Low-rank techniques applied to moment equations for the stochastic Darcy problem with lognormal permeability

Transcription

1 Low-rank techniques applied to moment equations for the stochastic arcy problem with lognormal permeability Francesca Bonizzoni 1,2 and Fabio Nobile 1 1 CSQI-MATHICSE, EPFL, Switzerland 2 MOX, Politecnico di Milano, Italy Acknowlegments:. Kressner (EPFL), C. Tobler (EPFL), R. Kumar (RICAM) Workshop on Partial ifferential Equations with Random Coefficients Berlin, November 13, 2013 Italian project FIRB-IEAS ( 09) Advanced Numerical Techniques for Uncertainty Quantification in Engineering and Life Science Problems F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

2 Outline 1 The lognormal arcy problem 2 Perturbation approach and moment equations 3 Approximation properties of the Taylor polynomial 4 Moment equations: well posedness and discretization 5 Tensor Train approximation 6 1 Numerical experiments F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

3 Outline The lognormal arcy problem 1 The lognormal arcy problem 2 Perturbation approach and moment equations 3 Approximation properties of the Taylor polynomial 4 Moment equations: well posedness and discretization 5 Tensor Train approximation 6 1 Numerical experiments F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

4 The lognormal arcy problem arcy problem with log-normal permeability We study the groundwater flow in a saturated heterogeneous medium where the permeability is described as a log-normal stochastic r.f. (model widely used in geophysical applications) div(e Y (ω,x) u(ω, x)) = f (x), a.e. in R d, d = 1, 2, 3 u(ω, x) = g(x), a.e. on Γ, e Y (ω,x) n u(ω, x) = h(x), a.e. on Γ N. Y (ω, x) : Gaussian r.f., E [Y ] (x) = µ(x), Cov [Y ] (x, y) = ρ(x, y), ( ) 1 σ := ρ(x, x)dx 2 < 1. 1 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

5 The lognormal arcy problem arcy problem with log-normal permeability We study the groundwater flow in a saturated heterogeneous medium where the permeability is described as a log-normal stochastic r.f. (model widely used in geophysical applications) div(e Y (ω,x) u(ω, x)) = f (x), a.e. in R d, d = 1, 2, 3 u(ω, x) = g(x), a.e. on Γ, e Y (ω,x) n u(ω, x) = h(x), a.e. on Γ N. Y (ω, x) : Gaussian r.f., E [Y ] (x) = µ(x), Cov [Y ] (x, y) = ρ(x, y), ( ) 1 σ := ρ(x, x)dx 2 < 1. 1 Assumption: Cov [Y ] C 0,t ( ) for some 0 < t 1. = Y a.s. continuous and Y L () Lp (Ω), p 1 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

6 The lognormal arcy problem arcy problem with log-normal permeability We study the groundwater flow in a saturated heterogeneous medium where the permeability is described as a log-normal stochastic r.f. (model widely used in geophysical applications) div(e Y (ω,x) u(ω, x)) = f (x), a.e. in R d, d = 1, 2, 3 u(ω, x) = g(x), a.e. on Γ, e Y (ω,x) n u(ω, x) = h(x), a.e. on Γ N. Y (ω, x) : Gaussian r.f., E [Y ] (x) = µ(x), Cov [Y ] (x, y) = ρ(x, y), ( ) 1 σ := ρ(x, x)dx 2 < 1. 1 Assumption: Cov [Y ] C 0,t ( ) for some 0 < t 1. = Y a.s. continuous and Y L () Lp (Ω), p 1 Under the above assump. the prb. admits a unique solution u L p (Ω; H 1 ()), p 1. [Galvis Sarkis, 2009, Gittelson, 2010, Charrier ebussche, 2013] F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

7 The lognormal arcy problem Goal: Compute statistical quantities for u, i.e. assess how the uncertainty on the permeability reflects on u. Expected value E [u] (x) := u(ω, x)dp(ω) Ω Variance Var [u] (x) := E [ u 2] (x) E [u] 2 (x) m-points correlation E [u m ] (x 1,..., x m ) := E [u(ω, x 1 )... u(ω, x m )] Method adopted: Moment equations erive, theoretically analyze and numerically solve the deterministic equations solved by the statistical moments of the stochastic solution F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

8 Outline Perturbation approach and moment equations 1 The lognormal arcy problem 2 Perturbation approach and moment equations 3 Approximation properties of the Taylor polynomial 4 Moment equations: well posedness and discretization 5 Tensor Train approximation 6 1 Numerical experiments F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

9 Perturbation approach and moment equations Perturbation approach and moment equations (see e.g. Hydrology literature: [Tartakovsky Neuman, 2008], [Riva Guadagnini e Simoni, 2006], Math. literature: [von Petersdorff Schwab, 2006], [Todor Ph, 2005], [Harbrecht Schneider Schwab, 2008]) The proposed approach to compute moments of the solution relies on the following 3 steps: F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

10 Perturbation approach and moment equations Perturbation approach and moment equations (see e.g. Hydrology literature: [Tartakovsky Neuman, 2008], [Riva Guadagnini e Simoni, 2006], Math. literature: [von Petersdorff Schwab, 2006], [Todor Ph, 2005], [Harbrecht Schneider Schwab, 2008]) The proposed approach to compute moments of the solution relies on the following 3 steps: Step 1. Formally write the Taylor polynomial of u(y, x) w.r.t. Y, centered in E [Y ]. u T K u(y, x) = K k=0 u k (Y, x), k! u k = k [E [Y ]](Y,..., Y ) k-th Gateaux derivative of u F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

11 Perturbation approach and moment equations Perturbation approach and moment equations (see e.g. Hydrology literature: [Tartakovsky Neuman, 2008], [Riva Guadagnini e Simoni, 2006], Math. literature: [von Petersdorff Schwab, 2006], [Todor Ph, 2005], [Harbrecht Schneider Schwab, 2008]) The proposed approach to compute moments of the solution relies on the following 3 steps: Step 1. Formally write the Taylor polynomial of u(y, x) w.r.t. Y, centered in E [Y ]. u T K u(y, x) = K k=0 u k (Y, x), k! u k = k [E [Y ]](Y,..., Y ) k-th Gateaux derivative of u The k-th Gateaux derivative satisfies a recursive problem (for simplicity here E [Y ] = 0) u k (x) v(x) dx = k l=1 ( k l ) Y l (x) u k l (x) v(x) dx F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

12 Perturbation approach and moment equations Perturbation approach and moment equations (see e.g. Hydrology literature: [Tartakovsky Neuman, 2008], [Riva Guadagnini e Simoni, 2006], Math. literature: [von Petersdorff Schwab, 2006], [Todor Ph, 2005], [Harbrecht Schneider Schwab, 2008]) The proposed approach to compute moments of the solution relies on the following 3 steps: Step 1. Formally write the Taylor polynomial of u(y, x) w.r.t. Y, centered in E [Y ]. u T K u(y, x) = K k=0 u k (Y, x), k! u k = k [E [Y ]](Y,..., Y ) k-th Gateaux derivative of u The k-th Gateaux derivative satisfies a recursive problem (for simplicity here E [Y ] = 0) u k (x) v(x) dx = k l=1 ( k l ) Y l (x) u k l (x) v(x) dx The derivatives u k are not directly computable (they are still -dimensional random fields) F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

13 Perturbation approach and moment equations Perturbation approach and moment equations Step 2. Approximate the moments of u using the Taylor expansion; e.g. for the first moment: E[u](x) E [ T K u(y, x) ] K E[u k ](x) = k! k=0 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

14 Perturbation approach and moment equations Perturbation approach and moment equations Step 2. Approximate the moments of u using the Taylor expansion; e.g. for the first moment: E[u](x) E [ T K u(y, x) ] K E[u k ](x) = k! k=0 Each correction term E[u k ] to the mean satisfies the recursion E[u k ](x) v(x) dx = k l=1 ( k l ) E[Y l (x) u k l (x)] v dx F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

15 Perturbation approach and moment equations Perturbation approach and moment equations Step 2. Approximate the moments of u using the Taylor expansion; e.g. for the first moment: E[u](x) E [ T K u(y, x) ] K E[u k ](x) = k! k=0 Each correction term E[u k ] to the mean satisfies the recursion E[u k ](x) v(x) dx = k l=1 ( k l ) E[Y l (x) u k l (x)] v dx efine the (l + 1)-points correlation E [ u k l Y l] : (l+1) R E [ u k l Y l] (x 1,..., x l+1 ) = E [ u k l (x 1 ) Y (x 2 ) Y (x l+1 ) ] F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

16 Perturbation approach and moment equations Perturbation approach and moment equations Step 2. Approximate the moments of u using the Taylor expansion; e.g. for the first moment: E[u](x) E [ T K u(y, x) ] K E[u k ](x) = k! k=0 Each correction term E[u k ] to the mean satisfies the recursion E[u k ](x) v(x) dx = k l=1 ( k l ) E[Y l (x) u k l (x)] v dx efine the (l + 1)-points correlation E [ u k l Y l] : (l+1) R E [ u k l Y l] (x 1,..., x l+1 ) = E [ u k l (x 1 ) Y (x 2 ) Y (x l+1 ) ] and evaluate it on the diagonal (x,..., x) (l+1) : E [ u k l (x)y l (x) ] ( = Id l) E [ u k l Y l] (x,..., x) F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

17 Perturbation approach and moment equations Perturbation approach and moment equations Step 3. Write the recursion for the (l + 1)-points correlations E [ u j Y l], j + l k. We start from the problem solved by the k-th derivative: u j (x) v(x) dx = j s=1 ( j s ) Y s (x) u j s (x) v(x) dx F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

18 Perturbation approach and moment equations Perturbation approach and moment equations Step 3. Write the recursion for the (l + 1)-points correlations E [ u j Y l], j + l k. We start from the problem solved by the k-th derivative: l.h.s. = u j (x) v(x) dx = r.h.s. = u j (x 1 ) v(x 1 ) dx 1 j s=1 ( j s Y s (x 1 ) u j s (x 1 ) v(x 1 ) dx 1 ) Y s (x) u j s (x) v(x) dx F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

19 Perturbation approach and moment equations Perturbation approach and moment equations Step 3. Write the recursion for the (l + 1)-points correlations E [ u j Y l], j + l k. We start from the problem solved by the k-th derivative: l.h.s. = u j (x) v(x) dx = r.h.s. = j s=1 ( j s ) ( ) Y (x 2 ) u j (x 1 ) v(x 1 ) dx 1 v(x 2 ) dx 2 Y s (x) u j s (x) v(x) dx ( ) Y (x 2 ) Y s (x 1 ) u j s (x 1 ) v(x 1 ) dx 1 v(x 2 ) dx 2 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

20 Perturbation approach and moment equations Perturbation approach and moment equations Step 3. Write the recursion for the (l + 1)-points correlations E [ u j Y l], j + l k. We start from the problem solved by the k-th derivative: l.h.s. = u j (x) v(x) dx = r.h.s. = j s=1 ( j s ) Y s (x) u j s (x) v(x) dx ( ) Y (x l+1 ) u j (x 1 ) v(x 1 ) dx 1 v(x l+1 ) dx l+1 ( ) Y (x l+1 ) Y s (x 1 ) u j s (x 1 ) v(x 1 ) dx 1 v(x l+1 ) dx l+1 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

21 Perturbation approach and moment equations Perturbation approach and moment equations Step 3. Write the recursion for the (l + 1)-points correlations E [ u j Y l], j + l k. We start from the problem solved by the k-th derivative: u j (x) v(x) dx = j s=1 ( j s ) Y s (x) u j s (x) v(x) dx [ ( ) ] l.h.s. = E Y (x l+1 ) u j (x 1 ) v(x 1 ) dx 1 v(x l+1 ) dx l+1 [ ( ) ] r.h.s. = E Y (x l+1 ) Y s (x 1 ) u j s (x 1 ) v(x 1 ) dx 1 v(x l+1 ) dx l+1 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

22 Perturbation approach and moment equations Perturbation approach and moment equations Step 3. Write the recursion for the (l + 1)-points correlations E [ u j Y l], j + l k. We start from the problem solved by the k-th derivative: u j (x) v(x) dx = j s=1 ( j s ) Y s (x) u j s (x) v(x) dx l.h.s. = E [ u j (x 1 )Y (x 2 ) Y (x l+1 ) ] v(x 1 ) v(x l+1 ) dx 1 dx l+1 (l+1) r.h.s. = E [ ( u j s Y s ) Y l] v(x 1 ) v(x l+1 ) dx 1 dx l+1 (l+1) F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

23 Perturbation approach and moment equations Perturbation approach and moment equations Step 3. Write the recursion for the (l + 1)-points correlations E [ u j Y l], j + l k. We start from the problem solved by the k-th derivative: u j (x) v(x) dx = j s=1 ( j s ) Y s (x) u j s (x) v(x) dx l.h.s. = Id l E [ u j Y l] v(x 1 ) v(x l+1 ) dx 1 dx l+1 (l+1) [ r.h.s. = Tr 1:s+1 E u j s Y (s+l)] v(x 1 ) v(x l+1 ) dx 1 dx l+1 (l+1) F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

24 Perturbation approach and moment equations Perturbation approach and moment equations Step 3. Id l E [ u j Y l] Id l v dx 1... dx l+1 (l+1) j ( ) j [ = Tr s 1:s+1 E u j s Y (s+l)] Id l v dx 1... dx l+1 (l+1) s=1 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

25 Perturbation approach and moment equations Perturbation approach and moment equations Step 3. Id l E [ u j Y l] Id l v dx 1... dx l+1 (l+1) j ( ) j [ = Tr s 1:s+1 E u j s Y (s+l)] Id l v dx 1... dx l+1 (l+1) s=1 This is a sequence of deterministic high dimensional problems. F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

26 Perturbation approach and moment equations Perturbation approach and moment equations Step 3. Id l E [ u j Y l] Id l v dx 1... dx l+1 (l+1) j ( ) j [ = Tr s 1:s+1 E u j s Y (s+l)] Id l v dx 1... dx l+1 (l+1) s=1 This is a sequence of deterministic high dimensional problems. A similar recursion can be written for higher order moments. For instance, the k-th order correction to the second moment will involve the computation of all the correlations E[u j1 u j2 Y l ], j 1 + j 2 + l k F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

27 Perturbation approach and moment equations The structure of the recursion for the first moment im. k = 0 k = 1 k = 2 d u 0 E [ u 1] E [ u 2] 2d E [ u 0 Y ] E [ u 1 Y ]... 3d E [ u 0 Y 2].... Recursive, triangular structure F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

28 Perturbation approach and moment equations The structure of the recursion for the first moment im. k = 0 k = 1 k = 2 d u 0 E [ u 1] E [ u 2] 2d E [ u 0 Y ] E [ u 1 Y ]... 3d E [ u 0 Y 2].... Recursive, triangular structure The Algorithm for k = 0,..., K Compute E [ u 0 Y k] for j = 1,..., k Solve the boundary value problem for E [ u j Y k j] end The result for j = k is the k-th order correction E [ u k] end F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

29 Perturbation approach and moment equations The structure of the recursion for the first moment im. k = 0 k = 1 k = 2 d u 0 E [ u 1] E [ u 2] 2d E [ u 0 Y ] E [ u 1 Y ]... 3d E [ u 0 Y 2].... Recursive, triangular structure The Algorithm for k = 0,..., K Compute E [ u 0 Y k] for j = 1,..., k Solve the boundary value problem for E [ u j Y k j] end The result for j = k is the k-th order correction E [ u k] end If E [Y ] = 0, E [ Y (2k+1)] = 0 k F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

30 Perturbation approach and moment equations A few relevant questions 1 The perturbation method relies on Taylor expansion. What is the accuracy of the Taylor approximation? F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

31 Perturbation approach and moment equations A few relevant questions 1 The perturbation method relies on Taylor expansion. What is the accuracy of the Taylor approximation? 2 The k-th order correction to the mean (or higher order moments) can be obtained by solving the recursion for the correlations E[u j Y l ], j, l k. Are these problems well posed? What is the smoothness of the correlations functions E[u j Y l ]? F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

32 Perturbation approach and moment equations A few relevant questions 1 The perturbation method relies on Taylor expansion. What is the accuracy of the Taylor approximation? 2 The k-th order correction to the mean (or higher order moments) can be obtained by solving the recursion for the correlations E[u j Y l ], j, l k. Are these problems well posed? What is the smoothness of the correlations functions E[u j Y l ]? 3 From the numerical point of view How can we effectively approximate and solve the equations for the correlations E[u j Y l ]? (given that they are high dimensional objects) F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

33 Outline Approximation properties of the Taylor polynomial 1 The lognormal arcy problem 2 Perturbation approach and moment equations 3 Approximation properties of the Taylor polynomial 4 Moment equations: well posedness and discretization 5 Tensor Train approximation 6 1 Numerical experiments F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

34 Approximation properties of the Taylor polynomial Local convergence of the Taylor series Let Y be a centered Gaussian random field (E [Y ] = 0). Consider the following map defined on the Banach space L () with values in H 1 (): u : L () H 1 () Y u(y ) and its Taylor polynomial T K u = K k=0 uk k!, where uk = k [0](Y,..., Y ). Problem: Is the Taylor series T K u convergent in H 1 -norm for K +? T K u K u k H 1 H 1 k! k=0 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

35 Approximation properties of the Taylor polynomial Local convergence of the Taylor series Let Y be a centered Gaussian random field (E [Y ] = 0). Consider the following map defined on the Banach space L () with values in H 1 (): u : L () H 1 () Y u(y ) and its Taylor polynomial T K u = K k=0 uk k!, where uk = k [0](Y,..., Y ). Problem: Is the Taylor series T K u convergent in H 1 -norm for K +? T K u K u k K ( ) k H 1 H 1 Y L C k! log 2 k=0 k=0 By a recursive argument we prove that ( ) k u k C Y L H1 () log 2 k! with C = C(C P, u 0 H 1), C P being the Poincaré constant. F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

36 Approximation properties of the Taylor polynomial Local convergence of the Taylor series Let Y be a centered Gaussian random field (E [Y ] = 0). Consider the following map defined on the Banach space L () with values in H 1 (): u : L () H 1 () Y u(y ) and its Taylor polynomial T K u = K k=0 uk k!, where uk = k [0](Y,..., Y ). Problem: Is the Taylor series T K u convergent in H 1 -norm for K +? T K u K u k K ( ) k H 1 H 1 Y L k! log 2 k=0 k=0 By a recursive argument we prove that ( ) k u k C Y L H1 () log 2 k! with C = C(C P, u 0 H 1), C P being the Poincaré constant. The Taylor series is convergent σ > 0 in the disk B := {Y L () : Y L < log 2} F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

37 Approximation properties of the Taylor polynomial Global conv. of the Taylor series? A counter example Let Y (ω, x) = ξ(ω)x, with ξ N (0, σ 2 ) Gaussian random variable (one-dimensional probability space). Consider the following one-dimensional PE (e ξ(ω)x u (ω, x)) = 0, a.e. in [0, 1] u(ω, 0) = 0, u(ω, 1) = 1 The exact solution is u(ξ, x) = 1 e ξx 1 e ξ. F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

38 Approximation properties of the Taylor polynomial Global conv. of the Taylor series? A counter example Let Y (ω, x) = ξ(ω)x, with ξ N (0, σ 2 ) Gaussian random variable (one-dimensional probability space). Consider the following one-dimensional PE (e ξ(ω)x u (ω, x)) = 0, a.e. in [0, 1] u(ω, 0) = 0, u(ω, 1) = 1 The exact solution is u(ξ, x) = 1 e ξx 1 e ξ. Observe that: On the real axis (ξ R), u(ξ, x) is analytic as a function of ξ. In the complex plane (ξ C), u(ξ, x) is not entire. Indeed, it admits countable many poles in ξ = 2πik, k Z \ {0}. F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

39 Approximation properties of the Taylor polynomial Global conv. of the Taylor series? A counter example Let Y (ω, x) = ξ(ω)x, with ξ N (0, σ 2 ) Gaussian random variable (one-dimensional probability space). Consider the following one-dimensional PE (e ξ(ω)x u (ω, x)) = 0, a.e. in [0, 1] u(ω, 0) = 0, u(ω, 1) = 1 The exact solution is u(ξ, x) = 1 e ξx 1 e ξ. Observe that: On the real axis (ξ R), u(ξ, x) is analytic as a function of ξ. In the complex plane (ξ C), u(ξ, x) is not entire. Indeed, it admits countable many poles in ξ = 2πik, k Z \ {0}. The Taylor series centered in ξ = 0 converges only in the disk of radius r < 2π and K 0 E [ T K u ] is not convergent to E [u] F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

40 Approximation properties of the Taylor polynomial A priori error upper bound Given the counter example, in general we do not expect E [ T K u ] to be convergent to E [u]. Nevertheless, for σ and K sufficiently small, E [ T K u ] is a good approximation of E [u]. The method we propose can be used even if the Taylor series is not globally convergent. F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

41 Approximation properties of the Taylor polynomial A priori error upper bound Given the counter example, in general we do not expect E [ T K u ] to be convergent to E [u]. Nevertheless, for σ and K sufficiently small, E [ T K u ] is a good approximation of E [u]. The method we propose can be used even if the Taylor series is not globally convergent. Problem: Let 0 < σ < 1 be fixed. Which is the optimal degree K σ opt (which depends in σ) of the Taylor polynomial to consider? F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

42 Approximation properties of the Taylor polynomial A priori error upper bound Given the counter example, in general we do not expect E [ T K u ] to be convergent to E [u]. Nevertheless, for σ and K sufficiently small, E [ T K u ] is a good approximation of E [u]. The method we propose can be used even if the Taylor series is not globally convergent. Problem: Let 0 < σ < 1 be fixed. Which is the optimal degree K σ opt (which depends in σ) of the Taylor polynomial to consider? A priori error estimate (0 < σ < 1) [Bonizzoni Nobile, 2013] E u T K u (K + 1)! σ j ( ) K+1 σ H C 1 () (log 2) K+1 j!! C K!! log 2 j=k+1 Remark: K!! = K(K 2)(K 4)... 1 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

43 Approximation properties of the Taylor polynomial The error upper bound as a function of K bound L1 norm of R K u σ=0.10 σ=0.15 σ=0.18 σ= K ivergence of error upper bound σ > 0 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

44 Approximation properties of the Taylor polynomial The error upper bound as a function of K bound L1 norm of R K u σ=0.10 σ=0.15 σ=0.18 σ= K ivergence of error upper bound σ > 0 ( ) 2 log 2 Estimated optimal K, Kopt σ = 4. (bullets in the picture) σ F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

45 Approximation properties of the Taylor polynomial Error upper bound: sketch of the proof Key ingredients: 1 We prove by a recursive argument that u k (ty, x) ( H1 () Cet Y Y L ) k L () log 2 k!, 0 t 1 2 E Y k L () C σk (k 1)!! (application of a result in [Adler Taylor, 2007, Charrier ebussche, 2013]). F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

46 Approximation properties of the Taylor polynomial Error upper bound: sketch of the proof Key ingredients: 1 We prove by a recursive argument that u k (ty, x) ( H1 () Cet Y Y L ) k L () log 2 k!, 0 t 1 2 E Y k L () C σk (k 1)!! (application of a result in [Adler Taylor, 2007, Charrier ebussche, 2013]). u T K u H1 () 1 K! 1 0 (1 t) K u K+1 (ty, x) H1 () dt F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

47 Approximation properties of the Taylor polynomial Error upper bound: sketch of the proof Key ingredients: 1 We prove by a recursive argument that u k (ty, x) ( H1 () Cet Y Y L ) k L () log 2 k!, 0 t 1 2 E Y k L () C σk (k 1)!! (application of a result in [Adler Taylor, 2007, Charrier ebussche, 2013]). u T K u H1 () 1 K! 1 0 C(K + 1) (1 t) K u K+1 (ty, x) H1 () dt ( ) K+1 Y 1 L (1 t) K e t Y L dt [use (1)] log 2 0 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

48 Approximation properties of the Taylor polynomial Error upper bound: sketch of the proof Key ingredients: 1 We prove by a recursive argument that u k (ty, x) ( H1 () Cet Y Y L ) k L () log 2 k!, 0 t 1 2 E Y k L () C σk (k 1)!! (application of a result in [Adler Taylor, 2007, Charrier ebussche, 2013]). u T K u H1 () 1 K! 1 0 C(K + 1) (1 t) K u K+1 (ty, x) H1 () dt ( ) K+1 Y 1 L (1 t) K e t Y L dt log 2 0 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

49 Approximation properties of the Taylor polynomial Error upper bound: sketch of the proof Key ingredients: 1 We prove by a recursive argument that u k (ty, x) ( H1 () Cet Y Y L ) k L () log 2 k!, 0 t 1 2 E Y k L () C σk (k 1)!! (application of a result in [Adler Taylor, 2007, Charrier ebussche, 2013]). u T K u H1 () 1 K! 1 0 C(K + 1) (1 t) K u K+1 (ty, x) H1 () dt ( ) K+1 Y L K! log 2 Y K+1 L j=k+1 Y j L j! F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

50 Approximation properties of the Taylor polynomial Error upper bound: sketch of the proof Key ingredients: 1 We prove by a recursive argument that u k (ty, x) ( H1 () Cet Y Y L ) k L () log 2 k!, 0 t 1 2 E Y k L () C σk (k 1)!! (application of a result in [Adler Taylor, 2007, Charrier ebussche, 2013]). u T K u H1 () 1 K! 1 0 C(K + 1)! (1 t) K u K+1 (ty, x) H1 () dt ( ) K+1 1 Y j L log 2 j! j=k+1 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

51 Approximation properties of the Taylor polynomial Error upper bound: sketch of the proof Key ingredients: 1 We prove by a recursive argument that u k (ty, x) ( H1 () Cet Y Y L ) k L () log 2 k!, 0 t 1 2 E Y k L () C σk (k 1)!! (application of a result in [Adler Taylor, 2007, Charrier ebussche, 2013]). u T K u H1 () 1 K! E [ u T K u H 1 () ] 1 0 C(K + 1)! (1 t) K u K+1 (ty, x) H1 () dt ( ) K+1 1 Y j L log 2 j! j=k+1 ( ) K+1 1 E C(K + 1)! log 2 j=k+1 [ Y jl ] j! F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

52 Approximation properties of the Taylor polynomial Error upper bound: sketch of the proof Key ingredients: 1 We prove by a recursive argument that u k (ty, x) ( H1 () Cet Y Y L ) k L () log 2 k!, 0 t 1 2 E Y k L () C σk (k 1)!! (application of a result in [Adler Taylor, 2007, Charrier ebussche, 2013]). u T K u H1 () 1 K! E [ u T K u H 1 () ] 1 0 C(K + 1)! (1 t) K u K+1 (ty, x) H1 () dt ( ) K+1 1 Y j L log 2 j! j=k+1 ( ) K+1 1 C(K + 1)! log 2 ( ) K+1 1 C(K + 1)! log 2 j=k+1 j=k+1 ] E [ Y jl σ j j!! j! [use (2)] F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

53 Approximation properties of the Taylor polynomial A numerical check: Single Gaussian random variable div(e cos(πx)ξ(ω) u(ω, x)) = x a.e. in = [0, 1] ξ(ω) N (0, σ 2 ), 0 < σ < 1 The Taylor polynomial is computable! Computed error vs K Computed error vs σ error (p=1) σ=0.1 σ=0.15 σ=0.2 σ=0.3 σ=0.4 σ=0.5 σ=0.8 error order 2 σ 3 order 4 σ 5 order 6 σ 7 order 8 σ 9 order 10 σ K σ We numerically show the divergence of the Taylor series for any value of the standard deviation σ > 0 The exponential behavior as function of σ is confirmed F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

54 Approximation properties of the Taylor polynomial How good is the a priori error estimate? Comp. err. and err. estimate error (p=1) σ=0.1 (FEM) σ=0.15 (FEM) σ=0.2 (FEM) σ=0.1 (bound) σ=0.15 (bound) σ=0.2 (bound) K The a priori error bound is very pessimistic F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

55 Approximation properties of the Taylor polynomial How good is the a priori error estimate? Comp. err. and err. estimate Comp. err. and fitted err. estimate error (p=1) σ=0.1 (FEM) σ=0.15 (FEM) σ=0.2 (FEM) σ=0.1 (bound) σ=0.15 (bound) σ=0.2 (bound) norm of the K th Taylor residual σ=0.1 (FEM) σ=0.15 (FEM) σ=0.2 (FEM) σ=0.3 (FEM) σ=0.4 (FEM) σ=0.5 (FEM) σ=0.8 (FEM) σ=0.1 (fit) σ=0.15 (fit) σ=0.2 (fit) σ=0.3 (fit) σ=0.4 (fit) σ=0.5 (fit) σ=0.8 (fit) K K The a priori error bound is very pessimistic It is possible to fit the parameter γ in the a priori error bound E u T K u ( ) K+1 γσ H 1 () C K!! log 2 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

56 Approximation properties of the Taylor polynomial Single bounded random variable 0 < α 1 a(ω, x) = E[a](x) + b(x)y (ω) α 2 < + Y (ω) [ γ, γ], 0 < γ < + The Taylor series is convergent provided that the variability of a is small enough [Babuška Chatzipantelidis, 2002, Todor Ph, 2005] error (p=1) γ=1(fem) γ=2(fem) γ=3(fem) γ=6(fem) γ=8(fem) γ=9(fem) γ=1(bound) γ=2(bound) γ=3(bound) γ=6(bound) γ=8(bound) γ=9(bound) Computed error and bound vs K K F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

57 Outline Moment equations: well posedness and discretization 1 The lognormal arcy problem 2 Perturbation approach and moment equations 3 Approximation properties of the Taylor polynomial 4 Moment equations: well posedness and discretization 5 Tensor Train approximation 6 1 Numerical experiments F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

58 Moment equations: well posedness and discretization Moment equations: well-posedness and regularity results Consider again the recursion for the correlations E[u k Y l ]: im. k = 0 k = 1 k = 2 d u 0 0 E [ u 2] 2d 0 E [ u 1 Y ]... 3d E [ u 0 Y 2].... Theorem: well-posedness [Bonizzoni Ph, 2013] Let Y be a Gaussian random field with Gaussian covariance function Cov Y C 0,t ( ), 0 < t 1. Then, all the problems in the recursion are well-posed. F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

59 Moment equations: well posedness and discretization Moment equations: well-posedness and regularity results Consider again the recursion for the correlations E[u k Y l ]: im. k = 0 k = 1 k = 2 d u 0 0 E [ u 2] 2d 0 E [ u 1 Y ]... 3d E [ u 0 Y 2].... Theorem: well-posedness [Bonizzoni Ph, 2013] Let Y be a Gaussian random field with Gaussian covariance function Cov Y C 0,t ( ), 0 < t 1. Then, all the problems in the recursion are well-posed. Theorem: regularity [Bonizzoni Ph, 2013] Let Y be a Gaussian random field with Gaussian covariance function Cov Y C 0,t ( ), 0 < t 1. Moreover, if the domain is convex and C 1,t/2 and u 0 C 1,t/2 ( ), then E [ u k Y l] C ( 0,t/2,mix l, C 1,t/2 ( ) ) F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

60 Moment equations: well posedness and discretization Problem for E [ u 1 Y ] Full TP discretization Given E [ u 0 Y 2] H 1 () ( L 2 () ) 2 [, find E u 1 Y ] H 1 () L 2 () s.t. [ ] ( Id) E u 1 Y (x 1, x 2) ( Id) v(x 1, x 2) dx 1 dx 2 [ = Tr 1:2 E u 0 Y 2] (x 1, x 2) ( Id) v(x 1, x 2) dx 1 dx 2 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

61 Moment equations: well posedness and discretization Problem for E [ u 1 Y ] Full TP discretization Given E [ u 0 Y 2] H 1 () ( L 2 () ) 2 [, find E u 1 Y ] H 1 () L 2 () s.t. [ ] ( Id) E u 1 Y (x 1, x 2) ( Id) v(x 1, x 2) dx 1 dx 2 [ = Tr 1:2 E u 0 Y 2] (x 1, x 2) ( Id) v(x 1, x 2) dx 1 dx 2 Let us introduce: {φ i } i linear FEM elements to discretize H 1 () {ψ j } j piecewise constants to discretize L 2 () A(n, m) = φn(x) φm(x)dx M(i, j) = ψ j(x)ψ i (x)dx B 1 (n, i 1, m) = φn(x)ψ i 1 (x) φ m(x)dx C 1,1(n, i) nodal repr. of E [ u 1 Y ] C 0,2(n, i 1, i 2) nodal repr. of E [ u 0 Y 2] A M C 1,1 = B 1 M C 0,2 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

62 Moment equations: well posedness and discretization Problem for E [ u 1 Y ] Full TP discretization Given E [ u 0 Y 2] H 1 () ( L 2 () ) 2 [, find E u 1 Y ] H 1 () L 2 () s.t. [ ] ( Id) E u 1 Y (x 1, x 2) ( Id) v(x 1, x 2) dx 1 dx 2 [ = Tr 1:2 E u 0 Y 2] (x 1, x 2) ( Id) v(x 1, x 2) dx 1 dx 2 Let us introduce: {φ i } i linear FEM elements to discretize H 1 () {ψ j } j piecewise constants to discretize L 2 () A(n, m) = φn(x) φm(x)dx M(i, j) = ψ j(x)ψ i (x)dx B 1 (n, i 1, m) = φn(x)ψ i 1 (x) φ m(x)dx C 1,1(n, i) nodal repr. of E [ u 1 Y ] C 0,2(n, i 1, i 2) nodal repr. of E [ u 0 Y 2] Simplifying the mass matrix: A M C 1,1 = B 1 M C 0,2 A 1:1 C 1,1 = B 1 1:2 C 0,2 where 1:s denotes the saturation of the first s indices of both the right and left hand side tensors. F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

63 Moment equations: well posedness and discretization Problem for E [ u k Y l] Full TP discretization Generalizing the previous equation: Tensorial equation k ( ) k A 1:1 C k,l = B s s 1:s+1 C k s,s+l s=1 Problem: Curse of the dimensionality. How to store all the tensors? im. k = 0 k = 1 k = 2 O(N h ) C 0,0 0 C 2,0 O(N 2 h ) 0 C 1,1... O(N 3 h ) C 0, F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

64 Outline Tensor Train approximation 1 The lognormal arcy problem 2 Perturbation approach and moment equations 3 Approximation properties of the Taylor polynomial 4 Moment equations: well posedness and discretization 5 Tensor Train approximation 6 1 Numerical experiments F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

65 Tensor Train approximation Tensor Train (TT) Format [Oseledets, 2011] Generalization of the SV decomposition of a matrix in more than 2 dimensions. SV of a matrix: let X R N1 N2 be a matrix X (i 1, i 2 ) = r 1 α 1=1 G 1 (i 1, α 1 )G 2 (α 1, i 2 ) F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

66 Tensor Train approximation Tensor Train (TT) Format [Oseledets, 2011] Generalization of the SV decomposition of a matrix in more than 2 dimensions. Tensor Train (TT) Format: let X R N1... Nn be a tensor of order n X (i 1,..., i n ) = r 1,...,r n 1 α 1,...,α n 1=1 G 1 (i 1, α 1 )G 2 (α 1, i 2, α 2 )... G n (α n 1, i n ) The (n + 1)-tupla (r 0,..., r n ) is called TT rank Idea: Storage of order 3 tensors in a (linear) linked format F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

67 Tensor Train approximation Tensor Train (TT) Format [Oseledets, 2011] Generalization of the SV decomposition of a matrix in more than 2 dimensions. Tensor Train (TT) Format: let X R N1... Nn be a tensor of order n X (i 1,..., i n ) = r 1,...,r n 1 α 1,...,α n 1=1 G 1 (i 1, α 1 )G 2 (α 1, i 2, α 2 )... G n (α n 1, i n ) The (n + 1)-tupla (r 0,..., r n ) is called TT rank Idea: Storage of order 3 tensors in a (linear) linked format Storage complexity: O(nNr 2 ) vs O((N) n ), r = max r i, N = max N i. Pro: It allows fast computations. Results obtained: Using the Matlab TT-toolbox 2.2 [Oseledets, 2012], we developed a code which solves the recursive problem for E [u] in TT-format. F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

68 The TT algorithm Tensor Train approximation What does it mean to solve a tensorial equation in TT-format? A 1:1 C 1,1 = B 1 1:2 C 0,2 C 1,1 = A 1 1:1 ( B 1 1:2 C 0,2 ) STEP1 saturation B 1 1:2 C 0,2 STEP2 saturation with A 1 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

69 Tensor Train approximation A 1:1 C k,l = k ( k s s=1 ) B s 1:s+1 C k s,s+l (1) Inputs needed: TT-format of the correlation C 0,s, C0,s TT, (nodal representation of E [ u 0 Y s] ) TT-format of the tensors B s TT,s, B Stiffness matrix A Operations needed: Saturation 1:s between two TT-tensors Lin. alg. operations and approximation (tt round) of tt-tensors [TT-toolbox] for k = 0,..., K Compute C TT 0,k with a tolerance tol TT for l = k 1,..., 0 Solve the tensorial equation (1) end The result for l = 0 is the k-th order correction C k,0 end F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

70 Tensor Train approximation TT-representation of E [ Y k] [Kumar Kressner Nobile Tobler, 2013] The starting point is the KL-expansion of the Gaussian random field Y : Y (ω, x) = E [Y ] (x) + i=1 λi ξ i (ω)φ i (x), x, ω Ω where ξ i i.i.d N (0, 1) and i=1 λ i = Var[Y (x)]dx. k-th correlation: E [ Y k] (x 1..., x k ) = i [ 1=1 i k =1 E k ] k η=1 λiη ξ iη η=1 φ i η (x η ) = i N C k k i 1...i k η=1 φ i η (x η ), where C i1...i k = [ ] l=1 λm i(l)/2 l E ξ m i(l) l, m i (l)=multiplicity of index l in i. C i1...i k is supersymmetric. An exact TT symmetric representation can be constructed: with U k/2 basis of Range(C (1,...,k/2) ). C (1,...,k/2) = U k/2 MU T k/2 Then the basis C (1,...,k/2) can be further truncated with a given tolerance tol TT : C C tol TT F F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

71 Outline 1 Numerical experiments 1 The lognormal arcy problem 2 Perturbation approach and moment equations 3 Approximation properties of the Taylor polynomial 4 Moment equations: well posedness and discretization 5 Tensor Train approximation 6 1 Numerical experiments F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

72 1 Numerical experiments Test 1 Analysis of the Taylor approximation Let Y (ω, x) be a centered Gaussian r. f. with Gaussian cov. function Cov Y (x 1, x 2) = σ 2 e x 1 x , (x 1, x 2) [0, 1] [0, 1] To compute the reference solution and the TT-solution we use: same spatial discretization N h = 100 of the physical domain = [0, 1] same KL-expansion: N = 11 r.v. (99% of variance captured) exact TT-computations: tol TT = We observe only the truncation error in the Taylor series Reference solution (collocation) Computed error vs k σ=0.05 σ=0.25 σ=0.45 σ=0.65 σ= σ=0.05 σ=0.15 σ=0.25 σ=0.35 E[u](x) 0.04 error x K F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

73 1 Numerical experiments Test 1 Analysis of the Taylor approx. 0.8 N obs = number of observations of the permeability field N = number of random variables considered conditional log conductivity mean coll. E[u](x) x N obs = 0, N = 11 N obs = 3, N = 9 N obs = 5, N = σ= σ=0.25 σ= σ=0.65 σ= E[u] N oss = σ= σ=0.25 σ= σ=0.65 σ= E[u] N oss = σ=0.05 σ= σ=0.45 σ= σ= x x x error vs K error σ=0.05 σ=0.15 σ=0.25 σ=0.35 error N oss =3 σ=0.05 σ=0.15 σ=0.25 σ=0.35 error N oss =5 σ=0.05 σ=0.15 σ=0.25 σ= K K As N obs increases, the variability of the field decreases: good for perturbation methods! K F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

74 1 Numerical experiments Test 2 Analysis of the dependence on the TT-precision Let Y (ω, x) be a centered Gaussian r. f. with Gaussian cov. function same spatial discretization N h = 100 of the physical domain = [0, 1] same KL-exp: N = 26 r.v. (100% of variance captured up to machine precision) different tolerances in the TT-computations 10 0 K= K= K= error 10 4 error 10 6 error tol=10 1 tol=10 4 tol=10 6 tol=10 8 σ σ tol=10 1 tol=10 4 tol=10 6 tol=10 8 σ σ tol=10 1 tol=10 4 tol=10 6 tol=10 8 σ σ 10 2 σ= σ=0.55 error tol=10 8 tol=10 1 tol= tol= tol=10 4 tol=10 6 tol=10 6 tol= K K error It is not always useful to consider small tol TT. There is an optimal tol opt depending on σ and K F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

75 1 Numerical experiments Test 2 The complexity of the TT-algorithm Complexity= number of linear systems to be solved We numerically studied how the error depends on the complexity of the TT-algorithm 10 0 σ=0.05 σ = 0.05 σ = 0.25 σ = σ= σ= error 10 6 tol= tol=10 4 tol= tol=10 8 tol=10 10 MC complexity error error 10 6 tol=10 1 tol=10 1 tol=10 4 tol=10 4 tol= tol= tol=10 8 tol=10 8 tol=10 10 tol=10 10 MC MC complexity complexity error vs complexity for different tol TT 10 2 If the optimal tol opt is chosen, the TT-algorithm is far superior to a standard Monte Carlo method (black line) F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

76 1 Numerical experiments Test 3 Storage requirements of the TT-algorithm Let Y (ω, x) be a centered Gaussian r. f. with Gaussian cov. function spatial discretization N h = 200 of the physical domain = [0, 1] exact KL-exp: N = 27 r.v. (100% of variance captured up to machine precision) different tolerances in the TT-computations TT-ranks of the TTcorrelations C TT Y for k different tol TT r p tol=10 1 tol=10 2 tol=10 3 tol=10 4 tol=10 6 bound ( ) N + p 1 r p p (black line) [Kumar Kressner Nobile Tobler] p 10 4 K=6, tol=10 10 E[u 0 Y 6 ] E[u 1 Y 5 ] TT-ranks of the correlations in the recursion for tol TT = r p E[u 2 Y 4 ] E[u 3 Y 3 ] E[u 4 Y 2 ] E[u 5 Y] E[u 6 ] bound the upper bound (black line) is valid p The storage requirement is a limiting aspect of our algorithm. Improvements could be obtained thanks to the implementation of sparse TT-tensors F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

77 1 Numerical experiments Conclusions We have applied the perturbation technique to the arcy problem with lognormal permeability. We have studied the approximation properties of the Taylor polynomial We have derived the moment equations, and proved their well-posedness and Hölder-type regularity results. We have developed an algorithm in TT-format able to solve the first statistical moment problem. Our TT-algorithm provide a valid solution both in the case where Y is parametrized by a small number of r.v. and if the entire random field is considered. If the optimal tol TT is considered, our TT-algorithm is far superior to a standard Monte Carlo method The main limitation is the storage requirement. F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

78 1 Numerical experiments Adler, R. J. and Taylor, J. E. Random fields and geometry, Springer, Babuška, I, and Chatzipantelidis, P. On solving elliptic stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg., 2002, Vol 191, pp Bonizzoni, F. Analysis and approximation of moment equations for PEs with stochastic data, Ph thesis, Politecnico di Milano, Italy, Bonizzoni, F and Nobile, F. Perturbation analysis for the arcy problem with log-normal permeability, MATHICSE Report 29/2013. Charrier, J. and ebussche, A. Weak truncation error estimates for elliptic PEs with lognormal coefficients Stochastic Partial ifferential Equations: Analysis and Computations, 2013, Vol 1, pp Galvis, J. and Sarkis, M. Approximating infinity-dimensional stochastic arcy s equations without uniform ellipticity SIAM J. Numer. Anal., 2009, Vol 47, pp Gittelson, C. J. Stochastic Galerkin discretization of the log-normal isotropic diffusion problem Math. Models Methods Appl. Sci., 2010, Vol 20, pp , Kumar, R. and Kressner,. and Nobile, F. and Tobler, C. Low-rank tensor approximation for high order correlation functions of Gaussian random fields In preparation Harbrecht, H. and Schneider, R. and Schwab, C. Sparse second moment analysis for elliptic problems in stochastic domains, Numerische Mathematik, 2008, Vol 109, pp Oseledets, I. V. Tensor-train decomposition, SIAM J. Sci. Comput., 2011, Vol 33, pp Riva, M. and Guadagnini, A. and e Simoni, M. Assessment of uncertainty associated with the estimation of well catchments by moment equations, Advances in Water Resources, 2006, Vol 29, 5, pp Tartakovsky,. M. and Neuman, S. P. Transient flow in bounded randomly heterogeneous domains: 1. Exact conditional moment equations and recursive approx, Water Resour. Res., Vol 34, pp Todor, R. A. Sparse Perturbation Algorithms for Elliptic PE s with Stochastic ata, Ph thesis, ETH Zurich, Switzerland, von Petersdorff, T. and Schwab, C. Sparse finite element methods for operator equations with stochastic data Appl. Math., 2006, Vol 51, pp F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

79 1 Numerical experiments Thank you for the attention! F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

80 1 Numerical experiments Well-posedness of the stochastic arcy problem find u L ( p Ω; H 1 () ) s.t. u Γ = g a.s., and a(ω, x) x u(ω, x) x v(x) dx = f (x)v(x) dx v HΓ 1 (), a.s. in Ω. A1 : The permeability field a L p ( Ω; C 0 ( ) ) for every p (0, ). Then, the quantities a min (ω) := min a(ω, x) (2) x a max (ω) := max a(ω, x) (3) x are well defined, and a max L p (Ω) for every p (0, + ). Moreover, we assume 1 A2 : a min (ω) > 0 a.s., a min (ω) Lp (Ω) for every p (0, ). Theorem If the permeability field a(ω, x) satisfies A1, A2, then the stochastic arcy problem is well-posed for every p (0, ), that is it admits a unique solution that depends continuously on the data. F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

81 1 Numerical experiments Upper bounds for the statistical moments of Y L () KL expansion: Y (ω, x) = E [Y ] (x) + σ 2 + j=1 λ j φ j (x) ξ j (ω) 1 φ j is Hölder continuous with exponent 0 < γ 1 for every j 1. 2 R γ := + j=1 λ j φ j 2 C 0,γ ( ) < +. Spectral technique [Charrier ebussche, 2013] [ ] E Y k L () C Rγ k/2 σ k (k 1)!!, k > 0 The domain is a d-dimensional rectangle = [0, T ] d. The centered Gaussian field Y (ω, x) is stationary and regular (C 2 ) Euler characteristic technique [Adler Taylor, 2007] [ ] E Y k L () C σ k 2 k (k 1)!!, k F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

82 1 Numerical experiments k l s=1... ( k l s ) Problem solved by E [ u k l Y l] Id l E [ u k l Y l] Id l v dx 1... dx l+1 =... E [ ( u k l s Y s ) Y l] Id l v dx 1... dx l+1 F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

83 1 Numerical experiments Hölder spaces with mixed regularity C 0,γ,mix ( k ), 0 < γ 1, is the space of all cont. funct. v : k R s.t. v C 0,γ,mix ( k ) := γ,mix h v(x 1,..., x k ) < +, where with sup x,x+h k h>0 γ,mix h v(x 1,..., x k ) := γ 1,h 1 γ k,h k v(x 1,..., x k ), γ i,h i v(x 1,..., x k ) := v(x 1,..., x i + h i,..., x k ) v(x 1,..., x k ) h i γ. C 0,γ,mix ( k ) is a Banach space with the norm v C 0,γ,mix ( k ) := v C 0 ( k ) + v C 0,γ,mix ( k ). C 0,γ,mix ( k ) C 0,γ ( k ) C 0,γ ( k ) C 0,γ/k,mix ( k ) F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

84 1 Numerical experiments Gaussian cov. function Truncated KL error vs σ Let Y (ω, x) be a centered Gaussian r. f. with Gaussian cov. function Cov Y (x 1, x 2) = σ 2 e x 1 x , (x 1, x 2) [0, 1] [0, 1] tol KL = 10 4 : N h = 100, N = 11 r.v. (99% of variance captured) tol TT = Reference solution (collocation) Computed error vs σ E[u](x) σ=0.05 σ=0.25 σ=0.45 σ=0.65 σ= x error K=2 σ 4 K=4 σ 6 K=6 σ σ Order of E [u(y, x)] E [ T K u(y, x) ] L as function of σ 2 () [ ] K = 0 K = 1 K = 2 K = 3 K = 4 K = 5 K = 6 E u T K L2 u F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

85 1 Numerical experiments Exponential cov. function Complete KL Let Y (ω, x) be a centered Guassian r. f. with exponential cov. function Cov Y (x 1, x 2) = σ 2 e x 1 x 2 0.2, (x 1, x 2) [0, 1] [0, 1] tol KL = 10 4 : N h = 100, N = 100 r.v. (100% of variance captured) tol TT = The collocation method is unusable. We compare the TT-solution with the Monte Carlo solution (M=10000 samples) σ = 0.05 σ = σ=0.05 MC TT 0.07 σ=0.05 MC TT E[u] E[u] x x F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

86 1 Numerical experiments ependence of the TT-ranks on the dimension Gauss. Cov. funct. Esp. Cov. funct. r max k=2 k=4 k=6 k=8 log(1/tol) (1/2) r max k=2 k=4 k=6 k=8 1/tol 1/tol (2/3) 1/tol tol tol F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,

87 1 Numerical experiments Comparison with the comp. of a truncated Taylor series M, tol= M 2, tol=10 2 M, tol= M 2, tol=10 4 M 2, tol=10 6 M K ( N + K/2 Truncated Taylor expansion: M 1 = K/2 TT-algorithm: M 2 = n 1 n=2:2:k p=0 r p + 1 ) F. Bonizzoni (EPFL) Low-rank techniques applied to SPEs November 13,