Optimal polynomial approximation of PDEs with stochastic coefficients by Galerkin and collocation methods

Transcription

1 Optimal polynomial approximation of PDEs with stochastic coefficients by Galerkin and collocation methods Fabio Nobile MOX, Department of Mathematics, Politecnico di Milano Seminaire du Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie (Paris VI), May 27, 2011 Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

2 Acknowledgments Collaborators: I. Babuška, L. Tamellini, M. Mohammad, R. Tempone, J. Bäck Funds: Italian project FIRB-IDEAS ( 09) Advanced Numerical Techniques for Uncertainty Quantification in Engineering and Life Science Problems Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

3 Outline 1 Problem setting: PDEs with random coefficients 2 Techniques for computing polynomial approximations Galerkin projection Collocation on sparse grids numerical comparison 3 Optimization of polynomial spaces 4 Numerical examples 5 Hyperbolic problems with random coefficients numerical results 6 Conclusions Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

4 Problem setting: PDEs with random coefficients Problem setting Consider a differential problem find u : L(y)(u) = F where the operator L(y) depends on a vector of N parameters: y = (y 1,..., y N ). The parameters are not perfectly known, or they have a lot of variability in repeated experiments Treat them as random variables with a given probability density fuction (estimated from experiments or from prior knowledge / expert opinion): ρ(y) : Γ R +, ρ(y)dy = 1 Γ We assume that for each y Γ the corresponding solution u = u(y) belongs to a given functional space V = u = u(y) : Γ V Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

5 Problem setting: PDEs with random coefficients Problem setting Goal: compute statistics of the solution u or some quantity of interest J(u). E.g. J = E[J(u)] = J(u(y))ρ(y)dy, mean value Γ Var[J] = E[J(u) 2 ] E[J(u)] 2 variance P[J(u) > J cr ] = 1 {J(u(y))>Jcr }ρ(y)dy exceedance prob. Γ Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

6 Problem setting: PDEs with random coefficients Multivariate polynomial approximation Computations of statistical quantities typically imply lots of evaluations of u(y) lots of problems to solve Idea: build a reduced model u Λ (y) u(y) that is cheap to evaluate and use it to compute statistical moments. E.g. J J Λ = E[J(u Λ )] In this talk we consider multivariate polynomial reduced models. Let Λ N N be an index set of cardinality Λ = M, and consider the multivariate polynomial space { N } P Λ (Γ N ) = span n=1 y n pn, with p = (p 1,..., p N ) Λ find M particular solutions u p V, p Λ and build u Λ (y) = p Λ u p y p 1 1 y p 2 2 y p N N Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

7 Problem setting: PDEs with random coefficients Examples of pol. spaces: N = 2, p = Tensor Product (TP) 16 Total Degree (TD) Tensor product: p n w p p Total degree: n p n w p p 1 16 Hyperbolic Cross (HC) 16 Smolyak (SM) Hyperbolic cross: n (p n + 1) w + 1 p p Smolyak: n f (p n) f (w) 8 >< 0, p = 0 f (p) = 1, p = 1 >: log 2 (p), p p p 1 Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

8 Problem setting: PDEs with random coefficients Anisotropic spaces: N = 2, p max = Aniso Tensor Product (ATP) 16 Aniso Total Degree (ATD) Tensor product: α n p n w p p Total degree: n α np n w p p 1 16 Aniso Hyperbolic Cross (AHC) 16 Aniso Smolyak (ASM) Hyperbolic cross: n (p n + 1) αn w + 1 p p 1 p p 1 Smolyak: α n f (p n ) f (w) n 8 >< 0, p = 0 f (p) = 1, p = 1 >: log 2 (p), p 2 Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

9 Problem setting: PDEs with random coefficients Accuracy requirements We look at mean square error control (strong convergence) err 2,ρ = u(y) u Λ (y) 2 V ρ(y)dy small Γ An easy implication: [N.-Tempone, IJNME 09] Assume u(y) V and u Λ (y) V uniformly bounded in Γ Let J : V R a locally Lipschitz functional, with J(0) = 0 Then E[J(u) q J(u Λ ) q ] C(q)E[ u u Λ 2 V ] 1 2 i.e. convergence in E[ u u Λ 2 V ] 1 2 implies convergence of all moments of the funciontal J. Weak convergence could (should) be considered as well. Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

10 Problem setting: PDEs with random coefficients Example I: Thermal conduction with inclusions of random conductivity (baked cookies problem) { div(a u) = f, u = 0 in D on D Each circular inclusion C i, i = 1,... 8 has a random conductivity coefficient y i. a(y 1,..., y N, x) = a 0 + N (y n a 0 )1 Cn (x), x D, y Γ n=1 8-dimensional parametric problem Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

11 Problem setting: PDEs with random coefficients Example II: Darcy flow in a medium with random permeability (groundwater flow problem) { u = a p div u = f a = a(ω, x): random permeability field (with a > 0 a.s.) Each realization of the stochastic process gives a spatially varying permeability field. The random field can be conditioned to available measurements (e.g. by Kriging techniques) Infinite-dimensional parametric problem! Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

12 Problem setting: PDEs with random coefficients Approximation of an -dimensional random field Let {b n (x)} be a complete orthonormal basis in L 2 (D) (trigon., wavelet, Karhunen-Loève,...) and a(ω, x) a -dimensional random field with finite second moments. Then, a can be expanded as a(ω, x) = E[a](x) + y n (ω)b n (x) with y n (ω) = D n=1 (a(ω, x) E[a](x))b n (x) dx If the basis {b n } has spectral approx. properties and the realizations of a are smooth, then Var[y n ] n 0 suff. fast and we can truncate the series a(ω, x) a N (ω, x) = E[a](x) + N y n (ω)b n (x) WARNING: the truncated expansion might not be positive almost surely! Possible remedy: a(ω, x) a N (ω, x) = a min + e b0(x)+p N n=1 yn(ω)bn(x) Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27, n=1

13 Techniques for computing polynomial approximations Galerkin projection Galerkin projection [Ghanem-Spanos, Karniadakis et al, Matthies-Keese, Schwab-Todor et al., Knio-Le Maître et al,babuska et al.,... ] Project the equation onto the subspace P Λ (Γ) Suitable for stochastic problems Let {ψ j } M j=1 be an orthonormal basis w.r.t. the probability density ρ(y). Expand u Λ (y) on the basis: u Λ (y) = M j=1 u jψ j (y) Galerkin formulation Find u j V, j = 1,... M s.t. [ E L(y)( M ] u j ψ j )ψ i = E[Fψ i ], i = 1,..., M j=1 This approach leads to solving M coupled deterministic problems; difficult to assemble and need good preconditioners. Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

14 Techniques for computing polynomial approximations Collocation on sparse grids Collocation on sparse grids [Smolyak 63, Griebel et al , Barthelmann-Novak-Ritter 00, Hesthaven-Xiu 05, N.-Tempone-Webster 08, Zabaras et al 07] 1 Choose a set of points y (j) Γ, j = 1,..., M 2 Compute the solutions u j V : L(y (j) )(u j ) = F 3 Interpolate the obtained values: u Λ (y) = e M j=1 u jφ j (y). φ j P Λ (Γ): suitable combinations of Lagrange polynomials Always leads to solving M uncoupled deterministic problems The number M of points needed is larger than the dimension M of the polynomial space (Except for tensor product spaces). Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

15 Techniques for computing polynomial approximations Collocation on sparse grids (Generalized) Sparse Grid approximation 1 Choose 1D abscissae. E.g. Clenshaw-Curtis (extrema on Chebyshev polynomials) Gauss points w.r.t. the weight ρ n, assuming that the probability density factorizes as ρ(y) = N n=1 ρ n(y n ) 2 Take a sequence of 1D polynomial interpolant operators : C 0 (Γ n ) P m(i) 1 (Γ n ) with increasing number of points U m(i) n The i-th interpolant uses m(i) abscissae ϑ i n = { y i n,1,..., y i n,m i }. 3 Take differences of consecutive operators: m(i) n = U m(i) n U m(i 1) n, U m(0) n = 0. 4 Multidimensional Smolyak approx.: let i = [i 1,..., i N ] N N +, and Λ N N an index set uλ SC = ( ) m(i 1) 1 m(i N) N (u) i Λ Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

16 Techniques for computing polynomial approximations Collocation on sparse grids By choosing properly the function m and the set Λ one can obtain a polynomial approximation in any given multivariate polynomial space ([Back-N.-Tamellini-Tempone LNCSE vol. 76, 2010]) Examples of sparse grids: N = 2, max. polynomial degree p = 16 1 Tensor Product (TP) 1 Total Degree (TD) y 2 0 y y y 1 1 Hyperbolic Cross (HC) 1 Smolyak Gauss (SM) 1 Smolyak CC (SM) y 2 0 y 2 0 y y y y 1 Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

17 Techniques for computing polynomial approximations numerical comparison Thermal conduction with random inclusions Conductivity coefficient: matrix k=1 circular inclusions: k Ωi U(0.01, 0.8) 8 iid uniform random variables forcing term f = 1001 F zero boundary conditions quantity of interest ψ(u) = F u mean std Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

18 Techniques for computing polynomial approximations Convergence plot for E[ψ(u)] numerical comparison E(ψ 1,p ) E(ψ 1,ovk ) MC SG TP SG TD SG HC SG SM SG: dim stoc * iter CG / MC: sample size Galerkin E(ψ 1,p ) E(ψ 1,ovk ) SG TD MC SC TP SC TD SC HC SC SM G SC SM CC SC: num pts / SG: dim stoc * iter CG / MC: sample size Collocation error versus estimated cost (see [Back-N.-Tamellini-Tempone, LNCSE 10]) Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

19 Techniques for computing polynomial approximations numerical comparison Thermal conduction with random inclusions anisotropic version Conductivity coefficient: matrix k=1 circular inclusions: k Ωi γ i U(0.01, 0.8) 4 indep. uniform random variables forcing term f = 100 zero boundary conditions quantity of interest ψ(u) = F u mean Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27, std

20 Techniques for computing polynomial approximations Convergence plot for E[ψ(u)] numerical comparison Anisotropic Total degree spaces with different anisotropy weights α n E(ψ 1 ) 10 3 E(ψ 1 ) MC SG isospaces 10 5 SG SG SG (exp) 10 6 SG (th) SG: dim stoc * iter CG / MC: sample size Galerkin 10 4 MC SC isospaces 10 5 SC SC SC (exp) 10 6 SC (th) SC: num pts / SG: dim stoc * iter CG / MC: sample size Collocation Error versus estimated cost (see [Back-N.-Tamellini-Tempone, LNCSE 10]) Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

21 Optimization of polynomial spaces Optimization of polynomial spaces Consider the diffusion problem { div(a(x, y 1,..., y N ) u) = f, u = 0, in D on D assume a(y) α > 0, y Γ (uniform coerciveness) Analyticity result [Back-N.-Tamellini-Tempone 11, Babuska-N.-Tempone 05, Cohen-DeVore-Schwab 09/ 10] let i = (i 1,..., i N ) N N and r = (r 1,..., r N ) > 0. Set r i = n r in n. assume 1 a i a y i L (D) r i uniformly in y Then i u y i V C i!( log 2 r) i uniformly in y u : Γ{ V is analytic and can be extended analyticially to Σ = z C N : } N n=1 r n z n ỹ n < log 2 for some ỹ Γ Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

22 Optimization of polynomial spaces Remark: The assumption is satisfied both for linear and exponential expansions of a random field: linear expansion: a(y, x) = a 0 + N n=1 b n(x)y n, with a min = a 0 N n=1 b n L (D) > 0. In this case r n = b n L (D)/a min ( N ) exponential expansion: a(y, x) = a 0 + exp n=1 b n(x)y n with a 0 > 0. In this case: r n = b n L (D). Better estimates on analyticity region can be obtained by complex analysis. Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

23 Optimization of polynomial spaces Optimization of polynomial spaces Galerkin u SG Λ = p Λ u p (x)ψ p (y) find uλ SG by Galerkin projection of the equation on P Λ = span{ψ p, p Λ}. Collocation u SC Λ = i Λ n=1,...,n m(in) n [u]. Compute uλ SC by collocation on the corresponding sparse grid Question: What is the best index set Λ in both cases? Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

24 Optimization of polynomial spaces Galerkin projection best M term approximation Galerkin optimality: u u SG Λ V L 2 ρ (Γ) C inf v Λ V P Λ u v Λ V L 2 ρ (Γ) Let {ψ p, p N N } be the orthonormal basis of Legendre multivariate polynomials and v Λ the truncated Legendre expansion of u v Λ = p Λ E[uψ p ]ψ p Parseval s identity: u v Λ 2 V L 2 ρ (Γ) = u p Λ E[uψ p]ψ p 2 V L 2 ρ (Γ) = p/ Λ E[uψ p] 2 V Best M terms approximation The optimal index set Λ of cardinality M is the one that contains the M largest Legendre coefficients E[uψ p ] V Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

25 Optimization of polynomial spaces Abstract construction of quasi optimal spaces Suppose we have an a priori estimate of the form E[uψ p ] V G(p) (1) Fix a threshold ɛ R +, and define the index set Λ as or equivalently Λ(ɛ) = { p N N : G(p) ɛ } Λ(w) = { p N N : log G(p) w, w = log ɛ } The sharper the estimate (1), the better Λ approximates the best M terms index set. Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

26 Optimization of polynomial spaces Estimate of Legendre coefficients For the diffusion problem with random coefficients, the solution u(y) is analytic in Γ and the following estimate of the Legendre coefficients holds [Cohen-DeVore-Schwab 10, Back-N.-Tamellini-Tempone 11] E[uψ p ] V C 0 e P p! n gnpn p! for some g n > 0, with p = n p n, p! = n p n!. Then the induced optimal index set is (TD-FC) { Λ(w) = p N N : g n p n log p! p! n } w (2) The factorial term p! p! accounts for the interaction between the random variables and is purely isotropic Estimate (2) is meaningful only if n e gn < 1! Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

27 Numerical tests Optimization of polynomial spaces We consider the 1D problem { (a(x, y)u(x, y) ) = 1 u(0, y) = u(1, y) = 0, x D = (0, 1), y Γ y Γ with several choices of a(x, y) and compute Θ(u) = u( 1 2 ). We compare: (Aniso) TD space: Λ(w) = (Aniso) TD-FC space: { Λ(w) = { p N N : p N N : N n=1 } g n p n w. n g n p n log p! p! w Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27, }.

28 Optimization of polynomial spaces Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27, Test 1: a(x, y) = y y Legendre coeff TD appr 10 5 TD FC appr best M term 10 3 TD iso TD TD FC Figure: Legendre coeffs of Θ(u) in lexicographic order, with TD and TD-FC estimates Figure: Convergence plot for Θ(u) Θ(uM) 2 L 2 ρ(γ) w.r.t. M = Λ The Legendre coefficients have been computed with a sufficiently high level sparse grids.

29 Optimization of polynomial spaces true Legendre coeffs iso-td estimate aniso TD estimate TD-FC estimate. Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

30 Test 2 Optimization of polynomial spaces log a(x, y) = y sin(πx)y sin(2πx)y sin(3πx)y 4 3 best M term 10 TD iso TD TD FC Convergence plot for Θ(u) Θ(u M ) 2 L 2 ρ(γ) w.r.t. M Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

31 Optimization of polynomial spaces Optimization of sparse grids u M = S m Λ [u] = i Λ N n=1 m(in) n [u]. We use a knapsack problem-approach [Griebel-Knapek 09, Gerstner-Griebel 03, Bungartz-Griebel 04]: for each multiindex i estimate E(i): how much error decreases if i is added to Λ (error contribution) W (i): how much work, i.e. number of evaluations, increases if i is added to Λ (work contribution) Then estimate the profit of each i as P(i) = E(i) W (i) and build the sparse grid using the set Λ of the M indices with the largest profit. Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

32 Estimate for W Optimization of polynomial spaces Suppose we use nested abscissae, e.g. Clenshaw Curtis. The number of points added to the grid by i is then W (i) = nb. new pts. in N n=1 m(in) n = N ( m(i n ) m(i n 1) ) n=1 Recall that for Clenshaw-Curtis 0 if i = 0 m(i) = 1 if i = 1 2 i 1 + 1, if i > 1, and the Lebesgue constant is L(m(i)) = 2 π log(m(i n) + 1) + 1 Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

33 Estimate for E 1 rewrite E(i) as Optimization of polynomial spaces E(i) = (u SΛ [u]) ( u S {Λ i} [u] ) = V L 2 ρ (Γ) m(j) [u] m(j) [u] V L 2 ρ (Γ) = m(i) [u]. V L 2 ρ (Γ) j {Λ i} j {Λ} 2 use the following estimate (numerically validated) E(i)[u] um(i 1) V N n=1 ( 1 + Lm(in 1)) where u m(i 1) is the corresponding Legendre coefficient. 3 estimate u m(i 1) as in the Galerkin case um(i 1) V C 0 e P n gnm(in 1) m(i 1)! m(i 1)! Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

34 Optimization of polynomial spaces Example: Comparison E vs. estimate: Let y 1, y 2 U( 1, 1). { [(1 + c 1 y 1 + c 2 y 2 ) u(x, y 1, y 2 ) ] = f (x) u(x) = 0 x D x D u(x, y 1, y 2 ) = ( Nested knots: Clenshaw-Curtis: m(i) = 2 i+1 1, y k = cos 1 f (x) 1+c 1y 1+c 2y 2 admits a Legendre expansion. kπ m(i) ) 10 0 m(i) 10 2 Leg m(i 1) f m(i 1) Leb(m(i)) 10 0 m(i) 10 2 Leg m(i 1) f m(i 1) Leb(m(i)) c 1 = 0.3, c 2 = c 1 = 0.1, c 2 = 0.5 Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

35 Optimization of polynomial spaces All the pieces together The set {i N N : P(i) ɛ} is then equivalent to { i N N + : N m(i n 1)g n log i=n (EW - Error Work grids) where m(i 1)! m(i 1)! N log n=1 2 π log(m(i n 1) + 1) + 2 w m(i n ) m(i n 1) Legendre coeff + Lebesgue constant = error estimate work estimate } Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

36 Numerical examples Numerical test 1 - Uniform case { (a(x, y)u(x, y) ) = 1 x D = (0, 1), u(0, y) = u(1, y) = 0 y Γ = [ 1, 1] N, N = 2, 4 different choices of diffusion coefficient a(x, y). We focus on a linear functional ψ : V R, ψ(v) = v( 1 2 ); Convergence: ψ(u SG ) ψ(u) L 2 ρ (Γ) vs. nb of points in sparse grid We compare standard isotropic Smolyak Sp. Grid, I = {i N N : N n=1 (i n 1) w} the Knapsack grid derived best M terms : knapsack grid, with computed profits P(i) dimension adaptive algorithm [Gerstner-Griebel 03, Klimke, PhD 06], Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

37 Numerical examples iso SM EW adaptive best M terms iso SM adaptive best M terms EW a = y y 2 a = y y Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

38 Numerical examples iso SM EW adaptive best M terms iso SM EW adaptive best M terms a(x, y) = 4 + y sin(πx)y sin(2πx)y sin(3πx)y log a(x, y) = y sin(πx)y sin(2πx)y sin(3πx)y 4 Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

39 Numerical examples Numerical test 2-1D lognormal field L = 1, D = [0, L] 2. a(y, x) u(y, x) = 0 u = 1 on x = 0, h = 0 on x = 1 no flux otherwise We approximate γ as γ(y, x) µ(x) + σa 0 Y 0 + σ with Y i N (0, 1), i.i.d. a(x, y) = e γ(x,y) µ γ (x) = 0 Cov γ (x, x ) = σ 2 e x 1 x LC 2 K ( π ) ( π )] a k [Y 2k 1 cos L kx 1 + Y 2k sin L kx 1 k=1 Given the Fourier series σ 2 e z 2 LC 2 = k=0 c k cos ( π L kz), a k = c k. 1 2 Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

40 Numerical examples Numerical test 2-1D lognormal field Quantity of interest: [ ] L u(, x) E[Φ(u)], with Φ = k(, x) dx x 0 Convergence: E[Φ(u SG )] E[Φ(u)] We compare Monte Carlo estimate with Knapsack grids Gauss-Hermite-Patterson points (nested Gauss-Hermite) Estimate of Hermite coefficients decay (heuristic) u i V N n=1 e gnin in! Estimate of Lebesgue constant (heuristic) L m(in) n 1 Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

41 Numerical examples Numerical test 2-1D lognormal field Here LC = 0.2, σ = 0.3. K = 6 N = 13 r.v., and 99% of total variability of e γ. K = 10 N = 21 r.v., and 99.99% of total variability of e γ mean MC MC MC MC MC MC slope sparse grid 21 var sparse grid 13 var Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

42 Numerical examples Numerical test 3-2D lognormal field L = 1, D = [0, L] 2. { a(x, y) h(y, x) = 0 B.C. : see figure We approximate γ as a(x, y) = e γ(y,x) µ γ (x) = 0 Cov γ (x, x ) = σ 2 e x x 2 LC 2 γ(x, y) µ(x) + σ X k K a k [Y k,1 cos (πk 1 x 1 ) cos (πk 2 x 2 ) + Y k,2 cos (πk 1 x 1 ) sin (πk 2 x 2 ) + Y k,3 sin (πk 1 x 1 ) cos (πk 2 x 2 ) + Y k,4 sin (πk 1 x 1 ) sin (πk 2 x 2 )] with Y i N (0, 1), i.i.d. Given the Fourier series σ 2 e z 2 LC 2 = k=0 c k cos ( π L kz), a k = c k1 c k2. Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

43 Numerical examples Numerical test 3-2D lognormal field Here LC = 0.4, σ = 0.3. N = 21 r.v., 92% of total variability of e γ mean MC MC MC 10 6 MC MC MC slope sparse grid Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

44 Hyperbolic problems with random coefficients Hyperbolic problems 2 u div(a 2 (x, y t 2 1 (ω),..., y N (ω)) u) = f, in D, t > 0 u = 0, on D, t > 0 u u t=0 = u 0, t t=0 = v 0, in D assume a(x, y(ω)) a max <, y Γ, x D boundedness) (uniform The solution is in general not smooth Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

45 Hyperbolic problems with random coefficients Example: 1D problem { 2 u t 2 y 2 2 u = 0, x 2 in R, t > 0 u(x, 0) = u 0 (x), (x, 0) = 0, in R D Alambert formula: u(x, t) = 1 2 u 0(x yt) u 0(x + yt) u t = u y (x, t) = t u 0 + t u 0 2 ξ ξ=x yt 2 ξ ξ=x+yt Remarks: u 0 ( ) C k (R) u(y) (x,t) C k (Γ) In particular, a discontinuous initial datum implies discontinuous dependendence on the parameter (wave speed) However, consider a functional J(y) = u(y; x, T )g(x)dx with R g C m (R) with compact support. Then, J(y) (x,t) C k+m (Γ). Linear functionals of the solution can be much smoother than the solution itself. It might still be good to approximate J(y) by polynomials (not u itself). Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

46 Hyperbolic problems with random coefficients Layered random medium Assume that the wave speed has the form a(x, y(ω)) = N y n (ω)ρ n (x)1 Dn (x) n=1 where 1 Dn are characteristic functions corresponding to a non-overlapping partition of the domain: D = N n=1d n with smooth interfaces, ρ n are smooth funtions and y n Γ n are random variables. Result 1 [Motamed-N.-Tempone 11] Given f L 2 (0, T ; H 1 0 (D)), u 0 H 1 0 (D), v 0 L 2 (D), the solution has in general only one bounded derivative yn u L 2 (0, T ; L 2 (D)). The solution might be smoother for smoother data not intersecting any interface between the layers. Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

47 Hyperbolic problems with random coefficients Layered random medium Result 2 [Motamed-N.-Tempone 11] Consider a finite dimensional approximation of the equation in space with discretization parameter h. The discrete solution u h (y) is always analytic with respect to y for all y R N. However, if we replace y with z Σ {z C N : dist(γ, z) τ}, and study the problem in the complex domain max z C N u h (z) L 2 (0,T ;H 1 0 ) C h τ exp{γt τ h } Consider a tensor product polynomial approximation if y of degree p. We have two regimes for hp CT exponential convergence in p for hp CT algebraic slow convergence in p due to small regularity of u w.r.t. y. Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

48 Hyperbolic problems with random coefficients numerical results Wave equation in two layered random medium layer 1 layer 2 y ~ U(0.2, 1) 1 y ~ U(0.3, 1.5) 2 wave equation in two-layered medium random wave speed in each layer smooth initial deformation across the interface initial solution E[u](x, t = 1) std[u](x, t = 1) Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

49 Hyperbolic problems with random coefficients numerical results Isotropic Smolyak grid approximation on Gauss-Legendre abscissae Finite difference approximation in space + leapfrog in time Maximum error in the expected value at t=1 versus # of collocation points M e2 ε h, x = 0.1 ε h, x = ε h, x = ε h, x = η 1/ e1 The solution has low regularity slow convergence (the convergence rate in M denpends on the mesh size x) Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

50 Smooth case: Hyperbolic problems with random coefficients numerical results The initial displacement is smooth and confined in the second layer Maximum error in the expected value at t=1 versus # of collocation points M. layer 1 layer 2 y 1~ U(0.2, 1) y 2~ U(0.3, 1.5) e ε, x = 0.1 h ε h, x = 0.05 ε h, x = ε, x = h e1 Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

51 Hyperbolic problems with random coefficients numerical results Conclusions 1 Solution may depend on a high number of parameters / random variables. An accurate choice of the approximation space is needed, to avoid unaffordable computational costs 2 Under regularity assumptions it makes sense to look for global polynomial approximations, either modal (galerkin procedure) or nodal (collocation procedure) 3 We propose a general procedure based on estimates of Legendre coefficients / profits of grids to build optimal polynomial approximations 4 There is no one for all recipe: the structure of the problem (hence of the solution) leads to the appropriate choice of approximation Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,

52 References Conclusions J. Bäck and F. Nobile and L. Tamellini and R. Tempone On the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocation methods, MOX Report 2011, submitted. J. Bäck and F. Nobile and L. Tamellini and R. Tempone Stochastic Spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison, LNCSE Springer, Vol. 76, I. Babuška, F. Nobile and R. Tempone. A stochastic collocation method for elliptic PDEs with random input data, SIAM Review, 52(2): , 2010 F. Nobile and R. Tempone Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients, IJNME, 80: , 2009 F. Nobile, R. Tempone and C. Webster An anisotropic sparse grid stochastic collocation method for PDEs with random input data, SIAM J. Numer. Anal., 46(5): , 2008 F. Nobile, R. Tempone and C. Webster A sparse grid stochastic collocation method for PDEs with random input data, SIAM J. Numer. Anal., 46(5), , 2008 Fabio Nobile (MOX, Politecnico di Milano) Polynomial approx. of stochastic PDEs LJLL, Paris, May 27,