pod/eim nonlinear model reduction and pod/4-d var with tr...
TRANSCRIPT
0.
POD/EIM nonlinear model reduction andPOD/4-D VAR with TR for models of the SWE
R. Stefanescu1 and I. M. Navon2
September 3, 2012
Florida State University, Tallahassee, Florida, USA
[email protected],[email protected]
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 1/101
Part1 - POD/DEIM nonlinear model reduction
1 POD/DEIM POD/EIM justification and methodology
2 Description of EIM algorithm
3 Limited - Area SWE
4 Generation of POD using FEM
5 POD FEM SWE Model
6 Numerical Results
7 Reduced order POD 4-D VAR
8 Trust Region POD
9 T-R POD algorithm
10 Illustration
11 Conclusions
1. POD/DEIM POD/EIM justification and methodology
POD/DEIM POD/EIM justification and methodology
Model order reduction : Reduce the computationalcomplexity/time of large scale dynamical systems byapproximations of much lower dimension with nearly the sameinput/output response characteristics.
Goal : Construct reduced-order model for different types ofdiscretization method (finite difference (FD), finite element(FEM), finite volume (FV)) of unsteady and/or parametrizednonlinear PDEs. E.g., PDE:
∂y
∂t(x , t) = L(y(x , t)) + F(y(x , t)), t ∈ [0,T ]
where L is a linear function and F a nonlinear one.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 3/101
POD/DEIM methodology applied to FD SCHEMES
The corresponding FD scheme is a n dimensional ordinarydifferential system
d
dty(t) = Ay(t) + F(y(t)), A ∈ Rn×n,
where y(t) = [y1(t), y2(t), .., yn(t)] ∈ Rn and yi (t) ∈ R arethe spatial components y(xi , t), i = 1, .., n. F is a nonlinearfunction evaluated at y(t) componentwise, i.e.F = [F(y1(t)), ..,F(yn(t))]T , F : I ⊂ R→ R.
A common model order reduction method involves theGalerkin projection with basis Vk ∈ Rn×k obtained fromProper Orthogonal Decomposition (POD), for k n, i.e.y ≈ Vk y(t), y(t) ∈ Rk . Applying an inner product to theODE discrete system we get
d
dty(t) = V T
k AVk︸ ︷︷ ︸k×k
y(t) + V Tk F(Vk y(t))︸ ︷︷ ︸
N(y)
(1)
1. POD/DEIM POD/EIM justification and methodology
POD/DEIM methodology applied to FD SCHEMES
The efficiency of POD - Galerkin technique is limited to thelinear or bilinear terms. The projected nonlinear term stilldepends on the dimension of the original system
N(y) = V Tk︸︷︷︸
k×n
F(Vk y(t))︸ ︷︷ ︸n×1
.
To mitigate this inefficiency we introduce ”Discrete EmpiricalInterpolation Method (DEIM) ” for nonlinear approximation.For m n
N(y) ≈ V Tk U(PTU)−1︸ ︷︷ ︸
precomputed k×m
F(PTVk y(t))︸ ︷︷ ︸m×1
.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 5/101
1. POD/DEIM POD/EIM justification and methodology
POD/EIM methodology applied to FE SCHEMES
The corresponding Finite Element (FE) scheme is a ndimensional ordinary differential system
Mhd
dty(t) = Khy(t) + Nh(y(t)), Mh,Kh ∈ Rn×n, (2)
y(t) = [y1(t), y2(t), .., yn(t)] ∈ Rn, yi (t) ∈ R.
y(t, x) 'n∑
j=1
ψj(x)yj(t) = Ψ(x)y(t), Ψ(x) ∈ R1×n.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 6/101
1. POD/DEIM POD/EIM justification and methodology
POD/EIM methodology applied to FE SCHEMES
Nh(y(t)) ∈ Rn is a nonlinear functional which can be of thefollowing form
[Nh(y(t))]i =
∫Ω
∂ψi (x)
∂xF (Ψ(x)y(t))dΩ, i = 1, ..n.
[Nh(y(t))]i =
∫Ωψi (x)F (Ψ(x)y(t))dΩ, i = 1, ..n.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 7/101
POD/EIM methodology applied to FE SCHEMES
Using the Galerkin projection with basis Φ(x) = Ψ(x)Uk ,Φ(x) ∈ R1×k , Uk ∈ Rn×k calculated via POD, for k n, i.e.y(t, x) ≈ Φ(x)y(t), y(t) ∈ Rk we apply the following innerproduct
< x , y >Mh= xTMhy .
One obtains the corresponding discretized reduced ordermodel:
UTk MhUk︸ ︷︷ ︸I∈Rk×k
d
dty(t) = UT
k KhUk︸ ︷︷ ︸k×k
y(t) + UTk Nh(y(t))︸ ︷︷ ︸
N(y(t))
. (3)
POD/EIM methodology applied to FE SCHEMES
The projected nonlinear term still depends on the dimensionof the original system
N(y(t)) = UTk︸︷︷︸
k×n
Nh(y(t))︸ ︷︷ ︸n×1
.
[Nh(y(t))]i =
∫Ωψi (x)F (Φ(x)y(t))dΩ, i = 1, ..n.
The Empirical Interpolation Method (EIM) approximation ofthe nonlinear function F (Φ(x)y(t)) is given by
F (Φ(x)y(t)) ' Q(x)ρ(t) = Q(x)(Q(z))−1F (Φ(z)y(t)),
Q(x) = [q1(x), ..., qm(x)], z = [z1, ..., zm], m n
Q(z) ∈ Rm×m, Φ(z) ∈ Rm×k ,
F (Φ(z)y(t)) ∈ Rm×1 − F is applied componentwise
POD/EIM methodology applied to FE SCHEMES
Thus
Nh(y(t)) '∫
ΩΨ(x)TQ(x)dΩ︸ ︷︷ ︸
n×m
(Q(z))−1︸ ︷︷ ︸m×m
F (Φ(z)y(t))︸ ︷︷ ︸m×1
Now we are able to separate the unknown y(t) from theintegrals allowing us the precomputation of the integralswhich then can be used in all of the time steps.
N(y(t)) ' UTk︸︷︷︸
k×n
∫Ω
Ψ(x)TQ(x)dΩ(Q(z))−1︸ ︷︷ ︸n×m
F (Φ(z)y(t))︸ ︷︷ ︸m×1
.
2. Description of EIM algorithm
Empirical Interpolation Method (EIM)
Empirical Interpolation method was proposed by Barrault et al.(2004).
M Barrault, Y Maday, NC Nguyen, AT Patera. An ’empiricalinterpolation’ method: application to efficient reduced basisdiscretization of partial differential equations. ComptesRendus Acad. Sci. Paris Series I, 339 (2004).
Grepl, Maday, Nguyen, Patera. Efficient reduced-basistreatment OF nonaffine and nonlinear partial differentialequations. Modelisation Mathematique et AnalyseNumerique, Vol.41 No.3 (2007).
Maday, Nguyen, Patera, Pau. A GENERAL MULTIPURPOSEINTERPOLATION PROCEDURE: THE MAGIC POINTS.Communications on Pure and Applied Analysis Vol.8 No.1(2009).
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 11/101
2. Description of EIM algorithm
Empirical Interpolation Method (EIM)
A rigorous a posteriori error estimation may be found in
JL Eftang, MA Grepl, and AT Patera. A Posteriori ErrorBounds for the Empirical Interpolation Method. CR Acad.Sci. Paris Series I, 348 (2010).
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 12/101
2. Description of EIM algorithm
Empirical Interpolation Method (EIM)
Let f (x ;µ) be a nonlinear parameterized real-valued function,x ∈ Ω ⊂ Rd and µ ∈ D ⊂ Rp.
The approximation
f (x ;µ) =m∑j=1
qj(x)cj(µ),
where qjmj=1 form a basis whose span gives a goodapproximation to
spanf (·;µ) : µ ∈ D
and cj(µ)mj=1 are obtained from the coefficient functionapproximation using a set of pre-specified points, calledinterpolation points zimi=1 ∈ Ω which are expected tocapture the parameter variation.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 13/101
2. Description of EIM algorithm
Empirical Interpolation Method (EIM)
Two ingredients are needed for constructing f .
A set of basis functions qjmj=1. The original paper for EIMuses this basis set constructed from a greedy selection processon the set of snapshots. Here the basis set is constructedfrom POD and EIM algorithm for interpolation points.
A set of interpolation points zimi=1 (EIM algorithm forinterpolation points) used in coefficient functionapproximation for cj(µ). For a fixed value of µ,c1(µ), .., cm(µ) satisfies
f (zi ;µ) = f (zi ;µ) =m∑j=1
qj(zi )cj(µ), i = 1, 2, ..m.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 14/101
2. Description of EIM algorithm
Empirical Interpolation Method (EIM)
Introducing the matrix vector notations
Q(x) = [q1(x), .., qm(x)], z = [z1, .., zm]T , c = [c1, .., cm]T
the approximation f of the function f can be written as:
f (x ;µ) = Q(x)c(µ)
and in particular
f (z ;µ) = Q(z)c(µ).
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 15/101
2. Description of EIM algorithm
Empirical Interpolation Method (EIM)
EIM algorithm for interpolation points chooses to use basisQ(x) and interpolation points z such that Q(z) is invertible.
Then the coefficient can be written as a function of parameterµ explicitly as follows:
c(µ) = (Q(z))−1f (z;µ).
The final function approximation has the following form
f (x ;µ) = Q(x)(Q(z))−1f (z;µ).
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 16/101
EIM: Algorithm for interpolation points
INPUT: ulml=1 ⊂ Rn (linearly independent generated by PODfrom the snapshots of f ):
OUTPUT: z and Q(x)
1 z1 = arg ess supx∈Ω|u1(x)|2 q1(x) = u1(x)
u1(z1)
3 z = [z1], Q(x) = [q1(x)].
4 For l = 2, ..,m do
a Solve Q(z)c = ul(z) for c
b rl(x) = ul(x)−Q(x)c
c zl = arg ess supx∈Ω|rl(x)|d ql(x) = rl (x)
rl (zl )
e z←[
zzl
], Q(x)← [Q(x) ql(x)].
5 end for.
EIM: Algorithm for interpolation points -discrete variant
INPUT: ulml=1 ⊂ Rn (linearly independent generated by PODfrom the snapshots of f ):
OUTPUT: z and Q(x)
1 [|ψ| ρ1] = max |u1|, ψ ∈ R and ρ1 is the component positionof the largest absolute value of u1, with the smallest indextaken in case of a tie. z1 = x(ρ1)
2 q1(x) = u1(x)u1(z1)
3 z = [z1], Q(x) = [q1(x)].4 For l = 2, ..,m do
a Solve Q(z)c = ul(z) for c
b rl(x) = ul(x)−Q(x)c
c [|ψ| ρl ] = max |rl |, zl = x(ρl)
d ql(x) = rl (x)rl (zl )
e z←[
zzl
], Q(x)← [Q(x) ql(x)].
5 end for.
2. Description of EIM algorithm
Empirical Interpolation Method (EIM)
The following example illustrates the efficiency of EIM inapproximating a highly nonlinear function defined on adiscrete 1D spatial domain. Consider a nonlinearparameterized function s : Ω×D→ R defined by
s(x ;µ) = (1− x)cos
(πµ(x + 1)
)e−(1+x)µ,
where x ∈ Ω = [−1, 1] and µ ∈ D = [0, π2 ] ⊂ R.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 19/101
2. Description of EIM algorithm
Empirical Interpolation Method (EIM)
Let [x1, x2, ..., xn] ∈ Rn, xi ∈ R being equally distributed in Ω,for i = 1, 2, .., n, n = 101. We introduce f : D→ Rn asfollows
f (µ) = [s(x1;µ), s(x2;µ), .., s(xn;µ)T ] ∈ Rn, µ ∈ D
We used 50 snapshots f (µj)50j=1 to construct POD basis
ulml=1 with µj equidistantly points in [0, π2 ].
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 20/101
2. Description of EIM algorithm
Empirical Interpolation Method (EIM)
0 5 10 15 20 25 30 35 40 45 50−40
−35
−30
−25
−20
−15
−10
−5
0
5Singular values of 50 snapshots
log
arit
hm
ic s
cale
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
1.5
2EIM Points and the first 6 basis functions
q1(x)q2(x)q3(x)q4(x)q5(x)q6(x)Eim ptsExact
Figure 1: Singular eigenvalues using logarithmic scale and thecorresponding first 6 basis functions with interpolation points generatedby EIM algorithm, µ = 1.13
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 21/101
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2EIM#1
q1(x)current point
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.4
−0.2
0
0.2
0.4
0.6
0.8
1EIM#2
q2(x)r=u2(x)−Q(x)ccurrent pointprevious points
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.4
−0.2
0
0.2
0.4
0.6
0.8
1EIM#3
q3(x)r=u3(x)−Q(x)ccurrent pointprevious points
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1EIM#4
q4(x)r=u4(x)−Q(x)ccurrent pointprevious points
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1EIM#5
q5(x)r=u5(x)−Q(x)ccurrent pointprevious points
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.4
−0.2
0
0.2
0.4
0.6
0.8
1EIM#6
q6(x)r=u6(x)−Q(x)ccurrent pointprevious points
Figure 2: The selection process of EIM interpolation points
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2EIM#1
exactEIM approx.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2EIM#2
exactEIM approx.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2EIM#3
exactEIM approx.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2EIM#4
exactEIM approx.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2EIM#5
exactEIM approx.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2EIM#6
exactEIM approx.
Figure 3: EIM approximation for different values of m
2. Description of EIM algorithm
Empirical Interpolation Method (EIM)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2 The Exact function and its EIM approximation for µ =1.13
Exact functionEIM approximation
0 5 10 15 20 25 30 35 40 45 50−35
−30
−25
−20
−15
−10
−5
0
5
m (Reduced dimension)
log
arit
hm
ic s
cale
Error in Euclidian Norm
EIM errorPOD error
Figure 4: The EIM approximate function for m = 20 compared with theexact function of dimension n = 101 at µ = 1.13 (left); Comparison ofthe spatial errors for POD and EIM approximations (right)
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 24/101
3. Limited - Area SWE
Limited - Area SWE
The shallow - water equations model on a β−plane
∂u
∂t+ u
∂u
∂x+ v
∂u
∂y+∂φ
∂x− fv = 0
∂v
∂t+ u
∂v
∂x+ v
∂v
∂y+∂φ
∂y+ fu = 0
∂φ
∂t+
∂
∂x(uφ) +
∂
∂x(vφ) = 0
(x , y) ∈ Ω = [0, L]× [0,D], t > 0
where L and D are the dimensions of a rectangular domain ofintegration, u and v are the velocity components in the x andy axis respectively, φ = gh is the geopotential height, h is thedepth of the fluid and g is the acceleration of gravity.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 25/101
3. Limited - Area SWE
Limited - Area SWE
The scalar function f is the Coriolis parameter.
f = f + β
(y − D
2
), β =
∂f
∂y
The f is the Coriolis frequency
f = f + β
(y − D
2
)The Coriolis parameter
f = 2Ωv sin θ
is defined at a mean latitude θ0, where Ωv is the angularvelocity of the earth’s rotation and θ is latitude.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 26/101
3. Limited - Area SWE
Limited - Area SWE
We impose initial conditions
w(x , y , 0) = ϕ(x , y), where state variables are
w = w(x , y , t) = (u(x , y , t), v(x , y , t), φ(x , y , t)) ,
with periodic boundary conditions are assumed in thex-direction:
w(0, y , t) = w(L, y , t)
whereas solid wall boundary condition are used in y -direction:
v(x , 0, t) = v(x ,D, t) = 0.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 27/101
3. Limited - Area SWE
Finite Element Limited - Area SWE
Finite element method (FEM) is used to obtain the discretemodel of the limited SWE
We employed linear piecewise polynomials on triangularelements in the formulation of Galerkin finite-elementshallow-water equations model, in which the global matrix wasstored into a compact matrix.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 28/101
3. Limited - Area SWE
Finite Element Limited - Area SWE
A time-extrapolated Crank-Nicholson time differencingscheme was applied for integrating in time the system ofordinary differential equations.
The Galerkin finite-element boundary conditions were treatedusing the approach suggested by Payne and Irons (1963) andmentioned by Huebner (1975), i.e. modifying the diagonalterms of the global matrix associated with the nodal variablesby multiplying them by a large number, say 1016, while thecorresponding term in the right-hand vector is replaced by thespecified boundary nodal variable multiplied by the same largefactor times the corresponding diagonal term.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 29/101
3. Limited - Area SWE
Variational form of the Limited - Area SWE
The variational form is obtained by multiplying each of themodel equations byw ∈ H1(Ω), w(0, y) = w(L, y), w(x , 0) = w(x ,D) = 0, andrequiring < w ,Ei >L2(Ω)= 0, i = 1, 2, 3 i.e.∫
Ω
∂u
∂twdΩ+
∫Ωu∂u
∂xwdΩ+
∫Ωv∂u
∂ywdΩ+
∫Ω
∂φ
∂xwdΩ−
∫ΩfvwdΩ = 0
∫Ω
∂v
∂twdΩ+
∫Ωu∂v
∂xwdΩ+
∫Ωv∂v
∂ywdΩ+
∫Ω
∂φ
∂ywdΩ+
∫ΩfuwdΩ = 0
∫Ω
∂φ
∂twdΩ−
∫Ωuφ∂w
∂xdΩ +
∫Ωvφ∂w
∂ydΩ = 0
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 30/101
3. Limited - Area SWE
Galerkin FE Limited - Area SWE
Let Ψ = [ψ1, .., ψN ] be a set of continuous and piecewiselinear functions which are formed by a sum of linear shapefunctions Ne
i , i = 1, 2, 3 i.e.
Nei (x , y) =
ai + bix + ciy
2∆
defined for each triangular element e of a N = nx × ny mesh.x , y are the coordinates of the mesh points.
ai = xjym − xmyj , bi = yj − ym, ci = xm − xj , i , j ,m = 1, 2, 3.
Each ψi , i = 1,N are constrained to satisfy the periodicboundary conditions on the x directions and the solidboundary conditions in the y directions.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 31/101
3. Limited - Area SWE
Galerkin FE Limited - Area SWE
By assuming that the SWE solutions have the following form
u 'N∑j=1
uj(t)ψj(x , y); v 'N∑j=1
vj(t)ψj(x , y); φ 'N∑j=1
φj(t)ψj(x , y)
and taking w = ψi , i = 1, 2, ..,N we obtain the Galerkin FEdiscrete SWE model.
Assume also
f ' fI ≡N∑j=1
fjψj .
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 32/101
3. Limited - Area SWE
Galerkin FE Limited - Area SWE
N∑j=1
u′j
∫ΩψiψjdΩ+
N∑j=1
uj
N∑p=1
up
∫Ωψiψj
∂ψp
∂xdΩ+
N∑j=1
vj
N∑p=1
up
∫Ωψiψj
∂ψp
∂ydΩ
+N∑j=1
φj
∫Ωψi∂ψj
∂xdΩ−
N∑j=1
fj
N∑p=1
vp
∫ΩψiψjψpdΩ = 0
N∑j=1
v′j
∫ΩψiψjdΩ+
N∑j=1
uj
N∑p=1
vp
∫Ωψiψj
∂ψp
∂xdΩ+
N∑j=1
vj
N∑p=1
vp
∫Ωψiψj
∂ψp
∂ydΩ
+N∑j=1
φj
∫Ωψi∂ψj
∂ydΩ +
N∑j=1
fj
N∑p=1
up
∫ΩψiψjψpdΩ = 0
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 33/101
3. Limited - Area SWE
Galerkin FE Limited - Area SWE
N∑j=1
φ′j
∫ΩψiψjdΩ−
N∑j=1
uj
N∑p=1
φp
∫Ω
∂ψi
∂xψjψpdΩ
−N∑j=1
vj
N∑p=1
φp
∫Ω
∂ψi
∂yψjψpdΩ = 0.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 34/101
3. Limited - Area SWE
Galerkin FE Limited - Area SWE
We introduce the following notation
Mij =
∫Ωψiψj , i , j = 1, ..,N,
A1ijp =
∫Ωψiψj
∂ψp
∂xdΩ, i , j , p = 1, ..,N,
A2ijp =
∫Ωψiψj
∂ψp
∂ydΩ, i , j , p = 1, ..,N,
Mdxij =
∫Ωψi∂ψj
∂x, i , j = 1, ..,N,
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 35/101
3. Limited - Area SWE
Galerkin FE Limited - Area SWE
A3ijp =
∫ΩψiψjψpdΩ, i , j , p = 1, ..,N,
Mdyij =
∫Ωψi∂ψj
∂y, i , j = 1, ..,N,
A4ijp =
∫Ω
∂ψi
∂xψjψpdΩ, i , j , p = 1, ..,N,
A5ijp =
∫Ω
∂ψi
∂yψjψpdΩ, i , j , p = 1, ..,N,
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 36/101
3. Limited - Area SWE
Galerkin FE Limited - Area SWE
Using the Silvestre formula∫e(Ne
1 )m(Ne2 )n(Ne
3 )lde =m!n!l!
(m + n + l + 2)!2∆,
the global matrices defined previous are calculated.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 37/101
3. Limited - Area SWE
The Matrix Formulation
M︸︷︷︸N×N
u′︸︷︷︸N×1
+ uT︸︷︷︸1×N
A1︸︷︷︸:,N×N
u︸︷︷︸N×1
+ vT︸︷︷︸1×N
A2︸︷︷︸:,N×N
u︸︷︷︸N×1
+Mdx︸︷︷︸N×N
φ︸︷︷︸N×1
− fT︸︷︷︸N×1
A3︸︷︷︸:,N×N
v︸︷︷︸N×1
= 0,
M︸︷︷︸N×N
v′︸︷︷︸N×1
+ uT︸︷︷︸1×N
A1︸︷︷︸:,N×N
v︸︷︷︸N×1
+ vT︸︷︷︸1×N
A2︸︷︷︸:,N×N
v︸︷︷︸N×1
+Mdy︸︷︷︸N×N
φ︸︷︷︸N×1
+ fT︸︷︷︸N×1
A3︸︷︷︸:,N×N
u︸︷︷︸N×1
= 0,
M︸︷︷︸N×N
φ′︸︷︷︸N×1
+ uT︸︷︷︸1×N
A4︸︷︷︸:,N×N
φ︸︷︷︸N×1
+ vT︸︷︷︸1×N
A5︸︷︷︸:,N×N
φ︸︷︷︸N×1
= 0.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 38/101
3. Limited - Area SWE
The Matrix Formulation
With the following notation
K11(u) = uT︸︷︷︸1×N
A1︸︷︷︸:,N×N
∈ RN×N,
K12(v) = vTA2, K13 = fTA3,
K31(u) = uTA4, K32(v) = vTAt ,
we finally obtain
Mu′ + K11(u)u + K12(v)u + Mdxφ− K13v = 0,
Mv′ + K11(u)v + K12(v)v + Mdyφ + K13u = 0,
Mu′ − K31(u)φ− K32(v)φ = 0.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 39/101
3. Limited - Area SWE
Crank - Nicolson scheme
The time-extrapolated Crank Nicolson method was used forintegrating in time the system of ordinary differentialequations resulting from the application of the Galerkin FEMto the SWE model.
This method was previously used by Douglas and Dupont,Hinsman, Navon and Muller etc.
An average is taken at time levels n and n + 1 of expressionsinvolving space de rivatives, while the non-linear advectiveterms are quasi- linearized by estimating them at time leveln + 1
2 using the following second-order approximation in time:
un+ 12 = u∗ =
3
2un − 1
2un−1 + O(∆t2),
vn+ 12 = v∗ =
3
2vn − 1
2vn−1 + O(∆t2).
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 40/101
3. Limited - Area SWE
Crank - Nicolson scheme
The algebraic systems of equations from bellow were solved byemploying the Gauss-Seidel iterative method. Let∆φ = φn+1 − φn, ∆u = un+1 − un, ∆v = vn+1 − vn[
2
∆tM − K31(u∗)− K32(v∗)
]∆φ = 2
[K31(u∗) + K32(v∗)
]φn,[
2
∆tM + K11(u∗) + K12(v∗)
]∆u = −2
[K11(u∗) + K12(v∗)
]un
−Mdx
(φn+1 + φn
)+ K13v∗,[
2
∆tM + K11(un+1) + K12(v∗)
]∆v = −2
[K11(un+1) + K12(v∗)
]vn
−Mdy
(φn+1 + φn
)− K13un+1.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 41/101
4. Generation of POD using FEM
Generation of POD using FEM
Proper orthogonal decomposition provides a technique forderiving low order model of dynamical systems. It can bethought of as a Galerkin approximation in the spatial variablebuilt from functions corresponding to the solution of thephysical system at specified time instances. These are calledsnapshots.
LetU = [u1, u2, .., uns ] ∈ RN×ns ,
V = [v1, v2, .., vns ] ∈ RN×ns ,
Φ = [φ1, φ2, .., φns ] ∈ RN×ns ,
be the snapshot sets, i.e. the numerical solution obtained withFE SWE at different time levels t1, t2, .., tns .
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 42/101
4. Generation of POD using FEM
Generation of POD using FEM
We introduce
Ψ(x , y) = [ψ1(x , y), ..., ψN(x , y)]
a 1- by -N matrix of FE basis functions.
The l th snapshots at time tl is of the form
u(x , y , tl) = Ψ(x , y)u(tl),
v(x , y , tl) = Ψ(x , y)v(tl),
φ(x , y , tl) = Ψ(x , y)φ(tl).
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 43/101
4. Generation of POD using FEM
Generation of POD using FEM
The POD bases of dimensions ki , i = u, v , φ, of the snapshots U,V , Φ can be constructed as follows
Compute the following matrices
Lu = UTMU, Lv = V TMV , Lφ = ΦTMΦ,
Lu, Lv , Lφ ∈ Rns×ns .
Find ku eigenvectors of Lu corresponding to the first kulargest eigenvalues
Luwi = wiλi , wi ∈ Rns×1, i = 1, 2, .., ku.
Then each POD basis function uPODi is given by
uPODi (x , y) =
1
λi
ns∑l=1
(wi )lu(x , y , tl), i = 1, 2, .., ku.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 44/101
4. Generation of POD using FEM
Generation of POD using FEM
To write each POD basis function uPODi in terms of the FE
basis Ψ(x , y) = [ψ1(x , y), ..., ψN(x , y)] in a matrix vectorform, let LuW = WΛ be the eigenvalue decomposition of Lu,where Λ = diag(λ1, ..λns) with λ1 ≥ ...λns ≥ 0 andW = [w1, ..,wns ] ∈ Rns×ns containing all the ns eigenvectorsof Lu.Denote by Wk the first ku columns of W and
Λku = diag(λ1, .., λku) ∈ Rku×ku
.The 1 -by-ku matrix of POD basis functions,UPOD(x , y) = [uPOD
1 (x , y), .., uPODku
(x , y)] can be written as
UPOD(x , y) = Ψ(x , y) U︸︷︷︸N×ns
Wku︸︷︷︸ns×ku
Λ−1/2ku︸ ︷︷ ︸
ku×ku
= Ψ(x , y)Uk .
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 45/101
4. Generation of POD using FEM
Generation of POD using FEM
In a similar way we obtain the POD basis corresponding to vand φ
V POD(x , y) = Ψ(x , y)Vk , ΦPOD(x , y) = Ψ(x , y)Φk ,
Vk ∈ RN×kv , Φk ∈ RN×kφ .
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 46/101
5. POD FEM SWE Model
POD FEM SWE Model
By using the POD bases UPOD , UPOD , ΦPOD theapproximate solution can be written as
u(x , y , t) = UPOD(x , y)u(t), v(x , y , t) = V POD(x , y)v(t),
φ(x , y , t) = ΦPOD(x , y)φ(t), u ∈ Rku , v ∈ Rkv , φ ∈ Rkφ .
From the weak form of the SWE model, instead of using thelarger set of finite element basis functions, we now use the
uPODi (x , y)kui=1, vPOD
i (x , y)kvi=1, φPODi (x , y)kφi=1 as the
test functions.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 47/101
5. POD FEM SWE Model
POD FEM SWE Model
Using the relationship between the reduced bases and the FEbasis we obtain the discretized system corresponding to FESWE model
UTk︸︷︷︸
ku×N
M︸︷︷︸N×N
Uk︸︷︷︸N×ku
d
dtu + UT
k︸︷︷︸ku×N
N11(u)︸ ︷︷ ︸N×1
+ UTk︸︷︷︸
ku×N
N12(u, v)︸ ︷︷ ︸N×1
+ UTk︸︷︷︸
ku×N
Mdx︸︷︷︸N×N
Φk︸︷︷︸N×kφ
φ− UTk︸︷︷︸
ku×N
K13︸︷︷︸N×N
Vk︸︷︷︸N×kv
v = 0.
V Tk︸︷︷︸
kv×N
M︸︷︷︸N×N
Vk︸︷︷︸N×kv
d
dtv + V T
k︸︷︷︸kv×N
N21(u, v)︸ ︷︷ ︸N×1
+ V Tk︸︷︷︸
kv×N
N22(v)︸ ︷︷ ︸N×1
+ V Tk︸︷︷︸
kv×N
Mdy︸︷︷︸N×N
Φk︸︷︷︸N×kφ
φ + V Tk︸︷︷︸
kv×N
K13︸︷︷︸N×N
Uk︸︷︷︸N×ku
u = 0.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 48/101
5. POD FEM SWE Model
POD FEM SWE Model
ΦTk︸︷︷︸
kφ×N
M︸︷︷︸N×N
Φk︸︷︷︸N×kφ
d
dtφ + ΦT
k︸︷︷︸kφ×N
N31(u, φ)︸ ︷︷ ︸N×1
+ ΦTk︸︷︷︸
kφ×N
N32(v, φ)︸ ︷︷ ︸N×1
= 0.
where
[N11(u)]i =
∫Ωψi (x , y)Nf11(UPOD(x , y)u)dΩ = uT︸︷︷︸
1×ku
UTk︸︷︷︸
ku×N
A1︸︷︷︸i ,N×N
Uk︸︷︷︸N×ku
u︸︷︷︸ku×1
.
[N12(u, v)]i =
∫Ωψi (x , y)Nf12(UPOD(x , y)u,V POD(x , y)v)dΩ =
vT︸︷︷︸1×kv
V Tk︸︷︷︸
kv×N
A2︸︷︷︸i ,N×N
Uk︸︷︷︸N×ku
u︸︷︷︸ku×1
.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 49/101
5. POD FEM SWE Model
POD FEM SWE Model
[N21(u, v)]i =
∫Ωψi (x , y)Nf21(UPOD(x , y)u,V POD(x , y)v)dΩ =
uT︸︷︷︸1×ku
UTk︸︷︷︸
ku×N
A1︸︷︷︸i ,N×N
Vk︸︷︷︸N×kv
v︸︷︷︸kv×1
.
[N22(v)]i =
∫Ωψi (x , y)Nf22(V POD(x , y)v)dΩ = vT︸︷︷︸
1×kv
V Tk︸︷︷︸
kv×N
A2︸︷︷︸i ,N×N
Vk︸︷︷︸N×kv
v︸︷︷︸kv×1
.
[N31(u, φ)]i =
∫Ω
∂
∂xψi (x , y)Nf31(UPOD(x , y)u,ΦPOD(x , y)φ)dΩ =
uT︸︷︷︸1×ku
UTk︸︷︷︸
ku×N
A4︸︷︷︸i ,N×N
Φk︸︷︷︸N×kφ
φ︸︷︷︸kφ×1
.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 50/101
5. POD FEM SWE Model
POD FEM SWE Model
[N32(v, φ)]i =
∫Ω
∂
∂yψi (x , y)Nf32(V POD(x , y)v,ΦPOD(x , y)φ)dΩ =
vT︸︷︷︸1×kv
V Tk︸︷︷︸
kv×N
A5︸︷︷︸i ,N×N
Φk︸︷︷︸N×kφ
φ︸︷︷︸kφ×1
.
Nf11(u) = u∂u
∂x, Nf12(u, v) = v
∂u
∂y,
Nf21(u, v) = u∂v
∂x, Nf22(v) = v
∂v
∂y,
Nf31(u, φ) = uφ, Nf22(v , φ) = vφ.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 51/101
5. POD FEM SWE Model
POD FEM SWE Model
Next we introduce
Mk = UTk︸︷︷︸
ku×N
Muk︸︷︷︸N×ku
,
K k11(u) = UT
k︸︷︷︸ku×N
uTUTk A1Uk︸ ︷︷ ︸
N×ku
, K k12(v) = UT
k︸︷︷︸ku×N
vTV Tk A2Uk︸ ︷︷ ︸N×ku
,
Mdxk = UTk︸︷︷︸
ku×N
MdxΦk︸ ︷︷ ︸N×kφ
, K k13 = UT
k︸︷︷︸ku×N
K13Vk︸ ︷︷ ︸N×kv
.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 52/101
5. POD FEM SWE Model
POD FEM SWE Model
K k21(u) = V T
k︸︷︷︸kv×N
uTUTk A1Vk︸ ︷︷ ︸
N×kv
, K k22(v) = V T
k︸︷︷︸kv×N
vTV Tk A2Vk︸ ︷︷ ︸
N×kv
,
Mdyk = V Tk︸︷︷︸
kv×N
MdyΦk︸ ︷︷ ︸N×kφ
, K k23 = V T
k︸︷︷︸kv×N
K13Uk︸ ︷︷ ︸N×ku
.
K k31(u) = ΦT
k︸︷︷︸kφ×N
uTUTk A4Φk︸ ︷︷ ︸
N×kφ
, K k32(v) = ΦT
k︸︷︷︸kφ×N
vTV Tk A5Φk︸ ︷︷ ︸
N×kφ
.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 53/101
5. POD FEM SWE Model
Matrix form of the POD FEM SWE
Mkd
dtu + K k
11(u)u + K k12(v)u + Mdxkφ− K k
13v = 0,
Mkd
dtv + K k
21(u)v + K k22(v)v + Mdykφ− K k
13u = 0,
Mkd
dtφ− K k
31(u)φ− K k32(v)φ = 0.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 54/101
5. POD FEM SWE Model
Crank - Nicolson scheme
Let us define
un+ 12 = u∗ =
3
2un − 1
2un−1 + O(∆t2),
vn+ 12 = v∗ =
3
2vn − 1
2vn−1 + O(∆t2),
∆φ = φn+1 − φn, ∆u = un+1 − un, ∆v = vn+1 − vn
than the Crank - Nicolson scheme corresponding to PODFEM SWE model is
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 55/101
5. POD FEM SWE Model
Crank - Nicolson scheme
[2
∆tMk − K k
31(u∗)− K k32(v∗)
]∆φ = 2
[K k
31(u∗) + K k32(v∗)
]φn,[
2
∆tMk + K k
11(u∗) + K k12(v∗)
]∆u = −2
[K k
11(u∗) + K k12(v∗)
]un
−Mdxk
(φn+1
+ φn)
+ K k13v∗,[
2
∆tMk + K21(un+1) + K k
22(v∗)
]∆v = −2
[K k
21(un+1) + K k12(v∗)
]vn
−Mdyk
(φn+1
+ φn)− K k
13un+1.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 56/101
5. POD FEM SWE Model
The POD/EIM version of SWE model
In POD FEM SWE model we still have computationalcomplexities depending on the dimension N of the originalsystem from both evaluating the nonlinear functions andperforming matrix multiplications to project on POD bases.EIM is used to remove this dependency.The projected nonlinear functions can be approximated byEIM in a form that enables precomputation so that thecomputational cost is decreased and independent of theoriginal system.Only a few entries of the nonlinear term corresponding to thespecially selected interpolation points from EIM must beevaluated at each time step.EIM approximation is applied to each of the nonlinearfunctions Nf11,Nf12,Nf21,Nf22,Nf31,Nf32 defined in PODreduced model.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 57/101
5. POD FEM SWE Model
The POD/EIM version of SWE model
Let UNf11 ∈ RN×m, m ≤ N, be the POD basis matrix of rankm for snapshots of the nonlinear function Nf11 (obtained fromFE SWE model).
Using the EIM algorithm we select a set of m EIM pointscorresponding to UNf11 , denoting byzNf11 = [zNf11
1 , .., zNf11m ]T ∈ Rm and generate the EIM basis
QNf1111 (x , y). The EIM approximation of Nf11 is
Nf11(x , y) ≈ QNf1111 (x , y) QNf11
11 (zNf11) Nf11(UPOD(zNf11)u),
whereQNf11
11 (x , y) = [qNf111 (x), ..., qm(x)Nf11 ],
Q(zNf11) ∈ Rm×m, UPOD(zNf11) ∈ Rm×ku ,
Nf11(UPOD(zNf11)u) ∈ Rm×1
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 58/101
5. POD FEM SWE Model
The POD/EIM version of SWE model
Now the projected nonlinear term UTk N11(u) in the POD
reduced system can be approximated as
UTk N11(u) =
K k−EIM11︷ ︸︸ ︷
UTk︸︷︷︸
ku×N
∫Ω
Ψ(x , y)TQNf1111 (x , y)dΩ︸ ︷︷ ︸
N×m
·QNf1111 (zNf11)︸ ︷︷ ︸
m×m
·Nf11(UPOD(zNf11)u)︸ ︷︷ ︸m×1
.
Similarly we obtain the EIM approximation for the rest of theprojected nonlinear terms
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 59/101
The POD/EIM version of SWE model
UTk N12(u, v) =
K k−EIM12︷ ︸︸ ︷
UTk︸︷︷︸
ku×N
∫Ω
Ψ(x , y)TQNf1212 (x , y)dΩ︸ ︷︷ ︸
N×m
·QNf1212 (zNf12)︸ ︷︷ ︸
m×m
·
Nf12(UPOD(zNf12)u,V POD(zNf12)v)︸ ︷︷ ︸m×1
,
V Tk N21(u, v) =
K k−EIM21︷ ︸︸ ︷
V Tk︸︷︷︸
kv×N
∫Ω
Ψ(x , y)TQNf2121 (x , y)dΩ︸ ︷︷ ︸
N×m
·QNf2121 (zNf21)︸ ︷︷ ︸
m×m
·
Nf21(UPOD(zNf21)u,V POD(zNf21)v)︸ ︷︷ ︸m×1
,
The POD/EIM version of SWE model
V Tk N22(v) =
K k−EIM22︷ ︸︸ ︷
V Tk︸︷︷︸
kv×N
∫Ω
Ψ(x , y)TQNf2222 (x , y)dΩ︸ ︷︷ ︸
N×m
·QNf2222 (zNf22)︸ ︷︷ ︸
m×m
·
Nf22(V POD(zNf22)v)︸ ︷︷ ︸m×1
,
ΦTk N31(u, φ) =
︷ ︸︸ ︷ΦTk︸︷︷︸
kφ×N
∫Ω
∂
∂xΨ(x , y)
T
QNf3131 (x , y)dΩ︸ ︷︷ ︸
N×m
·QNf3131 (zNf31)︸ ︷︷ ︸
m×m
·
K k−EIM31
Nf31(UPOD(zNf31)u,ΦPOD(zNf31)φ)︸ ︷︷ ︸m×1
,
5. POD FEM SWE Model
The POD/EIM version of SWE model
ΦTk N32(v, φ) =
K k−EIM32︷ ︸︸ ︷
ΦTk︸︷︷︸
kφ×N
∫Ω
∂
∂yΨ(x , y)
T
QNf3232 (x , y)dΩ︸ ︷︷ ︸
N×m
·QNf3232 (zNf32)︸ ︷︷ ︸
m×m
·
Nf32(V POD(zNf32)v,ΦPOD(zNf32)φ)︸ ︷︷ ︸m×1
.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 62/101
Matrix form of the POD FEM SWE
Each of the K k−EIM11 , K k−EIM
12 , K k−EIM21 , K k−EIM
22 , K k−EIM31 ,
K k−EIM32 matrices can be precomputed and re-used at all time
steps, so that the computational complexity of theapproximate nonlinear terms are independent of the full-orderdimension N .
the POD/EIM FE SWE reduced system is
Mkd
dtu + K k−EIM
11 Nf11(UPOD(zNf11)u) + K k−EIM12 ·
Nf12(UPOD(zNf12)u,V POD(zNf12)v) + Mdxkφ− K k13v = 0,
Mkd
dtv+K k−EIM
21 Nf21(UPOD(zNf21)u,V POD(zNf21)v)+K k−EIM22 ·
Nf22(V POD(zNf22)v) + Mdykφ− K k−EIM13 u = 0,
Mkd
dtφ+K k−EIM
31 Nf31(UPOD(zNf31)u,ΦPOD(zNf31)φ)+K k−EIM32 ·
Nf32(V POD(zNf32)v,ΦPOD(zNf32)φ) = 0.
6. Numerical Results
Numerical Results
The domain was discretized using a mesh of 30× 23 points.Thus the dimension of the full-order discretized model is 690.The integration time window was 24h.
We employed linear piecewise polynomials on triangularelements in the formulation of Galerkin Finite-elementshallow-water equations model, in which the global matrix wasstored into a compact matrix.
The initial condition were derived from the geopotentialheight formulation introduced by Grammelvedt (1969) usingthe geostrophic balance relationship.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 64/101
6. Numerical Results
Numerical Results
18000 18000 18000
18500 18500
18500
1850019000
1900019000
19500
1950019500
20000
2000020000
20500
20500 20500
21000
21000 2100021500
21500
21500 21500
22000 22000 22000
Contour of geopotential from 22000 to 18000 by 500
y(k
m)
x(km)0 1000 2000 3000 4000 5000
0
500
1000
1500
2000
2500
3000
3500
4000
0 1000 2000 3000 4000 5000 6000−500
0
500
1000
1500
2000
2500
3000
3500
4000
4500 Wind field
y(k
m)
x(km)
Figure 5: Initial condition: Geopotential height field for theGrammeltvedt initial condition (left). Wind field calculated from thegeopotential field by using the geostrophic approximation (right).
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 65/101
6. Numerical Results
Numerical Results
18000
18000 18000
18500
18500
18500
18500
19000
19000
19000
19500
19500
19500
20000
20000
20000
2050020500
20500
21000 21000
2100021500
21500
21500
2200022000
22000
Contour of geopotential from 22000 to 18000 by 500
y(k
m)
x(km)0 1000 2000 3000 4000 5000
0
500
1000
1500
2000
2500
3000
3500
4000
0 1000 2000 3000 4000 5000 60000
500
1000
1500
2000
2500
3000
3500
4000
4500 Wind field
y(k
m)
x(km)
Figure 6: The geopotential field (left) and the wind field at t = tf = 24hobtained using the FE SWE scheme for ∆t = 75s.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 66/101
6. Numerical Results
Numerical Results
The POD basis vectors were constructed using 1152snapshots obtained from the numerical solution of the full -order FE SWE model at equally spaced time steps in theinterval [0 24h].
The dimension of the POD bases for each variable was taken50, capturing more than 99.9% of the system energy.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 67/101
6. Numerical Results
Numerical Results
0 50 100 150 200 250 300 350 400−30
−20
−10
0
10
20
30 Singular Values of Snapshots Solution
Number of snapshots
log
arit
hm
ic s
cale
uvΦ
0 50 100 150 200 250 300 350 400−50
−40
−30
−20
−10
0
10
20 Singular Values of Nonlinear Snapshots Solution
Number of snapshots
log
arit
hm
ic s
cale
Nf11Nf12Nf21Nf22Nf31Nf32
Figure 7: The decay around the singular values of the snapshots solutionsfor u, v , φ and nonlinear functions (∆t = 75s).
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 68/101
Numerical Results
We applied the EIM algorithm for interpolation indices toimprove the efficiency of the POD approximation and toachieve a complexity reduction of the nonlinear terms with acomplexity proportional to the number of reduced variables.
0 1000 2000 3000 4000 5000 60000
500
1000
1500
2000
2500
3000
3500
4000
4500
1
23 4
5
6
7
8
9
10
11
1213
1415
16
17
18
19
20
21
22
23
24
25
26
2728
2930
31
32
33
34
3536
3738
39
40
EIM POINTS for NF31
x(km)
y(km
)
1
23 4
5
6
7
8
9
10
11
1213
1415
16
17
18
19
20
21
22
23
24
25
26
2728
2930
31
32
33
34
3536
3738
39
40
0 1000 2000 3000 4000 5000 60000
500
1000
1500
2000
2500
3000
3500
4000
4500
1234
5
67
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
2324
25
2627
28
29
30
31
32
33
3435
36
37
38
39
40
EIM POINTS for Nf32
x(km)
y(km
)
1234
5
67
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
2324
25
2627
28
29
30
31
32
33
3435
36
37
38
39
40
Figure 8: First 40 points selected by EIM for the nonlinear functions Nf31
(left) and Nf32 (right)
6. Numerical Results
Numerical Results
The dimension of EIM bases was chosen to be equal with 140
We emphasize the performances of POD - EIM method incomparison with the POD approach using the numericalsolution of the full FE SWE model. Next three slides depictthe space error behaviors between POD/POD - EIM solutionand FE SWE solution.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 70/101
6. Numerical Results
Numerical Results
POD errors φPOD
−φFEM
x(km)
y(km
)
0 1000 2000 3000 4000 50000
500
1000
1500
2000
2500
3000
3500
4000
−8
−6
−4
−2
0
2
4
6
8
10
x 10−6 POD/EIM errors φ
POD/EIM−φ
FEM
x(km)
y(km
)
0 1000 2000 3000 4000 50000
500
1000
1500
2000
2500
3000
3500
4000
−4
−2
0
2
4
6
x 10−3
Figure 9: Errors between the geopotential calculated with POD/POD-EIM and geopotential determined with the FE SWE model at t = 24h(∆t = 75s). The number of DEIM points was taken 140.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 71/101
6. Numerical Results
Numerical Results
POD errors uPOD
−uFEM
x(km)
y(km
)
0 1000 2000 3000 4000 50000
500
1000
1500
2000
2500
3000
3500
4000
−1.5
−1
−0.5
0
0.5
1
x 10−6 POD/EIM errors u
POD/EIM−u
FEM
x(km)
y(km
)
0 1000 2000 3000 4000 50000
500
1000
1500
2000
2500
3000
3500
4000
−1.5
−1
−0.5
0
0.5
1
1.5
2
x 10−4
Figure 10: Errors between u calculated with POD/POD -EIM and udetermined with the FE SWE model at t = 24h (∆t = 75s). Thenumber of EIM points was taken 140.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 72/101
6. Numerical Results
Numerical Results
POD errors vPOD
−vFEM
x(km)
y(km
)
0 1000 2000 3000 4000 50000
500
1000
1500
2000
2500
3000
3500
4000
−10
−8
−6
−4
−2
0
2
x 10−6 POD/EIM errors v
POD/EIM−v
FEM
x(km)
y(km
)
0 1000 2000 3000 4000 50000
500
1000
1500
2000
2500
3000
3500
4000
−1.5
−1
−0.5
0
0.5
1
1.5
2
x 10−3
Figure 11: Errors between v calculated with POD/POD -EIM and vdetermined with the FE SWE model at t = 24h (∆t = 75s). Thenumber of DEIM points was taken 140.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 73/101
6. Numerical Results
Numerical Results
FD SWE POD SWE POD-EIM SWE
CPU time 40.938 10.933 0.8785
φ - 1.8522 · 10−6 1.6532 · 10−3
u - 5.2427 · 10−7 7.0129 · 10−5
v - 1.2073 · 10−6 6.20619 · 10−4
Table 1: CPU time gains and the root mean square errors for each of themodel variables. The POD bases dimensions were taken 50 capturingmore than 99.9% of the system energy. 140 EIM points were chosen.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 74/101
7. Reduced order POD 4-D VAR
Reduced order POD 4-D VAR
The reduced-order cost functional can be expressed as
JPOD(yPOD
0
)=
1
2
(yPOD
0 − yb)T
B−1(yPOD
0 − yb)
+
1
2
k=n∑k=0
(Hky
PODk − yok
)TR−1k
(Hky
PODk − yok
)B background error covariance matrix
Rk observation error covariance matrix at time level k
Hk observation operator at time level k
yPOD0 control variables vector represented by POD basis
yPODk vector of variables obtained from the reduced-order
model at the time level k
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 75/101
7. Reduced order POD 4-D VAR
Reduced order POD 4-D VAR
The initial value yPOD0 and the reduced-order model solution
yPODk can be expressed as
yPOD0 =
∑i=Mi=1 αi (0)φi = Φα0
yPODk =
∑i=Mi=1 αi
(tk)φi = Φαk
where Φ =φ1, φ2, . . . , φM
is an ensemble of POD basis.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 76/101
7. Reduced order POD 4-D VAR
Reduced order POD 4-D VAR
We can rewrite the reduced - order cost functional as follows
JPODα (α0) =
1
2
(Φα0 − yb
)TB−1
(Φα0 − yb
)+
1
2
k=n∑k=0
(Hk (Φαk)− yok )T R−1k (Hk (Φαk)− yok )
The reduced model can be written as
αk = MPOD0→k (α0) ,∀k
αk = MPODk−1→k (αk−1) = MPOD
k (αk−1) , ∀k
and by recurrence
αk = MPODk · · ·MPOD
1 α0, ∀k
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 77/101
7. Reduced order POD 4-D VAR
Reduced order POD 4-D VAR
The reduced-order cost functional JPODα (α0) can be divided
into two components
JPODα = JPOD,b
α + JPOD,oα
and more,
JPODα = JPOD,b
α +n∑
k=0
JPOD,oα,k
where JPOD,oα,k = (Hk (Φαk)− yok )T dk and dk denotes the
’normalized departure’
dk = R−1k (Hk (Φαk)− yok ) .
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 78/101
7. Reduced order POD 4-D VAR
Reduced order POD 4-D VAR
Hence the gradient of the POD reduced-order cost functionalw.r.t. α0 is written as
∇α0JPODα = ΦTB−1
(Φα0 − yb
)+
n∑k=0
(MPOD
1
)T. . .(
MPODk
)TΦTHT
k dk
where(MPOD
k
)Tis the POD reduced-order adjoint model at
time step k .
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 79/101
Pseudo - Algorithmic form
1 Initialize reduced-order adjoint variables α∗ at final time tozero: α∗n = 0
2 For each step k − 1, adjoint variables α∗k−1 are obtained by
adding reduced-order adjoint forcing term ΦTHTk dk to α∗kand
performing reduced-order adjoint integration by multiplying
result by(MPOD
k
)T, i.e. α∗k−1 =
(MPOD
k
)T (α∗k + ΦTHT
k dk)
3 At the end of recurrence, the value of adjoint variableα∗0 = Joα0
yields the gradient of the observational costfunctional
4 Compute
∇α0JPOD,bα = ΦTB−1
(Φα0 − yb
)obtaining
∇α0JPODα = ∇α0J
POD,bα +∇α0J
POD,oα .
8. Trust Region POD
Trust region POD optimal control approach
The Trust - Region algorithm hooks direction of descent andstep-size simultaneously. It approximate a certain region, thetrust region (a sphere in Rn of the objective function with aquadratic model function
mk (p) = fk +∇f Tk +
1
2pTBkp, where
fk = f (xk) , ∇fk = ∇f (xk) and Bk is an approximation to the Hessian.
We seek a solution of
min mk (p) =fk +∇f Tk +
1
2pTBkp
s.t ‖p‖ ≤ δk ,
where δk > 0 is the trust-region radius.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 81/101
8. Trust Region POD
Trust region POD optimal control approach
The trust-region radius δk at each iteration is determined byanalyzing the following ratio
ρk =f (xk)− f (xk + pk)
mk (0)−mk (pk).
If ρk < 0, the new objective value is greater than the currentvalue so that the step must be rejected.
If ρk is close to 1, there is good agreement between theapproximate model mk and the object function fk over thisstep, so it is safe to expand the trust region radius for thenext iteration
If ρk is positive but not close to 1, we do not alter the trustregion radius, but if it is close to zero or negative, we shrinkthe trust region radius.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 82/101
9. T-R POD algorithm
Trust region POD 4D-VAR algorithm I
Let 0 < η1 < η2 < 1 , 0 < γ1 < γ2 < 1≤ γ3 and y(0)0 , δ0 be given,
set k = 0
1 Compute snapshot set YSNAPk based on initial condition y
(k)0
2 Compute the POD basis Φ(k) and build up the correspondingPOD based control model based on the initial conditionα
(0)0 =
⟨y
(0)0 ,Φ(0)
⟩3 Compute the minimizer sk of
min mk
(α
(k)0 + s
)= JPOD
α
(α
(k)0 + s
)subject to ‖s‖ ≤ δk
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 83/101
9. T-R POD algorithm
Trust region POD 4D-VAR algorithm II
4 Compute the new J(
Φ(k−1)(α
(k)0 + sk
))of the full model
and
ρk =J(
Φ(k−1)α(k)0
)− J
(Φ(k−1)
(α
(k)0 + sk
))mk
(α
(k)0
)−mk
(α
(k)0 + sk
)5 Update the trust-region radius:
If ρk ≥ η2: implement outer projection
y(k+1)0 = y + Φ(k−1)
(α
(k)0 + sk
)and increase trust-region
radius δk+1 = γ3δk and GOTO 1
If η1 < ρk < η2: implement outer iteration
y(k+1)0 = y + Φ(k−1)
(α
(k)0 + sk
)and decrease trust-region
radius δk+1 = γ2δk and GOTO 1
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 84/101
9. T-R POD algorithm
Trust region POD 4D-VAR algorithm III
If ρk ≤ η1: set y(k+1)0 = y
(k)0 and decrease trust-region radius
δk+1 = γ1δk and GOTO 3
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 85/101
10. Illustration
Numerical Results
18000 18000 18000
18500 18500
18500
19000 19000
19000
19500 19500
1950020000
20000
2000020500
20500
205002100021000
21000
21500
21500 21500
22000 22000 22000
Contour of geopotential from 22000 to 18000 by 500
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
Wind field calculated from the geopotential field by geostrophic approximation
Figure 12: Initial condition:(a) Geopotential field for the Grammeltvedtinitial condition. (b) Wind field calculated from the geopotential field bythe geostrophic approximation.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 86/101
Numerical Results
We employed linear piecewise polynomials on triangularelements in the formulation of Galerkin finite-elementshallow-water equations model, in which the global matrix wasstored into a compact matrix.
A time-extrapolated Crank-Nicholson time differencingscheme was applied for integrating in time the system ofordinary differential equations.
The Galerkin finite-element boundary conditions were treatedusing the approach suggested by Payne and Irons (1963) andmentioned by Huebner (1975), i.e. modifying the diagonalterms of the global matrix associated withthe nodal variablesby multiplying them by a large number, say 1016, while thecorresponding term in the right-hand vector is replaced by thespecified boundary nodal variable multiplied by the same largefactor times the corresponding diagonal term.
Numerical Results
We applied a 1% uniform random perturbations on the initialconditions in order to provide twin-experiment “observations”.
The data assimilation was carried on a 48 hours window usingthe ∆t = 1800s in time and a mesh of 30× 24 grid points inspace with ∆x = ∆y = 200km.
We generated 96 snapshots by integrating the fullfinite-element shallow-water equations model forward in time,from which we choose 10 POD bases for each of theu(x , y),v(x , y),and φ(x , y) to capture over 99.9% of theenergy.
The dimension of control variables vector for thereduced-order 4-D Var is 10× 3 = 30.
Numerical Results
The Polak Ribiere nonlinear conjugate gradient (CG)technique was employed for high-fidelity full model 4-D VARand all variants of ad-hoc POD 4-D Var, while thesteepest-descent strategy was used in the trust-region POD4-D Var.
In the ad-hoc POD 4-D Var, the POD bases are re-calculatedwhen the value of the cost function cannot be decreased bymore than 10−1 for ad-hoc POD 4-D Var and 10−2 for ad-hocDWPOD 4-D Var between the consecutive minimizationiterations.
In the trust-region 4-D Var, the POD bases are re-calculatedwhen the ratio ρk is larger than the trust-region parameter η1
in the process of updating the trust-region radius.
0 10 20 30 40 50 60 70 80−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
iterations
log(
cost
/cos
t 0)
Full 4D−VarUW ad−hoc POD 4D−VarDW ad−hoc POD 4D−VarUW TRPOD 4D−VarDW TRPOD 4D−Var
Figure 13: Comparison of the performance of minimization of costfunctional in terms of number of iterations for ad-hoc POD 4-D Var,ad-hoc dual weighed POD 4-D Var, trust-region POD 4-D Var,trust-region dual weighed POD 4-D Var and the full model 4-D Var.
To quantify the performance of the dual weighted trust-region4-D Var, we use two metrics namely the root mean squareerror (RMSE) and correlation of the difference between thePOD reduced-order simulation and high-fidelity model.
POD 4-D Var ADPOD DWAHPOD TRPOD DWTRPOD Full
Iterations 22 42 46 57 80
Outer projections 2 6 10 12 N/A
Error 10−1 10−2 10−5 10−8 10−10
CPU time (s) 15.2 38.7 121.2 142.8 222.6
Table 2: Comparison of iterations, outer projections, RMSE andCPU time for ad-hoc POD 4-D Var, ad-hoc dual weighed POD 4-DVar, trust-region POD 4-D Var, trust-region dual weighed POD 4-DVar and the full model 4-D Var.
Next image depicts the errors between the retrieved initialgeopotential and true initial geopotential applying dualweighted trust-region POD 4-D Var to the 5% uniformrandom perturbations of the true initial conditions taken asinitial guess.
17954
17954
17954
17954
18475
18475
18997
18997
19518
19518
20039
20039
20560
20560
21081
21081
21602
21602
22124 2212422124
22124
x−axis
y−ax
is
The contour of 5% perturbation of true initial geopotential
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−332
−332
−332
−115
−115
−115
−115
−115
−115
−115
−115
−115
−115
−115
−115
−115
−115−115
−115
−115
−115
−115
−115 −115
−115
102
102
102102
102
102
102
102
102
102
102
102
102 102102
102
102
102
102102
102
102
102
102
102
102102
102102319
319
319
319
319
319
319
319
319
319
319
319
319
319
319
x−axis
y−ax
is
The contour of difference between 5% perturbation of true initial geopotential
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1821718217
18666
18666
19115
19115
19565
19565
20014
20014
20463
20463
20913
209132136
2
21362
2181121811
x−axis
y−ax
is
The contour of retrieved initial geopotential(Window = 2(days), dt = 1800s)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−66
−66
−66
−66
−66
−66
−66
−66
−66
−66
−66
−22
−22
−22
−22
−22
−22
−22
−22
−22−22
−22
−22
−22
−22
−22
−22−22
−22
−22
−22
−22
−22
−22−
22
−22
−22
−22
−22
−22
−22
22
22
22
2222
22
22
2222
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
22
6666
66
66
66
66
66
66
66
66
66
6666
66
66
66
66
66
66
x−axis
y−ax
is
The contour of difference between retrieved initial geopotential and true initial geopotential
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Numerical Results
The correlation coefficient r used to evaluate quality of theinversion simulation is defined below
ri =cov i12
σi1σi2
,
where
σi1 =
j=N∑j=1
(Ui ,j − U j
)2, σ2 =
j=N∑j=1
(UPODi ,j − UPOD
j
)2, i , j = 1, . . . , n
cov12 =
j=N∑j=1
(Ui ,j − U j
) (UPODi ,j − UPOD
j
), i , j = 1, . . . , n
with U j and UPODj are the means over the simulation period
[0,T ] obtained from the full model and ones obtained byoptimal POD reduced-order model at node j , respectively.
10. Illustration
Numerical Results
0 20 40 60 80 1000.993
0.994
0.995
0.996
0.997
0.998
0.999
1
1.001
time steps
Cor
rela
tion
CORRELATION of geopotential before the DWTRPODCORRELATION of geopotential after the DWTRPOD
0 20 40 60 80 1000.982
0.984
0.986
0.988
0.99
0.992
0.994
0.996
0.998
1
1.002
time steps
Cor
rela
tion
CORRELATION of velocity before the DWTRPODCORRELATION of velocity after the DWTRPOD
Figure 14: Comparison of the correlation between the full model and theROM before and after data assimilation applying dual weightedtrust-region POD 4-D Var to the 5% uniform random perturbations ofthe true initial conditions serving as initial guess: geopotential (left),wind field (right).
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 94/101
11. Conclusions
Conclusion and future research
To obtain the approximate solution from both POD andPOD-EIM reduced systems, one must store POD or POD-EIMsolutions of order O(kNT ) and POD matrices of order O(Nk),k being the POD bases dimension, NT the number of timesteps in the integration window and N the space dimension.
The coefficient matrices that must be retained while solvingthe POD reduced system are of order of O(k2) for projectedlinear terms and O(Nk) for the nonlinear term.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 95/101
11. Conclusions
Conclusion and future research
In the case of solving the POD-EIM reduced system thecoefficient matrices that need to be stored are of order ofO(k2) for projected linear terms and O(mk) for the nonlinearterms, where m is the number of EIM points, m N.
EIM therefore improves the efficiency of the PODapproximation and achieves a complexity reduction of thenonlinear term with a complexity proportional to the numberof reduced variables.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 96/101
11. Conclusions
Conclusions
We compared several variants of POD 4-D Var, namelyunweighted ad-hoc POD 4-D Var, dual-weighed ad-hoc POD4-D Var, unweighted trust-region POD 4-D Var anddual-weighed trust-region POD 4-D Var, respectively.
We found that the ad-hoc POD 4-D Var version yielded theleast reduction of the cost functional compared with thetrust-region 4-D VAR . We assume that this result may beattributed to lack of feedbacks from the high-fidelity model.
The trust-region POD 4-D Var version yielded a sizably betterreduction of the cost functional, due to inherent properties ofTRPOD allowing local minimizer of the full problem to beattained by minimizing the TRPOD sub-problem. Thustrust-region 4-D Var resulted in global convergence to thehigh fidelity local minimum starting from any initial iterates.
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 97/101
11. Conclusions
Burger Equations
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 Finite Element Solution of 1D Burger Equation
y(x,1)y(x,2)y(x,3)y(x,5)y(x,51)
0 2 4 6 8 10 12 14 16 18−40
−35
−30
−25
−20
−15
−10
−5
0
5 Singular Values of Snapshots Solution
loga
rithm
ic s
cale
Number of Snapshots
y(x,t)
Figure 15
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 98/101
11. Conclusions
Burger Equations
0 2 4 6 8 10 12 14 16 18−18
−16
−14
−12
−10
−8
−6
−4
−2
0
2 Singular Values of Nonlinear Snapshots
loga
rithm
ic s
cale
Number of Snapshots
0.5*y(x,t)2
5 10 15 20 25 30 35 40 45−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1 EIM points for 0.5*y(x,t)2
EIM points
Figure 16
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 99/101
11. Conclusions
Burger Equations
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6x 10
−4 POD error and POD/EIM error
|yPOD(25,x)−yFEM(25,x)||yPOD/EIM(25,x)−yFEM(25,x)||yPOD(51,x)−yFEM(51,x)||yPOD/EIM(51,x)−yFEM(51,x)|
0 5 10 150
0.05
0.1
0.15
0.2
0.25 Euclidian Error of Burger Solution at t=0.48
POD dimension
Eu
clid
ian
No
rm
PODEIM5EIM25EIM45
Figure 17
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 100/101
11. Conclusions
Burger Equations
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 CPU time
POD dimension
tim
e(se
con
ds)
PODEIM5EIM25EIM45
0 5 10 150
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09 Euclidian Error of Burger Solution at t=1
POD dimension
Eu
clid
ian
No
rm
PODEIM5EIM25EIM45
Figure 18
R. Stefanescu1 and I. M. Navon2
POD/DEIM POD/EIM nonlinear model reduction and POD/4-D VAR with TR for models of the SWE 101/101