proper generalized decomposition for stochastic navier ......proper generalized decomposition for...
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Proper Generalized Decomposition for Stochastic
Navier–Stokes Equations
Lorenzo Tamellini],†
Olivier Le Maitre[, Anthony Nouy§
] MOX - Department of Mathematics, Politecnico di Milano, Italy† CSQI - MATHICSE, EPFL, Lausanne, Switzerland
[ LIMSI -CNRS, Orsay, France§ Ecole Centrale Nantes, France
SIAM Conference on Uncertainty Quantification
Raleigh, April 2-5, 2011
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 1 / 21
Outline
1 PGD general principles and algorithms
2 PGD for Navier–Stokes equations
3 Pressure reconstruction and residual computation
4 Conclusions
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 2 / 21
Notation and setting
a(u, v ; y) = b(v ; y), u, v : D ⊂ Rd1 → Rd2 , u, v ∈ V(D)
y is a set of N i.i.d. random parameters, y ∈ Γ = Γ1× Γ2× . . . ⊂ RN , jointp.d.f. %Γ(y) =
∏Nn=1 %n(yn).
Stochastic problem.
Find U ∈ V ⊗ L2%(Γ), such that
A(U,V) = B(V), ∀V ∈ V ⊗ L2%(Γ)
with A(U,V) = E[a(U,V ; y)], B(V) = E[b(V ; y)].
In many cases y→ U(x, y) is smooth. Calls for polynomial approximationof U! Beware of curse of dimensionality effect.
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 3 / 21
Notation and setting
a(u, v ; y) = b(v ; y), u, v : D ⊂ Rd1 → Rd2 , u, v ∈ V(D)
y is a set of N i.i.d. random parameters, y ∈ Γ = Γ1× Γ2× . . . ⊂ RN , jointp.d.f. %Γ(y) =
∏Nn=1 %n(yn).
Stochastic problem.
Find U ∈ V ⊗ P(Γ), such that
A(U,V) = B(V), ∀V ∈ V ⊗ P(Γ)
with A(U,V) = E[a(U,V ; y)], B(V) = E[b(V ; y)].
In many cases y→ U(x, y) is smooth. Calls for polynomial approximationof U! Beware of curse of dimensionality effect.
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 3 / 21
Modal polynomial approximation
KL UKL(x, y) =∑N
i=1 bi (x)y • L2%(Γ) optimal
• bi are the eigenvectors of corr(U)• need to know U
GPCE UGAL(x, y) =∑M
i=1 ui (x)Li (y) • Li %-orth., may not be optimal
(with Galerkin procedure) • no reuse of code• Need for suitable polynomial spaces• coupled (huge) system:
storage, preconditioners?
PGD UPGD(x, y) =∑m
i=1 ui(x)λi(y) • need to compute both ui ,λi
We aim at
reuse of code
lower computational costs than Galerkin
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 4 / 21
Principles and algorithmsPGD approximation
We look for Um(x, y) =∑m
i=1 ui (x)λi (y) approximating the solution of
Find U ∈ V ⊗ P(Γ) : A (U, vβ) = B(vβ), ∀(v, β) ∈ V × P(Γ)
We build solutions one couple at a time
Given m − 1 couples, the next one solves
A(Um−1 + umλm , umβ + vλm
)= B(umβ + vλm), ∀(v, β) ∈ V × P(Γ)
that is, simultaneously
det. pb: um = D(λm ; Um−1), A(Um−1 + umλm, vλm
)= B(vλm) ∀v ∈ V
stoc. pb: λm = S(um ; Um−1), A(Um−1 + umλm,um β
)= B(umβ) ∀β ∈ P(Γ)
Then, iterate until a fixed point is reached:
um = D(S(um)).
In practice, only a finite number of iterations is performed.
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 5 / 21
Observations
For linear symmetric positive definite problems the PGD solution is optimalw.r.t. the norm: ‖V‖A = E [a(V,V)]
Arbitrary normalization of one between ui and λi :cui ,
1c λi is also a solution, c 6= 0.
Symmetry: could think in terms of λm = Sm−1(Dm−1(λm))
Det. and stoc. pb. uncoupled! Can reuse most code
Let C be the cost of solving a det. pb.
PGD cost m ×#pow-it×[C + stoc. cost(M)
]Gal. cost #PCG-it×M × C
PGD cost possibly (much) lower!
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 6 / 21
Update problem
Once obtained Um, the quality of the solution can be improved by keeping ui andrecomputing λi
upd.pb : [λ1, . . . , λm] = Ui (u1, . . . ,um),
A
(m∑
i=1
uiλi ,ujβ
)= B(ujβ), ∀β ∈ P(Γ), ∀j = 1, . . . ,m.
1: U← 02: for i in 1, 2, . . . ,m do3: Initialize λ [e.g. at random]4: repeat5: Solve deterministic problem: u← Di−1(λ ; U)6: Normalize u: u← u/‖u‖V7: Solve stochastic problem: λ← Si−1(u ; U)8: until (u, λ) converged9: U← U + uλ
10: end for
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 7 / 21
Update problem
Once obtained Um, the quality of the solution can be improved by keeping ui andrecomputing λi
upd.pb : [λ1, . . . , λm] = Ui (u1, . . . ,um),
A
(m∑
i=1
uiλi ,ujβ
)= B(ujβ), ∀β ∈ P(Γ), ∀j = 1, . . . ,m.
1: U← 02: for i in 1, 2, . . . ,m do3: Initialize λ [e.g. at random]4: repeat5: Solve deterministic problem: u← Di−1(λ ; U)6: Normalize u: u← u/‖u‖V7: Solve stochastic problem: λ← Si−1(u ; U)8: until (u, λ) converged9: Solve update problem: [λ1, . . . , λi ]← Ui (u1, . . . ,ui )
10: U← U + uλ11: end for
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 7 / 21
Construction with Arnoldi procedure
Generate first m orthonormal modes ui (Gram–Schmidt), then the correspondingstoc. modes (as in update).
1: U← 02: Initialize λ [e.g. at random]3: for i = 1 to m do4: Solve deterministic problem u∗ ← Di−1(λ ; U)
5: Orthogonalize u∗: u← u∗ −∑l−1
k=1(uk ,u∗)V6: Normalize u: u← u/‖u‖V7: Solve stochastic problem: λ← Si−1(u ; U)8: Save ui ← u9: end for
10: Solve update problem: [λ1, . . . , λi ]← Ui (u1, . . . ,um)11: U←
∑mi=1 uiλi
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 8 / 21
More sophisticated version, handles the case ‖u∗‖ < ε (“stagnation”)
1: l ← 0 [initialize counter for modes]2: U← 03: Initialize λ [e.g. at random]4: while l < m do5: l ← l + 16: Solve deterministic problem u∗ ← D(λ,U)
7: Orthogonalize u∗: u← u∗ −Pl−1
k=1(uk , u∗)V
8: if ‖u∗‖V < ε then
9: l ← l − 1 [stagnation of Arnoldi detected]10: Solve update problem: [λ1, . . . , λi ]← Ui (u1, . . . , ui )
11: U←Pl
k=1 ukλk
12: else13: Normalize u: u← u/‖u‖V14: Solve stochastic problem: λ← S(u; U)15: Save ul ← u16: Save λl ← λ17: if l = m then18: Solve update problem: [λ1, . . . , λi ]← Ui (u1, . . . , ui )
19: U←Pl
k=1 ukλk
20: end if21: end if22: end while
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 9 / 21
Navier–Stokes equations
Navier–Stokes equations
Find (u, p) ∈ H10(D)× L2
0(D) such that
c(u,u, v) + ν v(u, v) + d(p, v) = b(v), ∀v ∈ H10(D)
d(q,u) = 0, ∀q ∈ L20(D),
Definitions
H10(D) =
{v ∈ H1(D), v = 0 on ∂D
}, L2
0(D) =
{q ∈ L2(D) :
∫D
qdx = 0
}.
c(u,w, v) =
∫D
(u ·∇w) · vdx v(u, v) =
∫D
∇u : ∇vdx,
d(p, v) = −∫
D
p∇ · vdx, b(v) =
∫D
f · vdx.
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 10 / 21
Navier–Stokes equations
Divergence-free Navier–Stokes equations
Find u ∈ H10,div (D) such that
c(u,u, v) + ν v(u, v) = b(v), ∀v ∈ H10,div (D).
Definitions
H10(D) =
{v ∈ H1(D), v = 0 on ∂D
}, L2
0(D) =
{q ∈ L2(D) :
∫D
qdx = 0
}.
c(u,w, v) =
∫D
(u ·∇w) · vdx v(u, v) =
∫D
∇u : ∇vdx,
d(p, v) = −∫
D
p∇ · vdx, b(v) =
∫D
f · vdx.
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 10 / 21
Navier–Stokes equations
Stochastic Navier–Stokes problem
Find U ∈ H10,div (D)⊗ P(Γ) such that
E [ c(U,U,V) ] + E [ ν(y) v(U,V) ] = E [ b(V) ] , ∀V ∈ H10,div (D)⊗ P(Γ).
Definitions
H10(D) =
{v ∈ H1(D), v = 0 on ∂D
}, L2
0(D) =
{q ∈ L2(D) :
∫D
qdx = 0
}.
c(u,w, v) =
∫D
(u ·∇w) · vdx v(u, v) =
∫D
∇u : ∇vdx,
d(p, v) = −∫
D
p∇ · vdx, b(v) =
∫D
f(y) · vdx.
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 10 / 21
Deterministic problem
Find um ∈ H10,div (D) such that
E[c(Um−1+ umλm, Um−1+ umλm, vλm )
]+ E
[ν(y) v(Um−1+ umλm, vλm)
]= E [ b(vλm) ] , ∀v ∈ H1
0,div (D).
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 11 / 21
Deterministic problem
Find um ∈ H10,div (D) such that
c(u,u, v) + c (u, vm−1c , v) + c (vm−1
c , u, v)
+ ν v(u, v) = b(v; Um−1), ∀v ∈ H10,div (D).
Each mode ui is divergence free
In practice
This is obtained solving the mixed-form problem for velocity and pressure
vm−1c (λ) =
m−1∑i=1
E[λ2λi
]E [λ3]
ui , ν =E[νλ2]
E [λ3]
b(v; Um−1, λ) =E [λ b(v ; f)]
E [λ3]−
m−1∑i=1
E [λνλi ]
E [λ3]v(ui , v)−
m−1∑i=1
m−1∑j=1
E [λλiλj ]
E [λ3]c(ui ,uj , v)
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 11 / 21
Stochastic problem
Find λm ∈ P(Γ) such that
E[c(Um−1+ umλm, Um−1+ umλm, umβ )
]+ E
[ν(y) v(Um−1+ umλm, umβ)
]= E [ b(umβ) ] , ∀β ∈ P(Γ).
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 12 / 21
Stochastic problem
Find λm ∈ P(Γ) such that
E[λ2β
]c(u,u,u) +
m∑i=1
E [λλiβ] ( c(ui ,u,u) + c(ui ,u,u) ) + E [νλβ] v(u,u) =
E [β b(u ; f)]−m∑
i,j=1
E [λiλjβ] c(ui ,uj ,u)−m∑
i=1
E [νλiβ] v(ui ,u) ∀β ∈ PM(Γ).
in practice
λ ∈ PM(Γ), λ =∑M
k=0 λkHk . Then, choose β = Hl and solve a set of M quadratic
equations in λk .
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 12 / 21
Problem setting
Arnoldi method
Space discretization: PNu − PNu−2 spectral method on tensorizedGauss–Lobatto points
Forcing term with random rotational:
∇ ∧ F = (0, 0, Φ(x, y))T .Next, we expand
Φ(x, ω) = Φ0 +N−1∑i=1
Φi (x)yi , yi ∼ N(0, 1)
Therefore
F(x, y) = F0 +N−1∑i=1
Fi (x)yi , Fi (x) = ∇ ∧ (0, 0, ∆−1[Φi (x)])T .
Random Reynolds: ν(y) = ν0 + ν′eσyN , yN ∼ N(0, 1)
N = 4, 8, 15
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 13 / 21
Results
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
−4
−3
−2
−1
0
1
2
mean of rotational field
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
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
st. dev. of rotational field
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 14 / 21
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
−20
−15
−10
−5
0
5
10
u1 and ∇ ∧ u1
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
−40
−30
−20
−10
0
10
20
30
40
50
u2 and ∇ ∧ u2
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
−20
−10
0
10
20
30
40
50
u5 and ∇ ∧ u5
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
−50
−40
−30
−20
−10
0
10
20
30
40
50
u8 and ∇ ∧ u8
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 15 / 21
0 10 20 30 40 5010
−10
10−8
10−6
10−4
10−2
100
m=M
ν=1/10, ||uG−uPGD||
ν=1/10, err. est.ν=1/50, ||uG−uPGD||
ν=1/50, err. est.ν=1/100, ||uG−uPGD||
ν=1/100, err. est.
N = 4,M = 150 10 20 30 40 50
10−10
10−8
10−6
10−4
10−2
100
m=M
ν=1/10, ||uG−uPGD||
ν=1/10, err. est.ν=1/50, ||uG−uPGD||
ν=1/50, err. est.ν=1/100, ||uG−uPGD||
ν=1/100, est.
N = 8,M = 45
“A posteriori” error estimate
‖Um −UGAL‖ ≈ ‖λm‖/√∑
i
‖λi‖2
0 10 20 30 40 50 6010
−8
10−6
10−4
10−2
100
ν=1/10, err. est.ν=1/50, err. est.ν=1/100, err. est.
N = 15,M = 861
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 16 / 21
Residual & Pressure
Pressure is needed for residual computation
More sound error estimate are based on residual computation. However, the residualin H1
0,div (D) is not easy to compute numerically.
We could measure the residual in H10(D) but we miss the pressure!
PGD representation of pressure
u =m∑
i=1
uiλi , ui divergence-free; p =
q∑i=1
piγi
µi → incompressibility Lagrange multipliers (from the det. solver).
How to choose q, pi , γi?
q pi γi
m = q pi = µi γi = λi
m = q pi = µi need equationsm 6= q need equations need equations
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 17 / 21
Idea: use p to minimize the residual
Continuity equation has no residual. Moment residual:
E [n(Um,V)] + E [d(Pm,V)] = 〈Rm,V〉 ∀V ∈ H10(D)
with n(Um,V) = c(U,U,V) + ν(y) v(U,V)− b(V).
Discretizing in space we get
N(m)h (y) + ET P
(m)h (y) = R
(m)h (y),
Minimizing ‖R(m)h (y)‖2 =
1
2E[‖R(m)
h (y)‖2Rdim(Vh)
], we get to
E ET P(m)h (y) = −EN
(m)h (y) ,
Use a PGD approach to compute pi and/or γi starting from here.
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 18 / 21
0 5 10 15 2010
−8
10−6
10−4
10−2
100
errorLM−residualPGD−residualλ norm
N = 4, ν = 100 5 10 15 20 25 30 35
10−8
10−6
10−4
10−2
100
errorLM−residualPGD−residualλ norm
N = 4, ν = 500 10 20 30 40 50
10−8
10−6
10−4
10−2
100
errorLM−residualPGD−residualλ norm
N = 4, ν = 100
0 5 10 15 20 25 3010
−8
10−6
10−4
10−2
100
errorLM−residualPGD−residualλ norm
N = 8, ν = 100 10 20 30 40 50
10−8
10−6
10−4
10−2
100
errorLM−residualPGD−residualλ norm
N = 8, ν = 500 10 20 30 40 50
10−8
10−6
10−4
10−2
100
errorLM−residualPGD−residualλ norm
N = 8, ν = 100
0 5 10 15 20 25 30 3510
−8
10−6
10−4
10−2
100
LM−residualPGD−residualλ normPGD−KL spectrum
N = 15, ν = 100 10 20 30 40 50 60
10−8
10−6
10−4
10−2
100
LM−residualPGD−residualλ normPGD−KL spectrum
N = 15, ν = 500 10 20 30 40 50 60
10−8
10−6
10−4
10−2
100
LM−residualPGD−residualλ normPGD−KL spectrum
N = 15, ν = 100
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 19 / 21
Conclusions
1 PGD allows to build a low-rank approximation of U2 algorithms:
I Power MethodI Arnoldi method
3 Results for Navier–Stokes equation are quite satisfactory
4 Non trivial pressure reconstruction
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 20 / 21
Bibliography
A. Nouy. A generalized spectral decomposition technique to solve a class of linearstochastic partial differential equations. Computer Methods in Applied Mechanics andEngineering, 196(45-48):4521–4537, 2007.
A. Nouy. Generalized spectral decomposition method for solving stochastic finite elementequations: invariant subspace problem and dedicated algorithms. Computer Methods inApplied Mechanics and Engineering, 197:4718–4736, 2008.
A. Nouy and O.P. Le Maıtre. Generalized spectral decomposition method for stochasticnon linear problems. Journal of Computational Physics, 228(1):202–235, 2009.
L. Tamellini, O. Le Maıtre, A. Nouy, Generalized Stochastic spectral decomposition for thesteady Navier–Stokes equations In preparation
Lorenzo Tamellini (EPFL/PoliMi) 3 April 2012 21 / 21