Proper Generalized Decomposition for Linear and Non-Linear Stochastic Models

Transcription

1 Proper Generalized Decomposition for Linear and Non-Linear Stochastic Models Olivier Le Maître 1 Lorenzo Tamellini 2 and Anthony Nouy 3 1 LIMSI-CNRS, Orsay, France 2 MOX, Politecnico Milano, Italy 3 GeM, Ecole Centrale Nantes, France Radon Special Semester, WS on Multiscale Simulation & Analysis in Energy and the Environment, Linz - (12-16)/12/2011

2 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

3 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 R N 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 u k Ψ k (ξ)

4 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) k(x) u(x) v(x)dx Ω b(v) f (x)v(x)dx (+ BC terms) Ω ɛ < k(x) and f (x) given V (= H0 1 (Ω)) deterministic space (bilinear form) (linear form) (problem data) (vector space).

5 Probabilistic framework Stochastic elliptic problem : Conductivity k, source field f (and BCs) uncertain Considered as random : Probability space (Θ, Σ, dµ) : E [h] h(θ)dµ(θ), h L 2 (Θ, dµ) = E [ h 2] <. Θ Assume 0 < ɛ 0 k a.e. in Θ Ω, k(x, ) L 2 (Θ, dµ) a.e. in Ω and f L 2 (Ω, Θ, dµ) Variational formulation : Find U V L 2 (Θ, dµ) s.t. A(U, V ) = B(V ) V V L 2 (Θ, dµ), where A(U, V ). = E [a(u, V )] and B(V ). = E [b(v )].

6 Stochastic Galerkin problem Stochastic expansion : Let {Ψ 0, Ψ 1, Ψ 2,...} be an orthonormal basis of L 2 (Θ, dµ) W V L 2 (Θ, dµ) has for expansion W (x, θ) = + α=0 w α (x)ψ α (θ), w α (x) V Truncated expansion : Galerkin problem : Find U V S P s.t. span {Ψ 0,..., Ψ P } = S P L 2 (Θ, dµ) A(U, V ) = B(V ) V V S P, with U = P α=0 u αψ α and V = P α=0 v αψ α

7 Stochastic Galerkin problem Stochastic expansion : Let {Ψ 0, Ψ 1, Ψ 2,...} be an orthonormal basis of L 2 (Θ, dµ) W V L 2 (Θ, dµ) has for expansion W (x, θ) = + α=0 w α (x)ψ α (θ), w α (x) V Truncated expansion : span {Ψ 0,..., Ψ P } = S P L 2 (Θ, dµ) Galerkin problem : Find {u 0,..., u P } s.t. for β = 0,..., P a α,β (u α, v β ) = b β (v β ), α v β V with a α,β (u, v) := Ω E [kψ αψ β ] u vdx, b β (v) := Ω E [f Ψ β] v(x)dx.

8 Stochastic Galerkin problem Stochastic expansion : Let {Ψ 0, Ψ 1, Ψ 2,...} be an orthonormal basis of L 2 (Θ, dµ) W V L 2 (Θ, dµ) has for expansion W (x, θ) = + α=0 w α (x)ψ α (θ), w α (x) V Truncated expansion : span {Ψ 0,..., Ψ P } = S P L 2 (Θ, dµ) Galerkin problem : Find {u 0,..., u P } s.t. for β = 0,..., P a α,β (u α, v β ) = b β (v β ), α v β V Large system of coupled linear problem, globally SDP.

9 Stochastic Discretization Stochastic parametrization Parameterization using N independent R-valued r.v. ξ(θ) = (ξ 1 ξ N ) Let Ξ R N be the range of ξ(θ) and p ξ its pdf The problem is solved in the image space (Ξ, B(Ξ), p ξ ) U(θ) U(ξ(θ)) Stochastic basis : Ψ α (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 S P - Curse of dimensionality

10 Stochastic Discretization Stochastic Galerkin solution U(x, ξ) P α=0 u α(x)ψ α (ξ) Find {u 0,... u P } s.t. α a α,β(u α, v β ) = b β (v β ), v β=0,...p V A priori selection of the subspace S P 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

11 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

12 Optimal Decomposition The m-terms PGD approximation of U is [Nouy, 2007, 2008, 2010] m<p U(x, θ) U m (x, θ) = u α (x)λ α (θ), λ α S P, u α V. α=1 separated representation Interpretation : U is approximated on the stochastic reduced basis {λ 1,..., λ m } of S P the deterministic reduced basis {u 1,..., u m } 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?

13 Optimal Decomposition Optimal L 2 -spectral decomposition : U m (x, θ) = m α=1 POD, KL decomposition [ ] u α (x)λ α (θ) minimizes E U m U 2 L 2 (Ω) The modes u α are the m dominant eigenvectors of the kernel E [U(x, )U(y, )] : E [U(x, )U(y, )] u α (y)dy = βu α (x), u α L 2 (Ω) = 1. Ω The modes are orthonormal : λ α (θ) = Ω U(x, θ)u α (x)dx However U(x, θ), so E [u(x, )u(y, )] is not known!

14 Optimal Decomposition Optimal L 2 -spectral decomposition : U m (x, θ) = m α=1 POD, KL decomposition [ ] u α (x)λ α (θ) minimizes E U m U 2 L 2 (Ω) Solve the Galerkin problem in V h S P <P to construct {u α }, and then solve for the { λ α S P}. Solve the Galerkin problem in V H S P to construct {λ α }, and then solve for the { u α V h} with dim V H dim V h. See works by groups of Ghanem and Matthies.

15 Optimal Decomposition Alternative definition of optimality A(, ) is symmetric positive definite, so U minimizes the energy functional J(V ) 1 A(V, V ) B(V ) 2 We define U m through ( m ) J(U m ) = min J u α λ α. {u α},{λ α} α=1 Equivalent to minimizing a Rayleigh quotient Optimality w.r.t the A-norm (change of metric) : V 2 A = E [a(v, V )] = A(V, V )

16 Optimal Decomposition Sequential construction : For i = 1, 2, 3... J(λ i u i ) = min J i 1 βv + λ j u j = min J ( βv + U i 1) v V,β S P v V,β S P j=1 The optimal couple (λ i, u i ) solves simultaneously a) deterministic problem u i = D(λ i, U i 1 ) A(λ i u i, λ i v) = B(λ i v) A ( U i 1, λ i v ), v V b) stochastic problem λ i = S(u i, U i 1 ) A(λ i u i, βu i ) = B(βu i ) A ( U i 1, βu i ), β S P

17 Optimal Decomposition Ω Deterministic problem : u i = D(λ i, U i 1 ) E [ λ 2 i k ] [ u i vdx = E λ i k U i 1 vdx + Stochastic problem : λ i = S(u i, U i 1 ) [ ] [ E λ i β k u i u i dx = E β k U i 1 u i dx + Ω Ω Ω Ω ] λ i fvdx, v. Ω ] fu i dx, β.

18 Optimal Decomposition Properties : The couple (λ i, u i ) is a fixed-point of : Homogeneity property : λ i = S D(λ i, ), u i = D S(u i, ) λ i c = S(cu i, ), u i c = D(cλ i, ), c R \ {0}. arbitrary normalization of one of the two elements. Algorithms inspired from dominant subspace methods Power-type, Krylov/Arnoldi,...

19 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, U l 1 ) b) Normalize u c) Solve : λ = S(u, U l 1 ) 4 Set u l = u, λ l = 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

20 Algorithms Power Iterations with Update 1 Same as Power Iterations, but after (u l, λ l ) is obtained (step 4) update of the stochastic coefficients : Orthonormalyze {u 1,..., u l } Find {λ 1,..., λ l } s.t. ( l A u i λ i, i=1 (optional) ) ( l l ) u i β i = B u i β i, β i=1,...,l S P i=1 2 Continue for next couple Comments : Improves the convergence i=1 Low dimensional stochastic linear system (l l) Cost of update increases linearly with the order l of the reduced representation

21 Algorithms Arnoldi (Full Update version) 1 Set l = 0 2 Initialize λ S P 3 For l = 1, 2,... (Arnoldi iterations) 4 l l + l Solve deterministic problem u = D(λ, U l ) Orthogonalize : u l+l = u l+l 1 j=1 (u, u j ) Ω If u l+l L 2 (Ω) ɛ or l + l = m then break Normalize u l+l Solve λ = S(u l, U l ) 5 Find {λ 1,..., λ l } s.t. (Update) ( l ) ( l l ) A u i λ i, u i β i = B u i β i, β i=1,...,l S P i=1 i=1 6 If l < m return to step 2. i=1

22 Algorithms Summary Resolution of a sequence of deterministic elliptic problems, with elliptic coefficients E [ λ 2 k ] and modified (deflated) rhs dimension is dim V h Resolution of a sequence of linear stochastic equations dimension is dim S P Update problems : system of linear equations for stochastic random variables dimension is m dim S P To be compared with the Galerkin problem dimension dim V h dim S P Weak modification of existing (FE/FV) codes (weakly intrusive)

23 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

24 An example Example definition z=695 z=695 z=595 M (Marl) z=595 z=295 z=200 z=0 L (Limestone) C (Clay) D (Dogger) z=350 z=200 z=0 x=0 x=25,000 Rectangular domain 25, (m) 4 Geological layers : D (Dogger), C (Clay), L (Limestone) and M (Marl)

25 An example Test case definition (cont.) : uncertain Dirichlet boundary conditions h4 Variation lineaire de h4 a h3 h3 M h5 L h2 C h6 D Head (m) Expectation Range distribution h 1,2 +51 ±10 Uniform h 1,3 +21 ±5 Uniform h 1,6-3 ±2 Uniform h 2,5-110 ±10 Uniform h 3,4-160 ±20 Uniform h1 Heads at boundaries are taken independent

26 An example Example definition (cont.) : Uncertain conductivities Layer k i median k i min k i max distribution Dogger LogUniform Clay LogUniform Limestone LogUniform Marl LogUniform Conductivities are taken independent Parameterization 9 independent r.v. {ξ 1,..., ξ 9 } U[0, 1] 9 Stochastic space S P : Legendre polynomial up to order No dim S P = P + 1 = (9 + No)!/(9!No!)

27 An example Deterministic discretization : P 1 finite-element Mesh conforming with the geological layers 700 z N e 30, 000 finite elements dim(v h ) 15, 000 Dimension of Galerkin problem : (No = 2), (No = 3) x

28 An example Norm of Galerkin residual as a function of m (No = 3) Power Power-Update Arnoldi-P-Update Arnoldi-F-Update Residual 1e-04 1e-05 1e-06 1e-07 1e-08 1e m

29 An example Norm of error (vs Galerkin) as a function of m (No = 3) Power Power-Update Arnoldi-P-Update Arnoldi-F-Update 0.01 Error e-04 1e-05 1e m

30 An example CPU times (No = 3) e-04 Power Power-Update Arnoldi-P-Update Arnoldi-F-Update Full-Galerkin Residual 1e-06 1e-08 1e-10 1e CPU time (s)

31 An example CPU times of PGD algorithms (No = 3) Power Power-Update Arnoldi-P-Update Arnoldi-F-Update Residual 1e-04 1e-05 1e-06 1e-07 1e-08 1e CPU time (s)

32 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

33 Extension to general problems Non-definite / non symmetric linear and nonlinear problems : no optimality results [Nouy, 2008] Algorithms : from numerical experiments methods & algorithms can be safely applied [Nouy & olm, 2009] Illustration on the Incompressible Navier-Stokes equations

34 Stochastic Navier-Stokes equations Test problem : steady, two dimensional, Newtonian fluid in a square domain Ω : U U + P ν 2 U = F, U = 0, U = 0, in Ω in Ω on Ω. U the velocity field, P the pressure field, uncertainty in the viscosity ν = ν(θ) and forcing F = F (x, θ).

35 Stochastic Navier-Stokes equations Functional spaces : L 2 (Ω) the space of functions that are square integrable over Ω, equipped with the inner product and norm (p, q) Ω := Ω pq dω, q L 2 (Ω) = (q, q) 1/2 Ω. Extended to vector valued functions by summation over vectors components, and to the constrained space { } L 2 0(Ω) = q L 2 (Ω) : q dω = 0. H 1 (Ω), Sobolev space of vector valued functions with all components and their first derivatives being square integrable over Ω, and H 1 0 (Ω) the constrained space of such vector functions vanishing on Ω, H 1 0(Ω) = { v H 1 (Ω), v = 0 on Ω }. Ω

36 Stochastic Navier-Stokes equations Weak formulation : Find U H 1 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 H 1 0(Ω) S, D(U, Q) = 0 Q L 2 0(Ω) S. where : C(U, V, W ) =. E [c(u, V, W )] = E [ (U V ) W dx] Ω D(V, Q) =. E [d(v, Q)] = E [ Q V dx] Ω A(U, V ) =. E [νa(u, V )] = E [ ν V : V dx] Ω

37 Stochastic Navier-Stokes equations Weak formulation in (weakly) Divergence Free space : Let H 1 0;div (Ω) H 1 0;div(Ω) = { v H 1 0(Ω), d(v, q) = 0 q L 2 0(Ω) }. The problem becomes : Find U H 1 0;div (Ω) S such that C(U, U, V ) + A(U, V ) = E [(F, V ) Ω ] V H 1 0;div(Ω) S. Remove the pressure Difficulties in the discretization of H 1 0;div (Ω)

38 Stochastic Navier-Stokes equations PGD approximation in H 1 0;div (Ω) : U(x, θ) U m (x, θ) = m u k (x)λ k (θ), The deterministic functions u k H 1 0;div (Ω) form a reduced basis of H 1 0;div (Ω) k=1 The stochastic coefficients λ k S form a reduced basis of S None of the two reduced bases are selected a priori Sequential construction of the approximation U 0 U 1 U 2 with Arnoldi algorithm U m+1 = U m + λu : successive couples (λ, u) are defined as previously

39 Stochastic Navier-Stokes equations The couple (λ m+1, u m+1 ) is sought as the fixed point of λ = S D(λ; U m ), u = D S(u; U m ), where Determinisic problem : u = D(λ; U m ) H 1 0;div (Ω) is the solution of C(λu, λu, λv) + C(λu, U m, λv) + C(U m, λu, λv) + A(λu, λv) = E [(F, λv) Ω ] C(U m, U m, λv) A(U m, λv), v H 1 0;div(Ω). Stochastic problem : λ = S(u; U m ) S is the solution of C(λu, λu, βu) + C(λu, U m, βu) + C(U m, λu, βu) + A(λu, βu) = E [(F, βu) Ω ] C(U m, U m, βu) A(U m, βu), β S.

40 Stochastic Navier-Stokes equations The couple (λ m+1, u m+1 ) is sought as the fixed point of λ = S D(λ; U m ), u = D S(u; U m ), where Determinisic problem : u = D(λ; U m ) H 1 0;div (Ω) is the solution of E [ λ 3] c(u, u, v) + E [ λ 2 ν ] a(u, v) + c(u, ũ, v) with ũ = E [ λ 2 U m] + c(ũ, u, v) = ( f, v) Ω, v H 1 0;div(Ω), Stochastic problem : λ = S(u; U m ) S is the solution of λ = S(u; U m ) E [ ( cλ 2 + Gλ + R)β ] = 0, β S, with c R and G, R S.

41 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 {λ k S} m k=1 such that C λ i u i, ( ) λ j u j, βu k + A λ i u i, βu k = E [(F, βu k )], i j i β S and k = 1,..., m. System of m quadratic stochastic equations : m m E q k,i,j λ i λ j + G k,i λ i R k β k = 0, β k S. i,j=1 i=1

42 Discretization Stochastic discretization : Parametrization of ν(θ) and F (θ) using N i.i.d. random variables : ξ = {ξ 1,..., ξ N } Stochastic basis : Polynomial Chaos λ(θ) = α λ α Ψ α (ξ(θ)), where the Ψs are orthonormal polynomials E [Ψ α (ξ)ψ β (ξ)] = δ α,β. Truncature to (total) polynomial degree No : dim S P = (No + N)!. No!N!

43 Discretization Spatial discretization P n -P n 2 Spectral Element Method H 1 (Ω) u(x) u h (x) = L 2 (Ω) p(x) p h (x) = n u i,j φ u i,j (x) Vh, i,j=1 n 2 i,j=1 p i,j φ p i,j (x) Πh. (Nodal basis on tenzorized Gauss-Lobatto grid) Resolution of the deterministic problem u = D(λ; U (i) ) H 1 0;div (Ω) achieved by solving in Vh 0 the constrained problem E [ λ 3] c(u h, u h, v h ) + E [ λ 2 ν ] a(u h, v h ) + c(u h, ũ h, v h ) + c(ũ h, u h, v h ) + d(v h, p h ) = ( f, v h ) Ω v h V h 0, d(u h, q h ) = 0 q Π h.

44 Results Case of a deterministic forcing and a random (Log-normal) viscosity : p ν pdf ( ν(θ) = exp σ ν N N i= ν ξ i (θ) ) (+10 4 ), ξ i N(0, 1) i.i.d. Same problem but for parametrization involving N Gaussian R.V.

45 Results Galerkin solution for N = 1 and No = 10 (Wiener-Hermite expansion) Mean and standard deviation of U G rotational.

46 Results POD modes of the Galerkin solution for N = 1 and No = Galerkin Rotational of the spatial POD modes of U G m

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

48 Results POD modes of the PGD solution for m = 15, N = 1 and No = Galerkin Reduced λ i Rotational of the spatial POD modes of U m= m

49 Results Convergence of PGD solution N = 1 and No = Galerkin Reduced λ i Residual Error m m Comparison of the norms of the POD coefficients at m = 15 (left), residual norm (center), error norm (right). m

50 Results Convergence of PGD solutions N = 1, 2 and 3 at No = Galerkin Reduced P=10 Reduced P=66 Reduced P= Reduced P=10 Reduced P=166 Reduced P= Reduced P=10 Reduced P=66 Reduced P= m m m 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

51 Results Stochastic forcing F : Hodge s decomposition F (x, θ) = Φ(x, θ) + Ψ(x, θ) = Ψ(x, θ)k. Define F ω = ( F ) k = Ψ and assume : F ω (x, θ) = fω 0 + F ω(x, θ), E [F ω(x, )] = 0, F ω(x, ) N(0, σ 2 ), E [F ω(x, )F ω(x, )] = σ 2 exp( x x /L). KL expansion of F ω : N F ω (x, θ) Fω N (x, ξ(θ)) = fω 0 + γk fω(x)ξ k k (θ) k=0 It comes N F (x, θ) F N (x, ξ(θ)) = f 0 + γk f k (x)ξ k (θ), with f i = ( 1 f i ω) k. k=0

52 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

53 Results PGD solution for ν = 1/50, N = 11, No = 3 and m = Mean and standard deviation of U m=45

54 Results First PGD-Arnoldi modes

55 Results Results at ν = 1/50 : No = 3, N = 11, P = Residual Error Galerkin Reduced m=45 λ i m m Residual (left), U m U G (center) and norm of POD modes for m = 45 (right). m

56 Results Results for ν = 1/50 : No = 3, N = 11, P = 364 m = 0 m = 2 m = e e e e e e e e e e e e e e e e e e e e e e e e+00 m = 26 m = 43 m = e e e e e e e e e e e e e e e e e e e e e e e e+00 Spatial distribution of the stochastic residual.

57 Results Residual computation : Residual computation in H 1 0;div (Ω) is difficult Projection of the H 1 0 (Ω)-residual into H1 0;div (Ω) amount to solving a stochastic linear equation for P m PGD can be used for the approximation of P m Re-use of the reduced basis of Arnoldi Lagrange multipliers {p i } associated to {u i } was found a effective approach However, monitoring convergence using the decay of λ i appears effective

58 Results Residual computation : 10 0 error 10 0 error 10 0 error 10 2 LM residual PGD residual λ norm 10 2 LM residual PGD residual λ norm 10 2 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).

59 On-going work : Evolution problems : PGD on each time step or global PGD with possibly alternative separated forms u(x, t, θ) α α u α (x, t)λ α (θ) α û α (x)τ α (t)γ α (θ) ũ α (x)β α (t, θ) Nested separation? Other definition of optimality for non definite linear operators (least square residual minimization).

60 Thanks for your attention Fundings : ANR-TYCHE project [ ] ANR-ASRMEI project [ ] MoMaS (Andra, Brgm, Cea, EdF, Irsn) [ ]