
0119870
0)
119866
119896
(73)
Here tr(119870119879
119896119870
0) = sum
119873119902
119894=1[119870
119896]
119894119894[119870
0]
119894119894and hence the coeffi-
cients in the above equality can be computed straightforwardIn addition since
119860 and 119870
0are symmetric and positive
definite thus 119866 and
119875 = 119866⨂119870
0have also these properties
[15 16]
Example 1 Consider the following distributed optimal con-trol problem
min119906119911
119869 (119906 119911) = 119864 [
1
2
119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω
(74)
where 119863 = [minus1 1]
2 Ω sim 119880[minus1 1] the boundary conditionsare given as
119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0
(75)
ISRN Applied Mathematics 9
005
1
005
10
05
1y d
xy
(a) Desired values
005
1
005
10
05
1
uM
ean
xy
times10minus3
(b) Initial state
005
1
005
10
05
1
u opt
xy
(c) Approximated solution (state)
005
1
005
10
200
400
z opt
xy
(d) Control
005
1
0
05
10
05
1
times 10minus3
xy
120583
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
05
1
15
2
25
times 10minus3
(f) Expected value
20 40
10
20
30
40
50
05
1
15
2
25
times 10minus3
(g) Standard dev
0 5 10times 104
0
5
10
times 104 0
(h) Sparsity of global matrix119899119911 = 1815980
Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))
2+ (119910 minus (12))
2)]
119873 = 3 119870 = 5 and 120572 = 000005
and the desired function is given by
(119909 119910) = exp [minus64 ((119909 minus
1
2
)
2
+ (119910 minus
1
2
)
2
)] (76)
The random field 119886(x 120596) is characterized by its mean andcovariance function
119864 [119886] = 119886 = 10
119877 (119909
1 119909
2) = 119890
minus|1199091minus1199092| 119909
1 119909
2isin [minus1 1]
(77)
Following Section 22 the truncated KLE can be representedas
119886 (119909 119910 120596) = 10 +
119873
sum
119895=0
radic120582
119895120579
119895120601
119895 (119909) (78)
The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867
1
0(119863)⨂119871
2(Ω) and all 119911isin 119871
2(119863)
119869 (119906opt 119911opt) le 119869 (119906 119911) (79)
The eigenpairs 120582119895 120579
119895 in truncated KLE solve the integral
equation
int
1
minus1
119890
minus|1199091minus1199092|120601
119895(119909
2) 119889119909
2= 120582
119895120601
119895(119909
1)
(80)
For this special case of the covariance function we haveexplicit expressions for 120601
119895and 120582
119895 [6] Let120596
119895evenand120596
119895oddsolve
the equations
1 minus 120596
119895eventan (120596
119895even) = 0
120596
119895odd+ tan (120596
119895odd) = 0
(81)
Then the even and odd indexed eigenfunctions are given by
120601
119895even(119909) =
cos (120596119895even
119909)
radic1 + (sin (2120596
119895even) 2120596
119895even)
120601
119895odd(119909)
=
sin (120596
119895odd119909)
radic1 minus (sin (2120596
119895odd) 2120596
119895odd)
(82)
10 ISRN Applied Mathematics
005
1
005
10
05
1y ha
nd
xy
(a) Desired values
005
1
005
1minus05
05
0
u Mea
n
xy
(b) Mean values
005
1
005
1
0
Num
sol
xy
0
1
(c) Approximated solution (State)
005
1
005
10
05
1
xy
(d) Control
005
1
005
10
05
1
Lend
a
xy
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
0
005
01
015
02
(f) Expected value-SGS
20 40
10
20
30
40
50
0
005
01
015
02
(g) Standard dev-SGS
0 2 4 6 8times 104
0
2
4
6
8
times 104
(h) Sparsity of global matrix119899119911 = 3046160
Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001
and corresponding eigenvalues
120582
119895even(119909) =
2
120596
2
119895even+ 1
120582
119895odd(119909) =
2
120596
2
119895odd+ 1
(83)
We choose 120579 = (120579
1 120579
119873)
119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]
First of all we find numerical approximations of120596119895even
and120596
119895oddwith bisection method and then with the eigenpairs
evaluated by this 120596119895even
and 120596
119895odd by choosing dimension 119873 =
3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1
In Figure 1(a) the desired function (119909 119910) is plottedfor 119909
119894= 119910
119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial
state is computed and plotted in Figure 1(b) by solving (37)where
119911 = 0 Approximated 119906opt corresponding to the
optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial
119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911
119900119901119905must approximate the
initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane
In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use
Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary
ISRN Applied Mathematics 11
conditions is replaced by (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2
4 Conclusion
The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996
[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987
[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005
[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947
[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946
[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991
[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007
[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009
[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938
[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993
[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008
[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999
[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006
[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010
[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010
[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007
Submit your manuscripts athttpwwwhindawicom
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Mathematical PhysicsAdvances in
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRN Applied Mathematics 5
Finally (35) can be considered as the following blockdiagonal system
(
(
(
119870
0+
119872
sum
119896=1
119862
1198961119870
119896sdot sdot sdot 0
d
0 sdot sdot sdot 119870
0+
119872
sum
119896=1
119862
119896119901119870
119896
)
)
)
(
1
119875
)
= (
119911
1
119911
119901
)
(37)
or briefly 119870 =
119911
3 Distributed Optimal Control Problem
Using the stochastic Galerkin method for general objectivefunctions computing the derivatives becomes very tediousand rather messy Consider the following optimal controlproblem
min119906isin119880 119911isinz
119869 (119906 119911) =
1
2
119864 [119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
st minus nabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) in 119863 times Ω
119906 (x 120596) = 0 on 120597119863 times Ω
(38)
The solutions to linear elliptic SPDEs live in the space119880 =
119867
1
0(119863)⨂119871
2(Ω) and the control space Z = 119871
2(119863) Denoting
the constraint equation by 119890(119906 119911) = 0 usually it is possibleto invoke the implicit function theorem to find a solution119906(119911) to the constraint equation This leads to an implicitlydefined objective function
119869(119911) = 119869(119906(119911) 119911) Now we focuson computing derivatives of 119869 [11] For any direction 119904 isin Z
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= ⟨119869
119906 (119906 (119911) 119911) 119906119911 (
119911) 119904⟩
119880lowast119880
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(39)
where119880lowast andZlowast are the dual space of119880 andZ respectivelyNow computing the derivative of the constraint 119890with respectto 119911 in the direction 119904 and applying it to the implicit solutionto 119890(119906(119911) 119911) = 0 yield
119890
119906 (119906 (119911) 119911) 119906119911 (
119911) 119904 + 119890
119911 (119906 (119911) 119911) 119904 = 0 (40)
So 119906
119911(119911) is
119906
119911 (119911) 119904 = minus119890
119906(119906 (119911) 119911)
minus1119890
119911 (119906 (119911) 119911) 119904
(41)
Hence
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= minus ⟨119869
119906 (119906 (119911) 119911) 119890119906(
119906 (119911) 119911)
minus1
times 119890
119911 (119906 (119911) 119911) 119904⟩
119880lowast119880
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(42)
Applying the adjoint to the 119906
119911(119911) term
⟨
119869
1015840(119911) 119904⟩
Zlowast Z= minus ⟨119890
119911(119906 (119911) 119911)
lowast119890
119906(119906 (119911) 119911)
minuslowast
times 119869
119906 (119906 (119911) 119911) 119904⟩
Zlowast Z
+ ⟨119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
= minus ⟨119890
119911(119906 (119911) 119911)
lowast119890
119906(119906 (119911) 119911)
minuslowast119869
119906(119906 (119911) 119911)
+119869
119911 (119906 (119911) 119911) 119904⟩
Zlowast Z
(43)
Considering the adjoint state 120583 = 120583(119911) isin Zlowast as the solutionto
119890
119906(119906 (119911) 119911)
lowast120583 = minus119869
119906 (119906 (119911) 119911) (44)
the derivative of 1198691015840 becomes
119869
1015840(119911) = 119890
119911(119906 (119911) 119911)
lowast120583 (119911) + 119869
119911 (119906 (119911) 119911)
(45)
Theweak formof the constraint function is defined as followsfor all V isin 119880
⟨119890 (119906 119911) V⟩119880lowast 119880 = int
Ω
120588 (120596)int
119863
(119886 (x 120596) nabla119906 (x 120596) sdot nablaV (x 120596)
minus119911 (x 120596) V (x 120596)) 119889x119889120596
(46)
Thus the constraint function 119890 119880 times Z rarr 119880
lowast and 119884 = 119880
lowastSince the constraint function 119890 acts linearly with respect to 119906
and 119911 thus the corresponding derivative of 119890 with respect to119911 in the direction 120575119911 isin Z for all V isin 119880 is given by
⟨119890
119911 (119906 119911) 120575119911 V⟩
119880lowast119880
= minusint
Ω
120588 (120596)int
119863
120575119911 (x 120596) V (x 120596) 119889x119889120596
(47)
Similarly the derivative of 119890 with respect to 119906 for anydirection 120575119906 isin 119880 and for all V isin 119880 is given by
⟨119890
119906 (119906 119911) 120575119906 V⟩
119880lowast119880
= int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla120575119906 (x 120596)
sdot nablaV (x 120596) 119889x119889120596
(48)
Both of these derivatives are symmetric so 119890
119906(119906 119911)
lowast=
119890
119906(119906 119911) and 119890
119911(119906 119911)
lowast= 119890
119911(119906 119911) From here the adjoint can
be computed as
119890
119906 (119906 (119911) 119911) 120583 = minus119869
119906 (119906 (119911) 119911) (49)
Using this to any direction V isin 119880
⟨119890
119906 (119906 (119911) 119911) 120583 V⟩
119880lowast119880
= int
Ω
120588 (120596)int
119863
119886 (x 120596) nabla120583 (x 120596)
sdot nablaV (x 120596) 119889x119889120596
= minus119869
119906 (119906 (119911) 119911) V
(50)
6 ISRN Applied Mathematics
By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is
⟨119869
119906 (119906 119911) 120575119906⟩
119880lowast119880
= 119864 [119906 (x 120596) minus (x)21198712(119863)
times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)
]
(51)
Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin
Z direction is
⟨119869
119911 (119906 119911) 120575119911⟩
Zlowast Z= 120572⟨119911 120575119911⟩Z (52)
With these computations one can compute the firstderivative of
119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883
ℎsub 119867
1
0(119863) is a finite dimensional
subspace of dimension 119873 and 119884
ℎsub 119871
2(Ω) is a finite
dimensional subspace of dimension 119875 then 119883
ℎ⨂119884
ℎis a
finite dimensional subspace of the state space 119880 Similarlylet Z
ℎsub Z be a finite dimensional subspace of the control
space (with dimension) For the stochastic Galerkin methodit is assumed that 119884
ℎ= P119875minus1 the space of polynomials with
highest degree 119875minus1 and119883
ℎis any finite element space here
119883
ℎis the space of linear functions built on a given meshT
ℎ
We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884
ℎ that is 119884
ℎ= spanΨ
1(120596) Ψ
119875(120596)
where
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596 = 120575
119898119899 (53)
The discretized optimization problem in the stochasticGalerkin framework is
min119911ℎisinZℎ
119869
ℎ119875(119911
ℎ) =
1
2
119864
[
[
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
119875
sum
119896=1
119873
sum
119895=1
(
119896(119911
ℎ))
119895Ψ
119896 (120596) 120601119895 (
x)
minus (x)1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
2
1198712(119863)
]
]
+
120572
2
1003817
1003817
1003817
1003817
119911
ℎ
1003817
1003817
1003817
1003817
2
1198712(119863)
(54)
where
119896(119911
ℎ) = (
1(119911
ℎ)
119879
119875(119911
ℎ)
119879)
119879
=
(55)
is the stochastic Galerkin solution to the state equation
(
119870
11sdot sdot sdot 119870
1119875
d
119870
1198751sdot sdot sdot 119870
119875119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119870
(
1
119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=
= (
119864 [Ψ
1]119872
119911
ℎ
119864 [Ψ
119875]119872
119911
ℎ
)
= (
120575
11119872
119911
ℎ
120575
1119875119872
119911
ℎ
)
(56)
since 119864[Ψ
119896] = 119864[1Ψ
119896] = 119864[Ψ
1Ψ
119896] = 120575
1119896 The blocks of the
above 119870matrix have the form
(119870
119896119897)
119894119895= int
Ω
120588 (120596)Ψ
119896(120596)Ψ119897 (
120596)
times int
119863
119886 (x 120596) nabla120601
119894 (x) sdot nabla120601
119895 (x) 119889x119889120596
(57)
First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations
minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))
in 119863 times Ω
120583 (x 120596) = 0 on 120597119863 times Ω
(58)
Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865
119879
1 119865
119879
119901)
119879 and 119865
119896is defined
as
(
119865
119896)
119894= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596
= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895120601
119895 (x) Ψ119897 (120596) 120601119894 (
x)
minus (x) 120601119894 (x))119889x119889120596
= minus
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
(x) 120601119894 (x) 119889x
=
119887
119894120575
1119896minus
119875
sum
119897=1
119873
sum
119895=1
119872
119894119895(
119897)
119895120575
119896119897
=
119887
119894120575
1119896minus
119873
sum
119895=1
119872
119894119895(
119897)
119895
(59)
in which
119887
119894= int 120601
119894 and119872
119894119895= int 120601
119894120601
119895
Thus
(
119865
119896)
119894=
119887 minus 119872119906
1if 119896 = 1
minus119872119906 if 119896 = 2 119875
(60)
where b= [
119887
1
119887
119873] and the matrix 119872 = (119872
119894119895) is of order
119873 times 119873
ISRN Applied Mathematics 7
Now the derivative of
119869(119911) in the direction 120601
119895(x) for all
119895 = 1 119873 with the computed adjoint state is
⟨
119869
1015840(119911) 120601119895 (
x)⟩Zlowast Z
= ⟨119890
119911(119906 (119911 sdot sdot) 119911)
lowast120583
+ 119869
119911 (119906 (119911 sdot sdot) 119911) 120601119895 (
x) ⟩Zlowast Z
= minusint
Ω
120588 (120596)int
119863
120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596
+ 120572int
119863
119911 (119909) 120601119895 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894int
Ω
120588 (120596)Ψ119896 (120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ 120572
119873
sum
119894=1
119911
119894int
119863
120601
119895 (x) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894119872
119894119895120575
1119896+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
= minus
119873
sum
119894=1
(
1)
119894119872
119894119895+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
(61)
Therefore the gradient can be computed as follows
119869
1015840(119911) = 119872(120572
119911
ℎminus
1)
(62)
Now for any vector V isin Z it is possible to write V as
V =
119873
sum
119896=1
V119896120601
119896 (x) (63)
In order to compute the Hessian times a vector that ismultiply the Hessian of the
119869 with a some vector V isin Z theequation 119890
119906(119906(119911) 119911)119908 = 119890
119911(119906(119911) 119911)V for 119908 must be solved
Similar to the adjoint computation119908 solves the linear system119870 = 119866 where
119866 = (
119866
119879
1
119866
119879
119875) and
119866
119896for 119896 = 1 119875 is
(
119866
119896)
119894= int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
119873
sum
119895=1
V119895120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895V119895120575
1119896
(64)
for all 119894 = 1 119873 Thus
119866
119896=
119872V if 119896 = 1
0 if 119896 = 2 119875
(65)
Now solving 119890
119906(119906(119911) 119911)119902 = 119869
119906119906(119906(119911) 119911)119908 for 119902 requires
the solution to the linear system 119870 119902 = 119867 where
(
119867
119896)
119894=
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895(
119896)
119895
(66)
or equivalently
119867
119896= 119872
119896for all 119896 = 1 119875 Hence in the
direction 120601
119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by
⟨
119869
10158401015840(119911) V 120601119894⟩Zlowast Z
asymp ⟨
119869
10158401015840
ℎ119901(119911
ℎ) V 120601
119894⟩
Zlowast Z
= minusint
Ω
120588 (120596)int
119863
119875
sum
119896=1
119873
sum
119895=1
( 119902
119896)
119895120601
119895 (x) Ψ119896 (120596)
times 120601
119894 (x) 119889x119889120596
+ 120572int
119863
V (119909) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119895=1
119872
119894119895( 119902
119896)
119895120575
1119896+ 120572
119873
sum
119895=1
119872
119894119895V119895
=
119873
sum
119895=1
119872
119894119895(120572V
119895minus ( 119902
1)
119895)
(67)
Therefore
119869
10158401015840(119911) V asymp 119872(120572V minus 119902
1)
(68)
Having (62) and (68) we can use the iterative solvers tothe Newton equation
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) (69)
Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572
119896and using this step size to generate
the iterate 119911
119896+1= 119911
119896+ 120572
119896119878
119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is
119869 (119911
119896+ 120572
119896119878
119896) le 119869 (119911
119896) + 119888120572
119896119869
1015840(119911
119896) 119878
119896 (70)
where 119888 isin (0 1) and typically is quite small for example 119888 =
10
minus4 [12 13]
8 ISRN Applied Mathematics
(1) Given 119911
0 and tol gt 0 set 119896 = 0
(2) Compute 119870119872 119860 and
119887
(3) Compute (
119865
119896)
119894=
119887 minus 119872119906 if 119896 = 1
minus119872119906 if 119896 = 2 119875
(4) Solve 119870 120583 = 119865
(5) Compute
119869
1015840(119911
119896) = 119872(120572119911
119896minus
1)
(6) Compute
119866
1= 119872V and
119866
119905= 0 for 119905 = 2 119875
(7) Solve 119870 = 119866 = (
119866
119879
1
119866
119879
119875)
119879
(8) Compute
119867
119896= 119872
119896for 119896 = 1 119875
(9) Solve 119870 119902 = 119867 = (
119867
119879
1
119867
119879
119875)
119879
(10) Compute
119869
10158401015840(119911)V asymp 119872(120572V minus 119902
1)
(11) If 1003817100381710038171003817
1003817
nabla
119869 (119911
119896)
1003817
1003817
1003817
1003817
1003817
lt tol stop(12) Solve
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) using PCG method with preconditioner
119875 = 119866⨂119870
0
(i) Set 1199090= 0
(ii) Set 1199030larr
119869
10158401015840(119911
119896) 119909
0+
119869
1015840(119911
119896)
(iii) Solve
119875119910
0= 119903
0for 119910
0
(iv) Set 1199010= minus119903
0 119905 larr 0
(v) While 1003817
1003817
1003817
1003817
119903
119905
1003817
1003817
1003817
1003817
gt tol
120572
119896larr
119903
119879
119905119910
119905
119901
119879
119905119860119901
119905
119909
119905+1larr 119909
119905+ 120572
119905119901
119905
119903
119905+1larr 119903
119905+ 120572
119905119860119901
119905
119875119910
119905+1larr 119903
119905+1
120573
119905+1larr
119903
119879
119905+1119910
119905+1
119903
119879
119905119910
119905
119901
119905+1larr minus119910
119905+1+ 120573
119905+1119901
119905
119905 larr 119905 + 1
End (while)(vi) 119904119896 = 119909
119905+1
(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911
119896+ 120572
119896119878
119896)
(ii) While 119869 (119911
119896+ 120572
119896119878
119896) gt 119869 (119911
119896) + 10
minus4120572
119896119878
119896nabla
119869 (119911
119896) do
(i) Set 120572119896 = 120572
1198962 and evaluate 119869 (119911
119896+ 120572
119896119878
119896)
(14) Set 119911119896+1 = 119911
119896+ 120572
119896119878
119896 119896 larr 119896 + 1 Go to (2)
Algorithm 1
31 Kronecker Product Preconditioners Let us to rewrite (37)in the form
119860 =
119911 where
119860 = (
119872
sum
119896=0
119866
119896⨂119870
119896) (71)
The strategy here is to introduce
119875 = 119866⨂119870
0as a
preconditioner such that
119866 = argmin 119867 isin R119875times119875
1003817
1003817
1003817
1003817
1003817
119860 minus 119867⨂119870
0
1003817
1003817
1003817
1003817
1003817119865 (72)
where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot
119865denotes the Frobe-
nius norm The closed form of the solution can be written asfollow [14]
119866 = 119868 + sum
119875
tr (119870119879
119896119870
0)
tr (119870119879
0119870
0)
119866
119896
(73)
Here tr(119870119879
119896119870
0) = sum
119873119902
119894=1[119870
119896]
119894119894[119870
0]
119894119894and hence the coeffi-
cients in the above equality can be computed straightforwardIn addition since
119860 and 119870
0are symmetric and positive
definite thus 119866 and
119875 = 119866⨂119870
0have also these properties
[15 16]
Example 1 Consider the following distributed optimal con-trol problem
min119906119911
119869 (119906 119911) = 119864 [
1
2
119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω
(74)
where 119863 = [minus1 1]
2 Ω sim 119880[minus1 1] the boundary conditionsare given as
119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0
(75)
ISRN Applied Mathematics 9
005
1
005
10
05
1y d
xy
(a) Desired values
005
1
005
10
05
1
uM
ean
xy
times10minus3
(b) Initial state
005
1
005
10
05
1
u opt
xy
(c) Approximated solution (state)
005
1
005
10
200
400
z opt
xy
(d) Control
005
1
0
05
10
05
1
times 10minus3
xy
120583
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
05
1
15
2
25
times 10minus3
(f) Expected value
20 40
10
20
30
40
50
05
1
15
2
25
times 10minus3
(g) Standard dev
0 5 10times 104
0
5
10
times 104 0
(h) Sparsity of global matrix119899119911 = 1815980
Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))
2+ (119910 minus (12))
2)]
119873 = 3 119870 = 5 and 120572 = 000005
and the desired function is given by
(119909 119910) = exp [minus64 ((119909 minus
1
2
)
2
+ (119910 minus
1
2
)
2
)] (76)
The random field 119886(x 120596) is characterized by its mean andcovariance function
119864 [119886] = 119886 = 10
119877 (119909
1 119909
2) = 119890
minus|1199091minus1199092| 119909
1 119909
2isin [minus1 1]
(77)
Following Section 22 the truncated KLE can be representedas
119886 (119909 119910 120596) = 10 +
119873
sum
119895=0
radic120582
119895120579
119895120601
119895 (119909) (78)
The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867
1
0(119863)⨂119871
2(Ω) and all 119911isin 119871
2(119863)
119869 (119906opt 119911opt) le 119869 (119906 119911) (79)
The eigenpairs 120582119895 120579
119895 in truncated KLE solve the integral
equation
int
1
minus1
119890
minus|1199091minus1199092|120601
119895(119909
2) 119889119909
2= 120582
119895120601
119895(119909
1)
(80)
For this special case of the covariance function we haveexplicit expressions for 120601
119895and 120582
119895 [6] Let120596
119895evenand120596
119895oddsolve
the equations
1 minus 120596
119895eventan (120596
119895even) = 0
120596
119895odd+ tan (120596
119895odd) = 0
(81)
Then the even and odd indexed eigenfunctions are given by
120601
119895even(119909) =
cos (120596119895even
119909)
radic1 + (sin (2120596
119895even) 2120596
119895even)
120601
119895odd(119909)
=
sin (120596
119895odd119909)
radic1 minus (sin (2120596
119895odd) 2120596
119895odd)
(82)
10 ISRN Applied Mathematics
005
1
005
10
05
1y ha
nd
xy
(a) Desired values
005
1
005
1minus05
05
0
u Mea
n
xy
(b) Mean values
005
1
005
1
0
Num
sol
xy
0
1
(c) Approximated solution (State)
005
1
005
10
05
1
xy
(d) Control
005
1
005
10
05
1
Lend
a
xy
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
0
005
01
015
02
(f) Expected value-SGS
20 40
10
20
30
40
50
0
005
01
015
02
(g) Standard dev-SGS
0 2 4 6 8times 104
0
2
4
6
8
times 104
(h) Sparsity of global matrix119899119911 = 3046160
Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001
and corresponding eigenvalues
120582
119895even(119909) =
2
120596
2
119895even+ 1
120582
119895odd(119909) =
2
120596
2
119895odd+ 1
(83)
We choose 120579 = (120579
1 120579
119873)
119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]
First of all we find numerical approximations of120596119895even
and120596
119895oddwith bisection method and then with the eigenpairs
evaluated by this 120596119895even
and 120596
119895odd by choosing dimension 119873 =
3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1
In Figure 1(a) the desired function (119909 119910) is plottedfor 119909
119894= 119910
119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial
state is computed and plotted in Figure 1(b) by solving (37)where
119911 = 0 Approximated 119906opt corresponding to the
optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial
119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911
119900119901119905must approximate the
initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane
In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use
Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary
ISRN Applied Mathematics 11
conditions is replaced by (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2
4 Conclusion
The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996
[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987
[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005
[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947
[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946
[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991
[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007
[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009
[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938
[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993
[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008
[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999
[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006
[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010
[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010
[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007
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Stochastic AnalysisInternational Journal of
6 ISRN Applied Mathematics
By the chain rule the derivative of 119869 with respect to 119906 inthe direction 120575119906 is
⟨119869
119906 (119906 119911) 120575119906⟩
119880lowast119880
= 119864 [119906 (x 120596) minus (x)21198712(119863)
times ⟨119906 (x 120596)minus (x) 120575119906 (x 120596)⟩1198712(119863)
]
(51)
Similarly the derivative of 119869 with respect to 119911 in the 120575119911 isin
Z direction is
⟨119869
119911 (119906 119911) 120575119911⟩
Zlowast Z= 120572⟨119911 120575119911⟩Z (52)
With these computations one can compute the firstderivative of
119869(119911) via the adjoint approach The aim is toderive the discrete versions of the objective function gradi-ent and Hessian times a vector calculation corresponding tothe stochastic Galerkin solution technique for SPDE that is119890(119906 119911) = 0 Suppose 119883
ℎsub 119867
1
0(119863) is a finite dimensional
subspace of dimension 119873 and 119884
ℎsub 119871
2(Ω) is a finite
dimensional subspace of dimension 119875 then 119883
ℎ⨂119884
ℎis a
finite dimensional subspace of the state space 119880 Similarlylet Z
ℎsub Z be a finite dimensional subspace of the control
space (with dimension) For the stochastic Galerkin methodit is assumed that 119884
ℎ= P119875minus1 the space of polynomials with
highest degree 119875minus1 and119883
ℎis any finite element space here
119883
ℎis the space of linear functions built on a given meshT
ℎ
We choose the system of polynomials that are 120588-orthonormalto be a basis for 119884
ℎ that is 119884
ℎ= spanΨ
1(120596) Ψ
119875(120596)
where
int
Ω
120588 (120596)Ψ
119899(120596)Ψ119898 (120596) 119889120596 = 120575
119898119899 (53)
The discretized optimization problem in the stochasticGalerkin framework is
min119911ℎisinZℎ
119869
ℎ119875(119911
ℎ) =
1
2
119864
[
[
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
119875
sum
119896=1
119873
sum
119895=1
(
119896(119911
ℎ))
119895Ψ
119896 (120596) 120601119895 (
x)
minus (x)1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
1003817
2
1198712(119863)
]
]
+
120572
2
1003817
1003817
1003817
1003817
119911
ℎ
1003817
1003817
1003817
1003817
2
1198712(119863)
(54)
where
119896(119911
ℎ) = (
1(119911
ℎ)
119879
119875(119911
ℎ)
119879)
119879
=
(55)
is the stochastic Galerkin solution to the state equation
(
119870
11sdot sdot sdot 119870
1119875
d
119870
1198751sdot sdot sdot 119870
119875119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=119870
(
1
119875
)
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=
= (
119864 [Ψ
1]119872
119911
ℎ
119864 [Ψ
119875]119872
119911
ℎ
)
= (
120575
11119872
119911
ℎ
120575
1119875119872
119911
ℎ
)
(56)
since 119864[Ψ
119896] = 119864[1Ψ
119896] = 119864[Ψ
1Ψ
119896] = 120575
1119896 The blocks of the
above 119870matrix have the form
(119870
119896119897)
119894119895= int
Ω
120588 (120596)Ψ
119896(120596)Ψ119897 (
120596)
times int
119863
119886 (x 120596) nabla120601
119894 (x) sdot nabla120601
119895 (x) 119889x119889120596
(57)
First in order to compute the derivatives of the objectivefunction we attempt to compute the adjoint state [8] Indeedthe adjoint state in the infinite dimensional formulationsolves the following equations
minus nabla sdot (119886 (x 120596) nabla120583 (x 120596)) = minus (119906 (x 120596) minus (x))
in 119863 times Ω
120583 (x 120596) = 0 on 120597119863 times Ω
(58)
Thus as in (37) the block system for (58) can bewritten inthe form 119870 = 119865 where 119865 = (119865
119879
1 119865
119879
119901)
119879 and 119865
119896is defined
as
(
119865
119896)
119894= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(119906 (x 120596) minus (x)) 120601119894 (x) 119889x119889120596
= minusint
Ω
120588 (120596)Ψ119896 (120596) int
119863
(
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895120601
119895 (x) Ψ119897 (120596) 120601119894 (
x)
minus (x) 120601119894 (x))119889x119889120596
= minus
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
(x) 120601119894 (x) 119889x
=
119887
119894120575
1119896minus
119875
sum
119897=1
119873
sum
119895=1
119872
119894119895(
119897)
119895120575
119896119897
=
119887
119894120575
1119896minus
119873
sum
119895=1
119872
119894119895(
119897)
119895
(59)
in which
119887
119894= int 120601
119894 and119872
119894119895= int 120601
119894120601
119895
Thus
(
119865
119896)
119894=
119887 minus 119872119906
1if 119896 = 1
minus119872119906 if 119896 = 2 119875
(60)
where b= [
119887
1
119887
119873] and the matrix 119872 = (119872
119894119895) is of order
119873 times 119873
ISRN Applied Mathematics 7
Now the derivative of
119869(119911) in the direction 120601
119895(x) for all
119895 = 1 119873 with the computed adjoint state is
⟨
119869
1015840(119911) 120601119895 (
x)⟩Zlowast Z
= ⟨119890
119911(119906 (119911 sdot sdot) 119911)
lowast120583
+ 119869
119911 (119906 (119911 sdot sdot) 119911) 120601119895 (
x) ⟩Zlowast Z
= minusint
Ω
120588 (120596)int
119863
120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596
+ 120572int
119863
119911 (119909) 120601119895 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894int
Ω
120588 (120596)Ψ119896 (120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ 120572
119873
sum
119894=1
119911
119894int
119863
120601
119895 (x) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894119872
119894119895120575
1119896+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
= minus
119873
sum
119894=1
(
1)
119894119872
119894119895+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
(61)
Therefore the gradient can be computed as follows
119869
1015840(119911) = 119872(120572
119911
ℎminus
1)
(62)
Now for any vector V isin Z it is possible to write V as
V =
119873
sum
119896=1
V119896120601
119896 (x) (63)
In order to compute the Hessian times a vector that ismultiply the Hessian of the
119869 with a some vector V isin Z theequation 119890
119906(119906(119911) 119911)119908 = 119890
119911(119906(119911) 119911)V for 119908 must be solved
Similar to the adjoint computation119908 solves the linear system119870 = 119866 where
119866 = (
119866
119879
1
119866
119879
119875) and
119866
119896for 119896 = 1 119875 is
(
119866
119896)
119894= int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
119873
sum
119895=1
V119895120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895V119895120575
1119896
(64)
for all 119894 = 1 119873 Thus
119866
119896=
119872V if 119896 = 1
0 if 119896 = 2 119875
(65)
Now solving 119890
119906(119906(119911) 119911)119902 = 119869
119906119906(119906(119911) 119911)119908 for 119902 requires
the solution to the linear system 119870 119902 = 119867 where
(
119867
119896)
119894=
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895(
119896)
119895
(66)
or equivalently
119867
119896= 119872
119896for all 119896 = 1 119875 Hence in the
direction 120601
119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by
⟨
119869
10158401015840(119911) V 120601119894⟩Zlowast Z
asymp ⟨
119869
10158401015840
ℎ119901(119911
ℎ) V 120601
119894⟩
Zlowast Z
= minusint
Ω
120588 (120596)int
119863
119875
sum
119896=1
119873
sum
119895=1
( 119902
119896)
119895120601
119895 (x) Ψ119896 (120596)
times 120601
119894 (x) 119889x119889120596
+ 120572int
119863
V (119909) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119895=1
119872
119894119895( 119902
119896)
119895120575
1119896+ 120572
119873
sum
119895=1
119872
119894119895V119895
=
119873
sum
119895=1
119872
119894119895(120572V
119895minus ( 119902
1)
119895)
(67)
Therefore
119869
10158401015840(119911) V asymp 119872(120572V minus 119902
1)
(68)
Having (62) and (68) we can use the iterative solvers tothe Newton equation
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) (69)
Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572
119896and using this step size to generate
the iterate 119911
119896+1= 119911
119896+ 120572
119896119878
119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is
119869 (119911
119896+ 120572
119896119878
119896) le 119869 (119911
119896) + 119888120572
119896119869
1015840(119911
119896) 119878
119896 (70)
where 119888 isin (0 1) and typically is quite small for example 119888 =
10
minus4 [12 13]
8 ISRN Applied Mathematics
(1) Given 119911
0 and tol gt 0 set 119896 = 0
(2) Compute 119870119872 119860 and
119887
(3) Compute (
119865
119896)
119894=
119887 minus 119872119906 if 119896 = 1
minus119872119906 if 119896 = 2 119875
(4) Solve 119870 120583 = 119865
(5) Compute
119869
1015840(119911
119896) = 119872(120572119911
119896minus
1)
(6) Compute
119866
1= 119872V and
119866
119905= 0 for 119905 = 2 119875
(7) Solve 119870 = 119866 = (
119866
119879
1
119866
119879
119875)
119879
(8) Compute
119867
119896= 119872
119896for 119896 = 1 119875
(9) Solve 119870 119902 = 119867 = (
119867
119879
1
119867
119879
119875)
119879
(10) Compute
119869
10158401015840(119911)V asymp 119872(120572V minus 119902
1)
(11) If 1003817100381710038171003817
1003817
nabla
119869 (119911
119896)
1003817
1003817
1003817
1003817
1003817
lt tol stop(12) Solve
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) using PCG method with preconditioner
119875 = 119866⨂119870
0
(i) Set 1199090= 0
(ii) Set 1199030larr
119869
10158401015840(119911
119896) 119909
0+
119869
1015840(119911
119896)
(iii) Solve
119875119910
0= 119903
0for 119910
0
(iv) Set 1199010= minus119903
0 119905 larr 0
(v) While 1003817
1003817
1003817
1003817
119903
119905
1003817
1003817
1003817
1003817
gt tol
120572
119896larr
119903
119879
119905119910
119905
119901
119879
119905119860119901
119905
119909
119905+1larr 119909
119905+ 120572
119905119901
119905
119903
119905+1larr 119903
119905+ 120572
119905119860119901
119905
119875119910
119905+1larr 119903
119905+1
120573
119905+1larr
119903
119879
119905+1119910
119905+1
119903
119879
119905119910
119905
119901
119905+1larr minus119910
119905+1+ 120573
119905+1119901
119905
119905 larr 119905 + 1
End (while)(vi) 119904119896 = 119909
119905+1
(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911
119896+ 120572
119896119878
119896)
(ii) While 119869 (119911
119896+ 120572
119896119878
119896) gt 119869 (119911
119896) + 10
minus4120572
119896119878
119896nabla
119869 (119911
119896) do
(i) Set 120572119896 = 120572
1198962 and evaluate 119869 (119911
119896+ 120572
119896119878
119896)
(14) Set 119911119896+1 = 119911
119896+ 120572
119896119878
119896 119896 larr 119896 + 1 Go to (2)
Algorithm 1
31 Kronecker Product Preconditioners Let us to rewrite (37)in the form
119860 =
119911 where
119860 = (
119872
sum
119896=0
119866
119896⨂119870
119896) (71)
The strategy here is to introduce
119875 = 119866⨂119870
0as a
preconditioner such that
119866 = argmin 119867 isin R119875times119875
1003817
1003817
1003817
1003817
1003817
119860 minus 119867⨂119870
0
1003817
1003817
1003817
1003817
1003817119865 (72)
where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot
119865denotes the Frobe-
nius norm The closed form of the solution can be written asfollow [14]
119866 = 119868 + sum
119875
tr (119870119879
119896119870
0)
tr (119870119879
0119870
0)
119866
119896
(73)
Here tr(119870119879
119896119870
0) = sum
119873119902
119894=1[119870
119896]
119894119894[119870
0]
119894119894and hence the coeffi-
cients in the above equality can be computed straightforwardIn addition since
119860 and 119870
0are symmetric and positive
definite thus 119866 and
119875 = 119866⨂119870
0have also these properties
[15 16]
Example 1 Consider the following distributed optimal con-trol problem
min119906119911
119869 (119906 119911) = 119864 [
1
2
119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω
(74)
where 119863 = [minus1 1]
2 Ω sim 119880[minus1 1] the boundary conditionsare given as
119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0
(75)
ISRN Applied Mathematics 9
005
1
005
10
05
1y d
xy
(a) Desired values
005
1
005
10
05
1
uM
ean
xy
times10minus3
(b) Initial state
005
1
005
10
05
1
u opt
xy
(c) Approximated solution (state)
005
1
005
10
200
400
z opt
xy
(d) Control
005
1
0
05
10
05
1
times 10minus3
xy
120583
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
05
1
15
2
25
times 10minus3
(f) Expected value
20 40
10
20
30
40
50
05
1
15
2
25
times 10minus3
(g) Standard dev
0 5 10times 104
0
5
10
times 104 0
(h) Sparsity of global matrix119899119911 = 1815980
Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))
2+ (119910 minus (12))
2)]
119873 = 3 119870 = 5 and 120572 = 000005
and the desired function is given by
(119909 119910) = exp [minus64 ((119909 minus
1
2
)
2
+ (119910 minus
1
2
)
2
)] (76)
The random field 119886(x 120596) is characterized by its mean andcovariance function
119864 [119886] = 119886 = 10
119877 (119909
1 119909
2) = 119890
minus|1199091minus1199092| 119909
1 119909
2isin [minus1 1]
(77)
Following Section 22 the truncated KLE can be representedas
119886 (119909 119910 120596) = 10 +
119873
sum
119895=0
radic120582
119895120579
119895120601
119895 (119909) (78)
The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867
1
0(119863)⨂119871
2(Ω) and all 119911isin 119871
2(119863)
119869 (119906opt 119911opt) le 119869 (119906 119911) (79)
The eigenpairs 120582119895 120579
119895 in truncated KLE solve the integral
equation
int
1
minus1
119890
minus|1199091minus1199092|120601
119895(119909
2) 119889119909
2= 120582
119895120601
119895(119909
1)
(80)
For this special case of the covariance function we haveexplicit expressions for 120601
119895and 120582
119895 [6] Let120596
119895evenand120596
119895oddsolve
the equations
1 minus 120596
119895eventan (120596
119895even) = 0
120596
119895odd+ tan (120596
119895odd) = 0
(81)
Then the even and odd indexed eigenfunctions are given by
120601
119895even(119909) =
cos (120596119895even
119909)
radic1 + (sin (2120596
119895even) 2120596
119895even)
120601
119895odd(119909)
=
sin (120596
119895odd119909)
radic1 minus (sin (2120596
119895odd) 2120596
119895odd)
(82)
10 ISRN Applied Mathematics
005
1
005
10
05
1y ha
nd
xy
(a) Desired values
005
1
005
1minus05
05
0
u Mea
n
xy
(b) Mean values
005
1
005
1
0
Num
sol
xy
0
1
(c) Approximated solution (State)
005
1
005
10
05
1
xy
(d) Control
005
1
005
10
05
1
Lend
a
xy
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
0
005
01
015
02
(f) Expected value-SGS
20 40
10
20
30
40
50
0
005
01
015
02
(g) Standard dev-SGS
0 2 4 6 8times 104
0
2
4
6
8
times 104
(h) Sparsity of global matrix119899119911 = 3046160
Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001
and corresponding eigenvalues
120582
119895even(119909) =
2
120596
2
119895even+ 1
120582
119895odd(119909) =
2
120596
2
119895odd+ 1
(83)
We choose 120579 = (120579
1 120579
119873)
119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]
First of all we find numerical approximations of120596119895even
and120596
119895oddwith bisection method and then with the eigenpairs
evaluated by this 120596119895even
and 120596
119895odd by choosing dimension 119873 =
3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1
In Figure 1(a) the desired function (119909 119910) is plottedfor 119909
119894= 119910
119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial
state is computed and plotted in Figure 1(b) by solving (37)where
119911 = 0 Approximated 119906opt corresponding to the
optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial
119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911
119900119901119905must approximate the
initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane
In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use
Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary
ISRN Applied Mathematics 11
conditions is replaced by (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2
4 Conclusion
The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996
[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987
[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005
[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947
[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946
[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991
[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007
[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009
[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938
[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993
[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008
[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999
[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006
[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010
[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010
[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007
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Stochastic AnalysisInternational Journal of
ISRN Applied Mathematics 7
Now the derivative of
119869(119911) in the direction 120601
119895(x) for all
119895 = 1 119873 with the computed adjoint state is
⟨
119869
1015840(119911) 120601119895 (
x)⟩Zlowast Z
= ⟨119890
119911(119906 (119911 sdot sdot) 119911)
lowast120583
+ 119869
119911 (119906 (119911 sdot sdot) 119911) 120601119895 (
x) ⟩Zlowast Z
= minusint
Ω
120588 (120596)int
119863
120583 (119911 sdot sdot) 120601119895 (x) 119889x 119889120596
+ 120572int
119863
119911 (119909) 120601119895 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894int
Ω
120588 (120596)Ψ119896 (120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
+ 120572
119873
sum
119894=1
119911
119894int
119863
120601
119895 (x) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119894=1
(
119896)
119894119872
119894119895120575
1119896+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
= minus
119873
sum
119894=1
(
1)
119894119872
119894119895+ 120572
119873
sum
119894=1
119872
119894119895
119911
119894
(61)
Therefore the gradient can be computed as follows
119869
1015840(119911) = 119872(120572
119911
ℎminus
1)
(62)
Now for any vector V isin Z it is possible to write V as
V =
119873
sum
119896=1
V119896120601
119896 (x) (63)
In order to compute the Hessian times a vector that ismultiply the Hessian of the
119869 with a some vector V isin Z theequation 119890
119906(119906(119911) 119911)119908 = 119890
119911(119906(119911) 119911)V for 119908 must be solved
Similar to the adjoint computation119908 solves the linear system119870 = 119866 where
119866 = (
119866
119879
1
119866
119879
119875) and
119866
119896for 119896 = 1 119875 is
(
119866
119896)
119894= int
Ω
120588 (120596)Ψ119896 (120596) 119889120596int
119863
119873
sum
119895=1
V119895120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895V119895120575
1119896
(64)
for all 119894 = 1 119873 Thus
119866
119896=
119872V if 119896 = 1
0 if 119896 = 2 119875
(65)
Now solving 119890
119906(119906(119911) 119911)119902 = 119869
119906119906(119906(119911) 119911)119908 for 119902 requires
the solution to the linear system 119870 119902 = 119867 where
(
119867
119896)
119894=
119875
sum
119897=1
119873
sum
119895=1
(
119897)
119895int
Ω
120588 (120596)Ψ119896 (120596)Ψ119897 (
120596) 119889120596
times int
119863
120601
119895 (x) 120601119894 (x) 119889x
=
119873
sum
119895=1
119872
119894119895(
119896)
119895
(66)
or equivalently
119867
119896= 119872
119896for all 119896 = 1 119875 Hence in the
direction 120601
119894for 119894 = 1 119873 11986910158401015840(119911)V can be approximated by
⟨
119869
10158401015840(119911) V 120601119894⟩Zlowast Z
asymp ⟨
119869
10158401015840
ℎ119901(119911
ℎ) V 120601
119894⟩
Zlowast Z
= minusint
Ω
120588 (120596)int
119863
119875
sum
119896=1
119873
sum
119895=1
( 119902
119896)
119895120601
119895 (x) Ψ119896 (120596)
times 120601
119894 (x) 119889x119889120596
+ 120572int
119863
V (119909) 120601119894 (x) 119889x
= minus
119875
sum
119896=1
119873
sum
119895=1
119872
119894119895( 119902
119896)
119895120575
1119896+ 120572
119873
sum
119895=1
119872
119894119895V119895
=
119873
sum
119895=1
119872
119894119895(120572V
119895minus ( 119902
1)
119895)
(67)
Therefore
119869
10158401015840(119911) V asymp 119872(120572V minus 119902
1)
(68)
Having (62) and (68) we can use the iterative solvers tothe Newton equation
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) (69)
Now by using the preconditioned conjugate gradient(PCG) method the Newton equation (69) is solved approx-imately When we access to the sufficiently small residual ofNewton system the PCG method is truncated In real worldcomputation it is possible to employ some globalizationtechnique for Newtonrsquos method Considering the case ofimplementation and relatively low computational cost linesearch techniques are popular choices in this way Findingan optimal step size 120572
119896and using this step size to generate
the iterate 119911
119896+1= 119911
119896+ 120572
119896119878
119896 are the mission of a line searchalgorithm The Armijo condition (or sufficient decreasecondition) that the step size is required to satisfy is
119869 (119911
119896+ 120572
119896119878
119896) le 119869 (119911
119896) + 119888120572
119896119869
1015840(119911
119896) 119878
119896 (70)
where 119888 isin (0 1) and typically is quite small for example 119888 =
10
minus4 [12 13]
8 ISRN Applied Mathematics
(1) Given 119911
0 and tol gt 0 set 119896 = 0
(2) Compute 119870119872 119860 and
119887
(3) Compute (
119865
119896)
119894=
119887 minus 119872119906 if 119896 = 1
minus119872119906 if 119896 = 2 119875
(4) Solve 119870 120583 = 119865
(5) Compute
119869
1015840(119911
119896) = 119872(120572119911
119896minus
1)
(6) Compute
119866
1= 119872V and
119866
119905= 0 for 119905 = 2 119875
(7) Solve 119870 = 119866 = (
119866
119879
1
119866
119879
119875)
119879
(8) Compute
119867
119896= 119872
119896for 119896 = 1 119875
(9) Solve 119870 119902 = 119867 = (
119867
119879
1
119867
119879
119875)
119879
(10) Compute
119869
10158401015840(119911)V asymp 119872(120572V minus 119902
1)
(11) If 1003817100381710038171003817
1003817
nabla
119869 (119911
119896)
1003817
1003817
1003817
1003817
1003817
lt tol stop(12) Solve
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) using PCG method with preconditioner
119875 = 119866⨂119870
0
(i) Set 1199090= 0
(ii) Set 1199030larr
119869
10158401015840(119911
119896) 119909
0+
119869
1015840(119911
119896)
(iii) Solve
119875119910
0= 119903
0for 119910
0
(iv) Set 1199010= minus119903
0 119905 larr 0
(v) While 1003817
1003817
1003817
1003817
119903
119905
1003817
1003817
1003817
1003817
gt tol
120572
119896larr
119903
119879
119905119910
119905
119901
119879
119905119860119901
119905
119909
119905+1larr 119909
119905+ 120572
119905119901
119905
119903
119905+1larr 119903
119905+ 120572
119905119860119901
119905
119875119910
119905+1larr 119903
119905+1
120573
119905+1larr
119903
119879
119905+1119910
119905+1
119903
119879
119905119910
119905
119901
119905+1larr minus119910
119905+1+ 120573
119905+1119901
119905
119905 larr 119905 + 1
End (while)(vi) 119904119896 = 119909
119905+1
(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911
119896+ 120572
119896119878
119896)
(ii) While 119869 (119911
119896+ 120572
119896119878
119896) gt 119869 (119911
119896) + 10
minus4120572
119896119878
119896nabla
119869 (119911
119896) do
(i) Set 120572119896 = 120572
1198962 and evaluate 119869 (119911
119896+ 120572
119896119878
119896)
(14) Set 119911119896+1 = 119911
119896+ 120572
119896119878
119896 119896 larr 119896 + 1 Go to (2)
Algorithm 1
31 Kronecker Product Preconditioners Let us to rewrite (37)in the form
119860 =
119911 where
119860 = (
119872
sum
119896=0
119866
119896⨂119870
119896) (71)
The strategy here is to introduce
119875 = 119866⨂119870
0as a
preconditioner such that
119866 = argmin 119867 isin R119875times119875
1003817
1003817
1003817
1003817
1003817
119860 minus 119867⨂119870
0
1003817
1003817
1003817
1003817
1003817119865 (72)
where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot
119865denotes the Frobe-
nius norm The closed form of the solution can be written asfollow [14]
119866 = 119868 + sum
119875
tr (119870119879
119896119870
0)
tr (119870119879
0119870
0)
119866
119896
(73)
Here tr(119870119879
119896119870
0) = sum
119873119902
119894=1[119870
119896]
119894119894[119870
0]
119894119894and hence the coeffi-
cients in the above equality can be computed straightforwardIn addition since
119860 and 119870
0are symmetric and positive
definite thus 119866 and
119875 = 119866⨂119870
0have also these properties
[15 16]
Example 1 Consider the following distributed optimal con-trol problem
min119906119911
119869 (119906 119911) = 119864 [
1
2
119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω
(74)
where 119863 = [minus1 1]
2 Ω sim 119880[minus1 1] the boundary conditionsare given as
119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0
(75)
ISRN Applied Mathematics 9
005
1
005
10
05
1y d
xy
(a) Desired values
005
1
005
10
05
1
uM
ean
xy
times10minus3
(b) Initial state
005
1
005
10
05
1
u opt
xy
(c) Approximated solution (state)
005
1
005
10
200
400
z opt
xy
(d) Control
005
1
0
05
10
05
1
times 10minus3
xy
120583
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
05
1
15
2
25
times 10minus3
(f) Expected value
20 40
10
20
30
40
50
05
1
15
2
25
times 10minus3
(g) Standard dev
0 5 10times 104
0
5
10
times 104 0
(h) Sparsity of global matrix119899119911 = 1815980
Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))
2+ (119910 minus (12))
2)]
119873 = 3 119870 = 5 and 120572 = 000005
and the desired function is given by
(119909 119910) = exp [minus64 ((119909 minus
1
2
)
2
+ (119910 minus
1
2
)
2
)] (76)
The random field 119886(x 120596) is characterized by its mean andcovariance function
119864 [119886] = 119886 = 10
119877 (119909
1 119909
2) = 119890
minus|1199091minus1199092| 119909
1 119909
2isin [minus1 1]
(77)
Following Section 22 the truncated KLE can be representedas
119886 (119909 119910 120596) = 10 +
119873
sum
119895=0
radic120582
119895120579
119895120601
119895 (119909) (78)
The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867
1
0(119863)⨂119871
2(Ω) and all 119911isin 119871
2(119863)
119869 (119906opt 119911opt) le 119869 (119906 119911) (79)
The eigenpairs 120582119895 120579
119895 in truncated KLE solve the integral
equation
int
1
minus1
119890
minus|1199091minus1199092|120601
119895(119909
2) 119889119909
2= 120582
119895120601
119895(119909
1)
(80)
For this special case of the covariance function we haveexplicit expressions for 120601
119895and 120582
119895 [6] Let120596
119895evenand120596
119895oddsolve
the equations
1 minus 120596
119895eventan (120596
119895even) = 0
120596
119895odd+ tan (120596
119895odd) = 0
(81)
Then the even and odd indexed eigenfunctions are given by
120601
119895even(119909) =
cos (120596119895even
119909)
radic1 + (sin (2120596
119895even) 2120596
119895even)
120601
119895odd(119909)
=
sin (120596
119895odd119909)
radic1 minus (sin (2120596
119895odd) 2120596
119895odd)
(82)
10 ISRN Applied Mathematics
005
1
005
10
05
1y ha
nd
xy
(a) Desired values
005
1
005
1minus05
05
0
u Mea
n
xy
(b) Mean values
005
1
005
1
0
Num
sol
xy
0
1
(c) Approximated solution (State)
005
1
005
10
05
1
xy
(d) Control
005
1
005
10
05
1
Lend
a
xy
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
0
005
01
015
02
(f) Expected value-SGS
20 40
10
20
30
40
50
0
005
01
015
02
(g) Standard dev-SGS
0 2 4 6 8times 104
0
2
4
6
8
times 104
(h) Sparsity of global matrix119899119911 = 3046160
Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001
and corresponding eigenvalues
120582
119895even(119909) =
2
120596
2
119895even+ 1
120582
119895odd(119909) =
2
120596
2
119895odd+ 1
(83)
We choose 120579 = (120579
1 120579
119873)
119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]
First of all we find numerical approximations of120596119895even
and120596
119895oddwith bisection method and then with the eigenpairs
evaluated by this 120596119895even
and 120596
119895odd by choosing dimension 119873 =
3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1
In Figure 1(a) the desired function (119909 119910) is plottedfor 119909
119894= 119910
119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial
state is computed and plotted in Figure 1(b) by solving (37)where
119911 = 0 Approximated 119906opt corresponding to the
optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial
119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911
119900119901119905must approximate the
initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane
In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use
Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary
ISRN Applied Mathematics 11
conditions is replaced by (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2
4 Conclusion
The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996
[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987
[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005
[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947
[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946
[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991
[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007
[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009
[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938
[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993
[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008
[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999
[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006
[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010
[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010
[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007
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Stochastic AnalysisInternational Journal of
8 ISRN Applied Mathematics
(1) Given 119911
0 and tol gt 0 set 119896 = 0
(2) Compute 119870119872 119860 and
119887
(3) Compute (
119865
119896)
119894=
119887 minus 119872119906 if 119896 = 1
minus119872119906 if 119896 = 2 119875
(4) Solve 119870 120583 = 119865
(5) Compute
119869
1015840(119911
119896) = 119872(120572119911
119896minus
1)
(6) Compute
119866
1= 119872V and
119866
119905= 0 for 119905 = 2 119875
(7) Solve 119870 = 119866 = (
119866
119879
1
119866
119879
119875)
119879
(8) Compute
119867
119896= 119872
119896for 119896 = 1 119875
(9) Solve 119870 119902 = 119867 = (
119867
119879
1
119867
119879
119875)
119879
(10) Compute
119869
10158401015840(119911)V asymp 119872(120572V minus 119902
1)
(11) If 1003817100381710038171003817
1003817
nabla
119869 (119911
119896)
1003817
1003817
1003817
1003817
1003817
lt tol stop(12) Solve
119869
10158401015840(119911
119896) 119904
119896= minus
119869
1015840(119911
119896) using PCG method with preconditioner
119875 = 119866⨂119870
0
(i) Set 1199090= 0
(ii) Set 1199030larr
119869
10158401015840(119911
119896) 119909
0+
119869
1015840(119911
119896)
(iii) Solve
119875119910
0= 119903
0for 119910
0
(iv) Set 1199010= minus119903
0 119905 larr 0
(v) While 1003817
1003817
1003817
1003817
119903
119905
1003817
1003817
1003817
1003817
gt tol
120572
119896larr
119903
119879
119905119910
119905
119901
119879
119905119860119901
119905
119909
119905+1larr 119909
119905+ 120572
119905119901
119905
119903
119905+1larr 119903
119905+ 120572
119905119860119901
119905
119875119910
119905+1larr 119903
119905+1
120573
119905+1larr
119903
119879
119905+1119910
119905+1
119903
119879
119905119910
119905
119901
119905+1larr minus119910
119905+1+ 120573
119905+1119901
119905
119905 larr 119905 + 1
End (while)(vi) 119904119896 = 119909
119905+1
(13) Perform Armijo line-search(i) Set 120572119896 = 1 and evaluate 119869(119911
119896+ 120572
119896119878
119896)
(ii) While 119869 (119911
119896+ 120572
119896119878
119896) gt 119869 (119911
119896) + 10
minus4120572
119896119878
119896nabla
119869 (119911
119896) do
(i) Set 120572119896 = 120572
1198962 and evaluate 119869 (119911
119896+ 120572
119896119878
119896)
(14) Set 119911119896+1 = 119911
119896+ 120572
119896119878
119896 119896 larr 119896 + 1 Go to (2)
Algorithm 1
31 Kronecker Product Preconditioners Let us to rewrite (37)in the form
119860 =
119911 where
119860 = (
119872
sum
119896=0
119866
119896⨂119870
119896) (71)
The strategy here is to introduce
119875 = 119866⨂119870
0as a
preconditioner such that
119866 = argmin 119867 isin R119875times119875
1003817
1003817
1003817
1003817
1003817
119860 minus 119867⨂119870
0
1003817
1003817
1003817
1003817
1003817119865 (72)
where as mentioned in PCE 119875 is equal with the degree ofpolynomial chaos truncation and sdot
119865denotes the Frobe-
nius norm The closed form of the solution can be written asfollow [14]
119866 = 119868 + sum
119875
tr (119870119879
119896119870
0)
tr (119870119879
0119870
0)
119866
119896
(73)
Here tr(119870119879
119896119870
0) = sum
119873119902
119894=1[119870
119896]
119894119894[119870
0]
119894119894and hence the coeffi-
cients in the above equality can be computed straightforwardIn addition since
119860 and 119870
0are symmetric and positive
definite thus 119866 and
119875 = 119866⨂119870
0have also these properties
[15 16]
Example 1 Consider the following distributed optimal con-trol problem
min119906119911
119869 (119906 119911) = 119864 [
1
2
119906 (x 120596) minus (x)21198712(119863)
]
+
120572
2
119911 (x)21198712(119863)
minusnabla sdot (119886 (x 120596) nabla119906 (x 120596)) = 119911 (x) x isin 119863 120596 isin Ω
(74)
where 119863 = [minus1 1]
2 Ω sim 119880[minus1 1] the boundary conditionsare given as
119906 (minus1 119910 120596) = 119906 (1 119910 120596) = 119906 (119909 1 120596) = 119906 (119910 minus1 120596) = 0
(75)
ISRN Applied Mathematics 9
005
1
005
10
05
1y d
xy
(a) Desired values
005
1
005
10
05
1
uM
ean
xy
times10minus3
(b) Initial state
005
1
005
10
05
1
u opt
xy
(c) Approximated solution (state)
005
1
005
10
200
400
z opt
xy
(d) Control
005
1
0
05
10
05
1
times 10minus3
xy
120583
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
05
1
15
2
25
times 10minus3
(f) Expected value
20 40
10
20
30
40
50
05
1
15
2
25
times 10minus3
(g) Standard dev
0 5 10times 104
0
5
10
times 104 0
(h) Sparsity of global matrix119899119911 = 1815980
Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))
2+ (119910 minus (12))
2)]
119873 = 3 119870 = 5 and 120572 = 000005
and the desired function is given by
(119909 119910) = exp [minus64 ((119909 minus
1
2
)
2
+ (119910 minus
1
2
)
2
)] (76)
The random field 119886(x 120596) is characterized by its mean andcovariance function
119864 [119886] = 119886 = 10
119877 (119909
1 119909
2) = 119890
minus|1199091minus1199092| 119909
1 119909
2isin [minus1 1]
(77)
Following Section 22 the truncated KLE can be representedas
119886 (119909 119910 120596) = 10 +
119873
sum
119895=0
radic120582
119895120579
119895120601
119895 (119909) (78)
The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867
1
0(119863)⨂119871
2(Ω) and all 119911isin 119871
2(119863)
119869 (119906opt 119911opt) le 119869 (119906 119911) (79)
The eigenpairs 120582119895 120579
119895 in truncated KLE solve the integral
equation
int
1
minus1
119890
minus|1199091minus1199092|120601
119895(119909
2) 119889119909
2= 120582
119895120601
119895(119909
1)
(80)
For this special case of the covariance function we haveexplicit expressions for 120601
119895and 120582
119895 [6] Let120596
119895evenand120596
119895oddsolve
the equations
1 minus 120596
119895eventan (120596
119895even) = 0
120596
119895odd+ tan (120596
119895odd) = 0
(81)
Then the even and odd indexed eigenfunctions are given by
120601
119895even(119909) =
cos (120596119895even
119909)
radic1 + (sin (2120596
119895even) 2120596
119895even)
120601
119895odd(119909)
=
sin (120596
119895odd119909)
radic1 minus (sin (2120596
119895odd) 2120596
119895odd)
(82)
10 ISRN Applied Mathematics
005
1
005
10
05
1y ha
nd
xy
(a) Desired values
005
1
005
1minus05
05
0
u Mea
n
xy
(b) Mean values
005
1
005
1
0
Num
sol
xy
0
1
(c) Approximated solution (State)
005
1
005
10
05
1
xy
(d) Control
005
1
005
10
05
1
Lend
a
xy
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
0
005
01
015
02
(f) Expected value-SGS
20 40
10
20
30
40
50
0
005
01
015
02
(g) Standard dev-SGS
0 2 4 6 8times 104
0
2
4
6
8
times 104
(h) Sparsity of global matrix119899119911 = 3046160
Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001
and corresponding eigenvalues
120582
119895even(119909) =
2
120596
2
119895even+ 1
120582
119895odd(119909) =
2
120596
2
119895odd+ 1
(83)
We choose 120579 = (120579
1 120579
119873)
119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]
First of all we find numerical approximations of120596119895even
and120596
119895oddwith bisection method and then with the eigenpairs
evaluated by this 120596119895even
and 120596
119895odd by choosing dimension 119873 =
3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1
In Figure 1(a) the desired function (119909 119910) is plottedfor 119909
119894= 119910
119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial
state is computed and plotted in Figure 1(b) by solving (37)where
119911 = 0 Approximated 119906opt corresponding to the
optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial
119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911
119900119901119905must approximate the
initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane
In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use
Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary
ISRN Applied Mathematics 11
conditions is replaced by (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2
4 Conclusion
The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996
[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987
[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005
[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947
[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946
[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991
[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007
[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009
[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938
[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993
[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008
[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999
[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006
[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010
[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010
[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRN Applied Mathematics 9
005
1
005
10
05
1y d
xy
(a) Desired values
005
1
005
10
05
1
uM
ean
xy
times10minus3
(b) Initial state
005
1
005
10
05
1
u opt
xy
(c) Approximated solution (state)
005
1
005
10
200
400
z opt
xy
(d) Control
005
1
0
05
10
05
1
times 10minus3
xy
120583
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
05
1
15
2
25
times 10minus3
(f) Expected value
20 40
10
20
30
40
50
05
1
15
2
25
times 10minus3
(g) Standard dev
0 5 10times 104
0
5
10
times 104 0
(h) Sparsity of global matrix119899119911 = 1815980
Figure 1 Illustration of the optimal control and state achieved by solving Example 1 taking (119909 119910) = exp[minus64((119909 minus (12))
2+ (119910 minus (12))
2)]
119873 = 3 119870 = 5 and 120572 = 000005
and the desired function is given by
(119909 119910) = exp [minus64 ((119909 minus
1
2
)
2
+ (119910 minus
1
2
)
2
)] (76)
The random field 119886(x 120596) is characterized by its mean andcovariance function
119864 [119886] = 119886 = 10
119877 (119909
1 119909
2) = 119890
minus|1199091minus1199092| 119909
1 119909
2isin [minus1 1]
(77)
Following Section 22 the truncated KLE can be representedas
119886 (119909 119910 120596) = 10 +
119873
sum
119895=0
radic120582
119895120579
119895120601
119895 (119909) (78)
The aim is to calculate 119906opt and 119911opt such that for all119906 isin 119867
1
0(119863)⨂119871
2(Ω) and all 119911isin 119871
2(119863)
119869 (119906opt 119911opt) le 119869 (119906 119911) (79)
The eigenpairs 120582119895 120579
119895 in truncated KLE solve the integral
equation
int
1
minus1
119890
minus|1199091minus1199092|120601
119895(119909
2) 119889119909
2= 120582
119895120601
119895(119909
1)
(80)
For this special case of the covariance function we haveexplicit expressions for 120601
119895and 120582
119895 [6] Let120596
119895evenand120596
119895oddsolve
the equations
1 minus 120596
119895eventan (120596
119895even) = 0
120596
119895odd+ tan (120596
119895odd) = 0
(81)
Then the even and odd indexed eigenfunctions are given by
120601
119895even(119909) =
cos (120596119895even
119909)
radic1 + (sin (2120596
119895even) 2120596
119895even)
120601
119895odd(119909)
=
sin (120596
119895odd119909)
radic1 minus (sin (2120596
119895odd) 2120596
119895odd)
(82)
10 ISRN Applied Mathematics
005
1
005
10
05
1y ha
nd
xy
(a) Desired values
005
1
005
1minus05
05
0
u Mea
n
xy
(b) Mean values
005
1
005
1
0
Num
sol
xy
0
1
(c) Approximated solution (State)
005
1
005
10
05
1
xy
(d) Control
005
1
005
10
05
1
Lend
a
xy
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
0
005
01
015
02
(f) Expected value-SGS
20 40
10
20
30
40
50
0
005
01
015
02
(g) Standard dev-SGS
0 2 4 6 8times 104
0
2
4
6
8
times 104
(h) Sparsity of global matrix119899119911 = 3046160
Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001
and corresponding eigenvalues
120582
119895even(119909) =
2
120596
2
119895even+ 1
120582
119895odd(119909) =
2
120596
2
119895odd+ 1
(83)
We choose 120579 = (120579
1 120579
119873)
119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]
First of all we find numerical approximations of120596119895even
and120596
119895oddwith bisection method and then with the eigenpairs
evaluated by this 120596119895even
and 120596
119895odd by choosing dimension 119873 =
3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1
In Figure 1(a) the desired function (119909 119910) is plottedfor 119909
119894= 119910
119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial
state is computed and plotted in Figure 1(b) by solving (37)where
119911 = 0 Approximated 119906opt corresponding to the
optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial
119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911
119900119901119905must approximate the
initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane
In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use
Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary
ISRN Applied Mathematics 11
conditions is replaced by (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2
4 Conclusion
The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996
[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987
[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005
[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947
[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946
[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991
[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007
[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009
[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938
[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993
[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008
[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999
[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006
[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010
[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010
[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 ISRN Applied Mathematics
005
1
005
10
05
1y ha
nd
xy
(a) Desired values
005
1
005
1minus05
05
0
u Mea
n
xy
(b) Mean values
005
1
005
1
0
Num
sol
xy
0
1
(c) Approximated solution (State)
005
1
005
10
05
1
xy
(d) Control
005
1
005
10
05
1
Lend
a
xy
(e) Adjoint
minus1 0 1minus1
minus05
0
05
1
0
005
01
015
02
(f) Expected value-SGS
20 40
10
20
30
40
50
0
005
01
015
02
(g) Standard dev-SGS
0 2 4 6 8times 104
0
2
4
6
8
times 104
(h) Sparsity of global matrix119899119911 = 3046160
Figure 2 Illustration of optimal control and state achieved by solving Example 2 taking (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910) 119873 = 3 119870 = 4 and120572 = 000001
and corresponding eigenvalues
120582
119895even(119909) =
2
120596
2
119895even+ 1
120582
119895odd(119909) =
2
120596
2
119895odd+ 1
(83)
We choose 120579 = (120579
1 120579
119873)
119879 to be independent randomvariables uniformly distributed over the interval [minus1 1]
First of all we find numerical approximations of120596119895even
and120596
119895oddwith bisection method and then with the eigenpairs
evaluated by this 120596119895even
and 120596
119895odd by choosing dimension 119873 =
3 and order 119870 = 5 which emphasize that 119875 = 55 weconstruct the KLE Taking 120572 = 000005 Algorithm 1 is usedto compute 119906opt and 119911opt Illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 1
In Figure 1(a) the desired function (119909 119910) is plottedfor 119909
119894= 119910
119894= minus1 + 004 lowast 119894 119894 = 0 1 2 50 Initial
state is computed and plotted in Figure 1(b) by solving (37)where
119911 = 0 Approximated 119906opt corresponding to the
optimal control 119911opt is calculated by Algorithm 1 and itsgraph is depicted in Figure 1(c) As it was expected by usingthe proposed Algorithm 1 even by very far initial state theapproximated state solution reaches to the desired functionIn Figure 1(d) the 119911opt is depicted Serious changes duringthe computation for initial
119911 = 0 happened reach to theoptimal control function 119911opt The adjoint state is computedand plotted in Figure 1(e) As it is expected from (58) theadjoint state corresponding to the 119911
119900119901119905must approximate the
initial state (see Figures 1(b) and 1(c)) Figures 1(f) and 1(g)represent counter plots of the optimal state expectation andstandard deviation Finally in Figure 1(h) the representationof coefficient matrix sparsity with the number of nonzerocomponent in the computation is plotted in 119909-119910 plane
In Example 1 we considered an exponential desired func-tion with v-sharp points In the next example the smoothdesired function and boundary conditions are consideredThis is the case that Algorithm 1 is more efficient to use
Example 2 Consider the optimal control of Example 1 inwhich the right hand side function as well as boundary
ISRN Applied Mathematics 11
conditions is replaced by (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2
4 Conclusion
The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996
[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987
[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005
[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947
[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946
[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991
[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007
[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009
[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938
[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993
[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008
[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999
[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006
[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010
[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010
[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRN Applied Mathematics 11
conditions is replaced by (119909 119910) = 2120587
2 sin(120587119909) sin(120587119910)119873 = 3 and 119870 = 4 which emphasize that 119875 = 34Taking 120572 = 000001 the illustration of computed optimalcontrol and corresponding state in addition to the globalmatrix sparsity standard deviation and expected values of thesolution adjoint values and initial state approximation of thedesired function is shown in Figure 2
4 Conclusion
The solution of SPDE-constrained optimization problems isa recently challenging computational task Here we considerdistributed control problems in which stochastic diffusionequation is the SPDE Since the saddle point system extractedfrom using KKT optimality condition of the problem is avery large system and more expensive to solve hence weuse the strategy of adjoint technique and preconditionedNewtonrsquos conjugate gradient method which iteratively solvethe problem and has low computational cost By separatingthe stochastic and deterministic parts of the SPDE usingKLE and discretizing each part by WCE and Galerkinfinite element method respectively we adjoint techniqueto compute the gradient and Hessian of the discretizedoptimization problem By using preconditioned Newtonrsquosconjugate gradient method the optimal control and stateof the problem are calculated numerically Two numericalexamples are given to illustrate the correspondence betweentheoretical and numerical approaches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M D Gunzburger and L S Hou ldquoFinite-dimensional approx-imation of a class of constrained nonlinear optimal controlproblemsrdquo SIAM Journal on Control and Optimization vol 34no 3 pp 1001ndash1043 1996
[2] M Gunzburger Perspectives in Flow Control and OptimizationSIAM 1987
[3] I Babuska R Tempone and G E Zouraris ldquoSolving ellipticboundary value problems with uncertain coefficients by thefinite element method the stochastic formulationrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 12ndash16 pp 1251ndash1294 2005
[4] K Karhunen ldquoUberlineare Methoden in der Wahrschein-lichkeitsrechnungrdquo Annales Academiaelig Scientiarum Fennicaeligvol 37 p 79 1947
[5] M Loeve ldquoFonctions aleatoire de second ordrerdquo La RevueScientifique vol 84 pp 195ndash206 1946
[6] R Ghanem and P D Spanos Stochastic Finite Elements ASpectral Approach Springer Berlin Germany 1991
[7] E Haber and L Hanson ldquoModel problems in PDE-constrainedoptimizationrdquo Tech Rep TR-2007-009 Emory Atlanta GaUSA 2007
[8] M Hinze R Pinnau M Ulbrich and S Ulbrich Optimizationwith PartialDifferential Equations vol 23 ofMathematicalMod-elling Theory and Applications Springer Heidelberg Germany2009
[9] N Wiener ldquoThe homogeneous chaosrdquoThe American Journal ofMathematics vol 60 pp 897ndash938 1938
[10] C F van Loan andN Pitsianis ldquoApproximationwith Kroneckerproductsrdquo in Linear Algebra for Large Scale and Real-TimeApplications M S Moonen G H Golub and B L R de MoorEds pp 293ndash314 Kluwer Academic Dordrecht Germany1993
[11] M Heinkenschloss ldquoNumerical solution of implicitly con-strained optimization problemsrdquo Tech Rep TR08-05 Depart-ment of Computational andAppliedMathematics RiceUniver-sity Houston Tex USA 2008
[12] C T Kelley IterativeMethods for Optimization SIAM Philadel-phia Pa USA 1999
[13] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2nd edition 2006
[14] E Ullmann ldquoA kronecker product preconditioner for stochasticgalerkin finite eleme nt discretizationsrdquo SIAM Journal onScientific Computing vol 32 no 2 pp 923ndash946 2010
[15] C E Powell and E Ullmann ldquoPreconditioning stochasticGalerkin saddle point systemsrdquo SIAM Journal on Matrix Anal-ysis and Applications vol 31 no 5 pp 2813ndash2840 2010
[16] J Schoberl and W Zulehner ldquoSymmetric indefinite precondi-tioners for saddle point problems with applications to PDE-constrained optimization problemsrdquo SIAM Journal on MatrixAnalysis and Applications vol 29 no 3 pp 752ndash773 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of