A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models

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1 A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models Rainer Niekamp and Martin Krosche. Institute for Scientific Computing TU Braunschweig ILAS: A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models ILAS: /11

2 Motivation (1) Diffusion equation with uncertain parameter: ( x κ(x, ω) x u(x, ω) ) = f (x, ω) κ, u and f are stochastic fields spatial dimension 2 or 3, stochastic dimension m Discretisation by Spectral Stochastic Finite Element Method (SSFEM) ansatz functions: N = N x N s DoFs Linear system to be solved: ( l i K i )u = f i=0 l K i U i = F A(U) = F i=0 u, f R Nx Ns Nx Ns U, F R A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models ILAS: /11

3 Motivation (2) Numerical complexity in SSFEM: U is dense and may become quite large e.g. N x N s = = classical construction of stochastic solution space suffers from the Curse of Dimension Recent approaches: Low-Rank approximation of U [Nouy 2007, Zander/Matthies 2008/2009] a priori solution space adaption [Doostan/Ghanem/Red-Horse 2007, Bieri/Schwab 2009] Low-Rank and a posteriori solution space adaption [Moselhy 2010, Krosche/Niekamp 2010] A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models ILAS: /11

4 Basic Low-Rank SSFEM (1) Minimisation of the expectation of the total potential energy: A(U) = F E(U) := 1 A(U) : U F : U min 2 with A : B := i,j A ij B ij, A, B R n 1 n 2 Ansatz: U = U + gh T (optimal rank-1 update) geometrical part g stochastic part h A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models ILAS: /11

5 Basic Low-Rank SSFEM (2) Differentiating E(U) with respect to U and... perturbing U by varying g leads to H g (h) g = b g (h) perturbing U by varying h leads to H h (g) h = b h (g) alternating iterative process Output: U r = r i=1 g i h T i = G H T A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models ILAS: /11

6 Optimisation in Low-Rank SSFEM Optimisation: obtain rank-1 updates: G = G + gv T and H = H + hw T perform the tensor products and add rank-1 updates stand-alone optimiser A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models ILAS: /11

7 Adaption in Low-Rank SSFEM Adaptive construction of the solution space: add new stochastic basis functions: ( i i := i i + i ) computation of the extended residual with current solution U rating of new basis functions maintain best rated basis functions A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models ILAS: /11

8 Basic Low-Rank SSFEM =0 geometrical dimension 2 stochastic dimension 5 rank-1 updates till to full rank u = u = expected value 2nd moment 3rd moment variance =0 relative error e-05 1e-06 1e rank A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models ILAS: /11

9 Optimisation Comparison: basic vs. optimised Low-Rank approach on the fly optimisation: during each rank-1 update maximum number of optimisation steps per rank: basic optimised relative error e-05 1e-06 1e-07 1e rank A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models ILAS: /11

10 Adaption Comparison: basic vs. adaptive Low-Rank approach basic approach: till order 3 adaptive approach: possible till order 5; chosen till order 4 the same number of stochastic DoFs at final rank post optimisation: at the final rank optimised: order 3 adaptive, optimised: order 4 relative error number of iterations A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models ILAS: /11

11 Conclusion Problem: large, dense solution U Curse of Dimension in classical solution space construction Proposal: Low-Rank SSFEM to tackle problem of large, dense U stand-alone optimiser to tackle problem of sub-optimal approximations residual-based, a posteriori solution space adaption to tackle Curse of Dimension A Posteriori Adaptive Low-Rank Approximation of Probabilistic Models ILAS: /11