Proper Generalized Decomposition for Linear and Non-Linear - - PowerPoint PPT Presentation

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks

Proper Generalized Decomposition for Linear and Non-Linear Stochastic Models

Olivier Le Maître 1

Lorenzo Tamellini 2 and Anthony Nouy 3

1LIMSI-CNRS, Orsay, France 2MOX, Politecnico Milano, Italy 3GeM, Ecole Centrale Nantes, France

Radon Special Semester, WS on Multiscale Simulation & Analysis in Energy and the Environment, Linz - (12-16)/12/2011

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks

1

Motivation Linear elliptic problem Probabilistic framework Stochastic Galerkin problem Stochastic Discretization

2

Proper Generalized Decomposition Optimal Decomposition Algorithms An example

3

Nonlinear problems : N-S Stochastic Navier-Stokes equations Discretization Results

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks

Parametric model uncertainty : A model M involving uncertain input parameters D Treat uncertainty in a probabilistic framework : D(θ) ∈ (Θ, Σ, dµ) Assume D = D(ξ(θ)), where ξ ∈ RN with known probability law The model solution is stochastic and solves : M(U(ξ); D(ξ)) = 0 a.s. Uncertainty in the model solution : U(ξ) can be high-dimensional U(ξ) can be analyzed by sampling techniques, solving multiple deterministic problems (e.g. MC) We would like to construct a functional approximation of U(ξ) U(ξ) ≈

• k

ukΨk(ξ)

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Linear elliptic problem

Consider the deterministic linear scalar elliptic problem (in Ω) Find u ∈ V s.t. : a(u, v) = b(v), ∀v ∈ V where a(u, v) ≡

• Ω

k(x)∇u(x) · ∇v(x)dx (bilinear form) b(v) ≡

• Ω

f(x)v(x)dx (+ BC terms) (linear form) ǫ < k(x) and f(x) given (problem data) V (= H1

0(Ω)) deterministic space

(vector space).

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Probabilistic framework

Stochastic elliptic problem : Conductivity k, source field f (and BCs) uncertain Considered as random : Probability space (Θ, Σ, dµ) : E [h] ≡

• Θ

h(θ)dµ(θ), h ∈ L2(Θ, dµ) = ⇒ E

• h2

< ∞. Assume 0 < ǫ0 ≤ k a.e. in Θ × Ω, k(x, ·) ∈ L2(Θ, dµ) a.e. in Ω and f ∈ L2(Ω, Θ, dµ) Variational formulation : Find U ∈ V ⊗ L2(Θ, dµ) s.t. A(U, V) = B(V) ∀V ∈ V ⊗ L2(Θ, dµ), where A(U, V) . = E [a(U, V)] and B(V) . = E [b(V)].

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Galerkin problem

Stochastic expansion : Let {Ψ0, Ψ1, Ψ2, . . .} be an orthonormal basis of L2(Θ, dµ) W ∈ V ⊗ L2(Θ, dµ) has for expansion W(x, θ) =

+∞

• α=0

wα(x)Ψα(θ), wα(x) ∈ V Truncated expansion : span {Ψ0, . . . , ΨP} = SP ⊂ L2(Θ, dµ) Galerkin problem : Find U ∈ V ⊗ SP s.t. A(U, V) = B(V) ∀V ∈ V ⊗ SP, with U = P

α=0 uαΨα and V = P α=0 vαΨα

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Galerkin problem

Stochastic expansion : Let {Ψ0, Ψ1, Ψ2, . . .} be an orthonormal basis of L2(Θ, dµ) W ∈ V ⊗ L2(Θ, dµ) has for expansion W(x, θ) =

+∞

• α=0

wα(x)Ψα(θ), wα(x) ∈ V Truncated expansion : span {Ψ0, . . . , ΨP} = SP ⊂ L2(Θ, dµ) Galerkin problem : Find {u0, . . . , uP} s.t. for β = 0, . . . , P

• α

aα,β(uα, vβ) = bβ(vβ), ∀vβ ∈ V with aα,β(u, v) :=

• Ω E [kΨαΨβ] ∇u · ∇vdx,

bβ(v) :=

• Ω E [fΨβ] v(x)dx.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Galerkin problem

Stochastic expansion : Let {Ψ0, Ψ1, Ψ2, . . .} be an orthonormal basis of L2(Θ, dµ) W ∈ V ⊗ L2(Θ, dµ) has for expansion W(x, θ) =

+∞

• α=0

wα(x)Ψα(θ), wα(x) ∈ V Truncated expansion : span {Ψ0, . . . , ΨP} = SP ⊂ L2(Θ, dµ) Galerkin problem : Find {u0, . . . , uP} s.t. for β = 0, . . . , P

• α

aα,β(uα, vβ) = bβ(vβ), ∀vβ ∈ V Large system of coupled linear problem, globally SDP.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Discretization

Stochastic parametrization Parameterization using N independent R-valued r.v. ξ(θ) = (ξ1 · · · ξN) Let Ξ ⊆ RN be the range of ξ(θ) and pξ its pdf The problem is solved in the image space (Ξ, B(Ξ), pξ) U(θ) ≡ U(ξ(θ)) Stochastic basis : Ψα(ξ) Spectral polynomials (Hermite, Legendre, Askey scheme, . . . )

[Ghanem and Spanos, 1991], [Xiu and Karniadakis 2001]

Piecewise continuous polynomials (Stochastic elements, multiwavelets, . . . )

[Deb et al, 2001], [olm et al, 2004]

Truncature w.r.t. polynomial order : advanced selection strategy

[Nobile et al, 2010]

Size of dim SP - Curse of dimensionality

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Discretization

Stochastic Galerkin solution U(x, ξ) ≈ P

α=0 uα(x)Ψα(ξ)

Find {u0, . . . uP} s.t.

α aα,β(uα, vβ) = bβ(vβ), ∀vβ=0,...P ∈ V

A priori selection of the subspace SP Is the truncature / selection of the basis well suited ? Size of the Galerkin problem scales with P + 1 : iterative solver Memory requirements may be an issue for large bases Paradigm : Decouple the modes computation (smaller size problems, complexity reduction) Use reduced basis representation : find important components in U (reduce complexity and memory requirements) Proper Generalized Decomposition∗

∗Also GSD : Generalized Spectral Decomposition

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Discretization

1

Motivation Linear elliptic problem Probabilistic framework Stochastic Galerkin problem Stochastic Discretization

2

Proper Generalized Decomposition Optimal Decomposition Algorithms An example

3

Nonlinear problems : N-S Stochastic Navier-Stokes equations Discretization Results

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Optimal Decomposition

The m-terms PGD approximation of U is

[Nouy, 2007, 2008, 2010]

U(x, θ) ≈ Um(x, θ) =

m<P

• α=1

uα(x)λα(θ), λα ∈ SP, uα ∈ V. separated representation Interpretation : U is approximated on the stochastic reduced basis {λ1, . . . , λm} of SP the deterministic reduced basis {u1, . . . , um} of V none of which is selected a priori The questions are then : how to define the (deterministic or stochastic) reduced basis ? how to compute the reduced basis and the m-terms PGD of U ?

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Optimal Decomposition

Optimal L2-spectral decomposition : POD, KL decomposition Um(x, θ) =

m

• α=1

uα(x)λα(θ) minimizes E

• Um − U2

L2(Ω)

• The modes uα are the m dominant eigenvectors of the kernel

E [U(x, ·)U(y, ·)] :

• Ω

E [U(x, ·)U(y, ·)] uα(y)dy = βuα(x), uαL2(Ω) = 1. The modes are orthonormal : λα(θ) =

• Ω

U(x, θ)uα(x)dx However U(x, θ), so E [u(x, ·)u(y, ·)] is not known !

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Optimal Decomposition

Optimal L2-spectral decomposition : POD, KL decomposition Um(x, θ) =

m

• α=1

uα(x)λα(θ) minimizes E

• Um − U2

L2(Ω)

• Solve the Galerkin problem in Vh ⊗ SP′<P to construct {uα}, and

then solve for the

• λα ∈ SP

. Solve the Galerkin problem in VH ⊗ SP to construct {λα}, and then solve for the

• uα ∈ Vh

with dim VH ≪ dim Vh.

See works by groups of Ghanem and Matthies.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Optimal Decomposition

Alternative definition of optimality A(·, ·) is symmetric positive definite, so U minimizes the energy functional J(V) ≡ 1 2A(V, V) − B(V) We define Um through J(Um) = min

{uα},{λα} J

m

• α=1

uαλα

• .

Equivalent to minimizing a Rayleigh quotient Optimality w.r.t the A-norm (change of metric) : V2

A = E [a(V, V)] = A(V, V)

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Optimal Decomposition

Sequential construction : For i = 1, 2, 3 . . . J(λiui) = min

v∈V,β∈SP J

 βv +

i−1

• j=1

λjuj   = min

v∈V,β∈SP J

• βv + Ui−1

The optimal couple (λi, ui) solves simultaneously a) deterministic problem ui = D(λi, Ui−1) A(λiui, λiv) = B(λiv) − A

• Ui−1, λiv
• ,

∀v ∈ V b) stochastic problem λi = S(ui, Ui−1) A(λiui, βui) = B(βui) − A

• Ui−1, βui
• ,

∀β ∈ SP

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Optimal Decomposition

Deterministic problem : ui = D(λi, Ui−1)

• Ω

E

• λ2

i k

• ∇ui·∇vdx = E
• −
• Ω

λik∇Ui−1 · ∇vdx +

• Ω

λifvdx

• ,

∀v. Stochastic problem : λi = S(ui, Ui−1) E

• λiβ
• Ω

k∇ui · ∇uidx

• = E
• −β
• Ω

k∇Ui−1 · ∇uidx +

• Ω

fuidx

• ,

∀β.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Optimal Decomposition

Properties : The couple (λi, ui) is a fixed-point of : λi = S ◦ D(λi, ·), ui = D ◦ S(ui, ·) Homogeneity property : λi c = S(cui, ·), ui c = D(cλi, ·), ∀c ∈ R \ {0}. ⇒ arbitrary normalization of one of the two elements. Algorithms inspired from dominant subspace methods Power-type, Krylov/Arnoldi, . . .

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Algorithms

Power Iterations

1

Set l = 1

2

initialize λ (e.g. randomly)

3

While not converged, repeat (power iterations) a) Solve : u = D(λ, Ul−1) b) Normalize u c) Solve : λ = S(u, Ul−1)

4

Set ul = u, λl = λ

5

l ← l + 1, if l < m repeat from step 2 Comments : Convergence criteria for the power iterations (subspace with dim > 1 or clustered eigenvalues)

[Nouy, 2007,2008]

Usually few (4 to 5) inner iterations are sufficient

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Algorithms

Power Iterations with Update

1

Same as Power Iterations, but after (ul, λl) is obtained (step 4) update of the stochastic coefficients : Orthonormalyze {u1, . . . , ul} (optional) Find {λ1, . . . , λl} s.t. A

• l
• i=1

uiλi,

l

• i=1

uiβi

• = B
• l
• i=1

uiβi

• ,

∀βi=1,...,l ∈ ×SP

2

Continue for next couple Comments : Improves the convergence Low dimensional stochastic linear system (l × l) Cost of update increases linearly with the order l of the reduced representation

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Algorithms

Arnoldi (Full Update version)

1

Set l = 0

2

Initialize λ ∈ SP

3

For l′ = 1, 2, . . . (Arnoldi iterations) Solve deterministic problem u′ = D(λ, Ul) Orthogonalize : ul+l′ = u′ − l+l′−1

j=1

(u′, uj)Ω If ul+l′L2(Ω) ≤ ǫ or l + l′ = m then break Normalize ul+l′ Solve λ = S(ul′, Ul)

4

l ← l + l′

5

Find {λ1, . . . , λl} s.t. (Update) A

• l
• i=1

uiλi,

l

• i=1

uiβi

• = B
• l
• i=1

uiβi

• ,

∀βi=1,...,l ∈ SP

6

If l < m return to step 2.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Algorithms

Summary Resolution of a sequence of deterministic elliptic problems, with elliptic coefficients E

• λ2k
• and modified (deflated) rhs

dimension is dim Vh Resolution of a sequence of linear stochastic equations dimension is dim SP Update problems : system of linear equations for stochastic random variables dimension is m × dim SP To be compared with the Galerkin problem dimension dim Vh × dim SP Weak modification of existing (FE/FV) codes (weakly intrusive)

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Algorithms

1

Motivation Linear elliptic problem Probabilistic framework Stochastic Galerkin problem Stochastic Discretization

2

Proper Generalized Decomposition Optimal Decomposition Algorithms An example

3

Nonlinear problems : N-S Stochastic Navier-Stokes equations Discretization Results

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks An example

Example definition

D (Dogger) L (Limestone) M (Marl) C (Clay) z=200 z=0 z=295 z=595 z=695 z=695 z=595 z=200 z=0 z=350 x=0 x=25,000

Rectangular domain 25,000×695 (m) 4 Geological layers : D (Dogger), C (Clay), L (Limestone) and M (Marl)

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks An example

Test case definition (cont.) : uncertain Dirichlet boundary conditions

D C L M h1 h2 h3 h5 Variation lineaire de h4 a h3 h6 h4

∆ Head (m) Expectation Range distribution ∆h1,2 +51 ±10 Uniform ∆h1,3 +21 ±5 Uniform ∆h1,6

• 3

±2 Uniform ∆h2,5

• 110

±10 Uniform ∆h3,4

• 160

±20 Uniform Heads at boundaries are taken independent

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks An example

Example definition (cont.) : Uncertain conductivities Layer ki median ki min ki max distribution Dogger 25 5 125 LogUniform Clay 3 10−6 3 10−7 3 10−5 LogUniform Limestone 6 1.2 30 LogUniform Marl 3 10−5 1 10−5 1 10−4 LogUniform Conductivities are taken independent Parameterization 9 independent r.v. {ξ1, . . . , ξ9} ∼ U[0, 1]9 Stochastic space SP : Legendre polynomial up to order No dim SP = P + 1 = (9 + No)!/(9!No!)

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks An example

Deterministic discretization : P − 1 finite-element Mesh conforming with the geological layers

200 400 600 800 1000 1200 1400 700 x z

Ne ≈ 30, 000 finite elements dim(Vh) ≈ 15, 000 Dimension of Galerkin problem : 8.2 105 (No = 2), 3.3 106 (No = 3)

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks An example

Norm of Galerkin residual as a function of m (No = 3)

1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 25 30 35 40 Residual m Power Power-Update Arnoldi-P-Update Arnoldi-F-Update

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks An example

Norm of error (vs Galerkin) as a function of m (No = 3)

1e-06 1e-05 1e-04 0.001 0.01 0.1 1 5 10 15 20 25 30 35 40 Error m Power Power-Update Arnoldi-P-Update Arnoldi-F-Update

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks An example

CPU times (No = 3)

1e-12 1e-10 1e-08 1e-06 1e-04 0.01 1 10000 20000 30000 40000 Residual CPU time (s) Power Power-Update Arnoldi-P-Update Arnoldi-F-Update Full-Galerkin

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks An example

CPU times of PGD algorithms (No = 3)

1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 500 1000 1500 2000 2500 3000 3500 Residual CPU time (s) Power Power-Update Arnoldi-P-Update Arnoldi-F-Update

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks An example

1

Motivation Linear elliptic problem Probabilistic framework Stochastic Galerkin problem Stochastic Discretization

2

Proper Generalized Decomposition Optimal Decomposition Algorithms An example

3

Nonlinear problems : N-S Stochastic Navier-Stokes equations Discretization Results

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks

Extension to general problems Non-definite / non symmetric linear and nonlinear problems : no

• ptimality results

[Nouy, 2008]

Algorithms : from numerical experiments methods & algorithms can be safely applied

[Nouy & olm, 2009]

Illustration on the Incompressible Navier-Stokes equations

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Navier-Stokes equations

Test problem : steady, two dimensional, Newtonian fluid in a square domain Ω : U∇U + ∇P − ν∇2U = F, in Ω ∇ · U = 0, in Ω U = 0,

• n ∂Ω.

U the velocity field, P the pressure field, uncertainty in the viscosity ν = ν(θ) and forcing F = F(x, θ).

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Navier-Stokes equations

Functional spaces : L2(Ω) the space of functions that are square integrable over Ω, equipped with the inner product and norm (p, q)Ω :=

• Ω

pq dΩ, qL2(Ω) = (q, q)1/2

Ω .

Extended to vector valued functions by summation over vectors components, and to the constrained space L2

0(Ω) =

• q ∈ L2(Ω) :
• Ω

q dΩ = 0

• .

H1(Ω), Sobolev space of vector valued functions with all components and their first derivatives being square integrable over Ω, and H1

0(Ω)

the constrained space of such vector functions vanishing on ∂Ω, H1

0(Ω) =

• v ∈ H1(Ω), v = 0 on ∂Ω
• .

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Navier-Stokes equations

Weak formulation : Find U ∈ H1

0(Ω) ⊗ S and P ∈ L2 0(Ω) ⊗ S, such that

C(U, U, V) + D(V, P) + A(U, V) = E [(F, V)Ω] ∀V ∈ H1

0(Ω) ⊗ S,

D(U, Q) = 0 ∀Q ∈ L2

0(Ω) ⊗ S.

where : C(U, V, W) . = E [c(U, V, W)] = E

• Ω(U∇V) · Wdx
• D(V, Q) .

= E [d(V, Q)] = E

• Ω Q∇Vdx
• A(U, V) .

= E [νa(U, V)] = E

• ν
• Ω ∇V : ∇Vdx

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Navier-Stokes equations

Weak formulation in (weakly) Divergence Free space : Let H1

0;div(Ω)

H1

0;div(Ω) =

• v ∈ H1

0(Ω), d(v, q) = 0 ∀q ∈ L2 0(Ω)

• .

The problem becomes : Find U ∈ H1

0;div(Ω) ⊗ S such that

C(U, U, V) + A(U, V) = E [(F, V)Ω] ∀V ∈ H1

0;div(Ω) ⊗ S.

Remove the pressure Difficulties in the discretization of H1

0;div(Ω)

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Navier-Stokes equations

PGD approximation in H1

0;div(Ω) :

U(x, θ) ≈ Um(x, θ) =

m

• k=1

uk(x)λk(θ), The deterministic functions uk ∈ H1

0;div(Ω) form a reduced basis

• f H1

0;div(Ω)

The stochastic coefficients λk ∈ S form a reduced basis of S None of the two reduced bases are selected a priori Sequential construction of the approximation U0 → U1 → U2 → · · · with Arnoldi algorithm Um+1 = Um + λu : successive couples (λ, u) are defined as previously

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Navier-Stokes equations

The couple (λm+1, um+1) is sought as the fixed point of λ = S ◦ D(λ; Um), u = D ◦ S(u; Um), where Determinisic problem : u = D(λ; Um) ∈ H1

0;div(Ω) is the solution of

C(λu, λu, λv) + C(λu, Um, λv) + C(Um, λu, λv) + A(λu, λv) = E [(F, λv)Ω] − C(Um, Um, λv) − A(Um, λv), ∀v ∈ H1

0;div(Ω).

Stochastic problem : λ = S(u; Um) ∈ S is the solution of C(λu, λu, βu) + C(λu, Um, βu) + C(Um, λu, βu) + A(λu, βu) = E [(F, βu)Ω] − C(Um, Um, βu) − A(Um, βu), ∀β ∈ S.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Navier-Stokes equations

The couple (λm+1, um+1) is sought as the fixed point of λ = S ◦ D(λ; Um), u = D ◦ S(u; Um), where Determinisic problem : u = D(λ; Um) ∈ H1

0;div(Ω) is the solution of

E

• λ3

c(u, u, v) + E

• λ2ν
• a(u, v) + c(u, ˜

u, v) + c(˜ u, u, v) = (˜ f, v)Ω, ∀v ∈ H1

0;div(Ω),

with ˜ u = E

• λ2Um

Stochastic problem : λ = S(u; Um) ∈ S is the solution of λ = S(u; Um) ⇔ E

• (˜

cλ2 + Gλ + R)β

• = 0,

∀β ∈ S, with ˜ c ∈ R and G, R ∈ S.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Navier-Stokes equations

Update problem : fix one of the two reduced bases and solve the Galerkin problem for the other component. Updating of the stochastic coefficients : Find {λk ∈ S}m

k=1 such that

C  

i

λiui,

• j

λjuj, βuk   + A

• i

λiui, βuk

• = E [(F, βuk)] ,

∀β ∈ S and k = 1, . . . , m. System of m quadratic stochastic equations : E    

m

• i,j=1

˜ qk,i,j λiλj +

m

• i=1

Gk,iλi − Rk   βk   = 0, ∀βk ∈ S.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Discretization

Stochastic discretization : Parametrization of ν(θ) and F(θ) using N i.i.d. random variables : ξ = {ξ1, . . . , ξN} Stochastic basis : Polynomial Chaos λ(θ) =

• α

λαΨα(ξ(θ)), where the Ψs are orthonormal polynomials E [Ψα(ξ)Ψβ(ξ)] = δα,β. Truncature to (total) polynomial degree No : dim SP = (No + N)! No!N! .

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Discretization

Spatial discretization Pn-Pn−2 Spectral Element Method H1(Ω) ∋ u(x) ≈ uh(x) =

n

• i,j=1

ui,jφu

i,j(x) ∈ Vh,

L2(Ω) ∋ p(x) ≈ ph(x) =

n−2

• i,j=1

pi,jφp

i,j(x) ∈ Πh.

(Nodal basis on tenzorized Gauss-Lobatto grid) Resolution of the deterministic problem u = D(λ; U(i)) ∈ H1

0;div(Ω) achieved by solving in Vh 0 the

constrained problem E

• λ3

c(uh, uh, vh) + E

• λ2ν
• a(uh, vh) + c(uh, ˜

uh, vh) + c(˜ uh, uh, vh) + d(vh, ph) = (˜ f, vh)Ω ∀vh ∈ Vh

0,

d(uh, qh) = 0 ∀q ∈ Πh.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

Case of a deterministic forcing and a random (Log-normal) viscosity :

0.0025 0.005 0.0075 0.01 pdf ν pν

ν(θ) = 1 200 exp

• σν

√ N

N

• i=1

ξi(θ)

• (+10−4),

ξi ∼ N(0, 1) i.i.d. Same problem but for parametrization involving N Gaussian R.V.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

Galerkin solution for N = 1 and No = 10 (Wiener-Hermite expansion)

• 6
• 5
• 4
• 3
• 2
• 1

1 2 3 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Mean and standard deviation of UG rotational.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

POD modes of the Galerkin solution for N = 1 and No = 10

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 2 4 6 8 10 m Galerkin

Rotational of the spatial POD modes of UG.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

First PGD-Arnoldi modes for N = 1 and No = 10

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

POD modes of the PGD solution for m = 15, N = 1 and No = 10

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 m Galerkin Reduced ||λi||

Rotational of the spatial POD modes of Um=15.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

Convergence of PGD solution N = 1 and No = 10

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 m Galerkin Reduced ||λi|| 10-5 10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 16 m Residual 10-6 10-5 10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 16 m Error

Comparison of the norms of the POD coefficients at m = 15 (left), residual norm (center), error norm (right).

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

Convergence of PGD solutions N = 1, 2 and 3 at No = 10

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 m Galerkin Reduced P=10 Reduced P=66 Reduced P=286 10-5 10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 16 m Reduced P=10 Reduced P=166 Reduced P=286 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 2 4 6 8 10 12 14 m Reduced P=10 Reduced P=66 Reduced P=286

Comparison of the norms of the POD coefficients at m = 15 (left), residual norm (center), error norm (right). PGD captures the essential features of the stochastic solution

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

Stochastic forcing F : Hodge’s decomposition F(x, θ) = ∇Φ(x, θ) + ∇ ∧ Ψ(x, θ) = ∇ ∧ Ψ(x, θ)k Define Fω . = (∇ ∧ F) · k = −△Ψ and assume : Fω(x, θ) = f 0

ω + F ′ ω(x, θ),

E [F ′

ω(x, ·)] = 0,

F ′

ω(x, ·) ∼ N(0, σ2),

E [F ′

ω(x, ·)F ′ ω(x′, ·)] = σ2 exp(−|x − x′|/L).

KL expansion of Fω : Fω(x, θ) ≈ F N

ω (x, ξ(θ)) = f 0 ω + N

• k=0

√γkf k

ω(x)ξk(θ)

It comes F(x, θ) ≈ F N(x, ξ(θ)) = f 0 +

N

• k=0

√γkf k(x)ξk(θ), with f i = ∇ ∧

• △−1f i

ω

• k.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

KL modes of the forcing :

scale = 1 scale = 5 scale = 5 scale = 5 scale = 15 scale = 15 scale = 15 scale = 15 scale = 25 scale = 25

Forcing modes for L = 1, σ/f 0

ω = 0.2

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

PGD solution for ν = 1/50, N = 11, No = 3 and m = 45

• 6
• 5
• 4
• 3
• 2
• 1

1 2 3 4 0.2 0.4 0.6 0.8 1 1.2 1.4

Mean and standard deviation of ∇ ∧ Um=45

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

First PGD-Arnoldi modes

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

Results at ν = 1/50 : No = 3, N = 11, P = 364

10-4 10-3 10-2 10-1 100 10 20 30 40 50 m Residual 10-6 10-5 10-4 10-3 10-2 10-1 100 10 20 30 40 50 m Error 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 10 20 30 40 50 m Galerkin Reduced m=45 ||λi||

Residual (left), Um − UG (center) and norm of POD modes for m = 45 (right).

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

Results for ν = 1/50 : No = 3, N = 11, P = 364 m = 0 m = 2 m = 14

0.0e+00 5.0e-05 1.0e-04 1.5e-04 2.0e-04 2.5e-04 3.0e-04 3.5e-04 4.0e-04 4.5e-04 5.0e-04 0.0e+00 2.0e-04 4.0e-04 6.0e-04 8.0e-04 1.0e-03 1.2e-03 0.0e+00 5.0e-05 1.0e-04 1.5e-04 2.0e-04 2.5e-04

m = 26 m = 43 m = 45

0.0e+00 5.0e-06 1.0e-05 1.5e-05 2.0e-05 2.5e-05 3.0e-05 3.5e-05 0.0e+00 5.0e-07 1.0e-06 1.5e-06 2.0e-06 2.5e-06 0.0e+00 2.0e-07 4.0e-07 6.0e-07 8.0e-07 1.0e-06 1.2e-06 1.4e-06 1.6e-06 1.8e-06

Spatial distribution of the stochastic residual.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

Residual computation : Residual computation in H1

0;div(Ω) is difficult

Projection of the H1

0(Ω)-residual into H1 0;div(Ω) amount to solving

a stochastic linear equation for Pm PGD can be used for the approximation of Pm Re-use of the reduced basis of Arnoldi Lagrange multipliers {pi} associated to {ui} was found a effective approach However, monitoring convergence using the decay of λi appears effective

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Results

Residual computation :

5 10 15 20 25 30 10

−8

10

−6

10

−4

10

−2

10 error LM−residual PGD−residual λ norm 10 20 30 40 50 10

−8

10

−6

10

−4

10

−2

10 error LM−residual PGD−residual λ norm 10 20 30 40 50 10

−8

10

−6

10

−4

10

−2

10 error LM−residual PGD−residual λ

Comparison of different error measures of the PGD solution at ν = 1/10, 1/50 and 1/100 (from left to right).

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks

On-going work : Evolution problems : PGD on each time step or global PGD with possibly alternative separated forms u(x, t, θ) ≈

• α

uα(x, t)λα(θ) ≈

• α
• uα(x)βα(t, θ)

≈

• α

ˆ uα(x)τα(t)γα(θ) Nested separation ? Other definition of optimality for non definite linear operators (least square residual minimization).

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks

Thanks for your attention

Fundings : ANR-TYCHE project [2010-2013] ANR-ASRMEI project [2008-2011] MoMaS (Andra, Brgm, Cea, EdF, Irsn) [2010-2011]