proper generalized decomposition for stochastic navier ......proper generalized decomposition for...

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


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.


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.


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


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.


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!


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


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


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


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


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.



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.



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.


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).


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)


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).


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 .


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


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


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


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


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


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.


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


N = 4, ν = 500 10 20 30 40 50

10−8

10−6

10−4

10−2

100


N = 4, ν = 100

0 5 10 15 20 25 3010

−8

10−6

10−4

10−2

100


N = 8, ν = 100 10 20 30 40 50

10−8

10−6

10−4

10−2

100


N = 8, ν = 500 10 20 30 40 50

10−8

10−6

10−4

10−2

100


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


N = 15, ν = 500 10 20 30 40 50 60

10−8

10−6

10−4

10−2

100


N = 15, ν = 100


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


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


proper generalized decomposition for stochastic navier ......proper generalized decomposition for...

Documents