Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs

Abstract

We investigate the convergence rate of approximations by finite sums of rank-1 tensors of solutions of multi-parametric elliptic PDEs. Such PDEs arise, for example, in the parametric, deterministic reformulation of elliptic PDEs with random field inputs, based for example, on the M-term truncated Karhunen-Loève expansion. Our approach could be regarded as either a class of compressed approximations of these solution or as a new class of iterative elliptic problem solvers for high dimensional, parametric, elliptic PDEs providing linear scaling complexity in the dimension M of the parameter space. It is based on rank-reduced, tensor-formatted separable approximations of the highdimensional tensors and matrices involved in the iterative process, combined with the use of spectrally equivalent low-rank tensor-structured preconditioners to the parametric matrices resulting from a Finite Element discretization of the high-dimensional parametric, deterministic problems. Numerical illustrations for the M-dimensional parametric elliptic PDEs resulting from sPDEs on parameter spaces of dimensions M ≤ 100 indicate that the gain from employing low-rank tensor-structured matrix formats in the numerical solution of such problems might be substantial.

Tensors-structured numerical methods in scientific computing: Survey on recent advances

“…Fully tensorized numerical approach for solving the Hartree-Fock equation, discretized over N × N × N grid, by tensor truncated iteration of complexity O(N log N ), was recently presented in [58]. Other successful applications to high-dimensional eigenvalue problems [37,54] and to stochastic PDEs [64,62] are reported. A class of low tensor rank preconditioners for the multidimensional elliptic problems with jumping coefficients in R d is proposed in [18].…”

“…(Tensor-truncated solvers in the case of parameter-dependent coefficients). Consider a class of high dimensional parametric elliptic problems, arising, for example, in stochastic PDEs (parameter-dependent coefficients in L, [88,64]). The governing equation is formulated as follows: Given an elliptic operator…”

“…Concerning the coefficient function a M (y, x), we assume that there exists a min > 0, such that (see [64]), In the case d = 1, under some technical assumptions, we are able to derive an estimate on the separation rank in variables y 1 , ..., y M , in the weakly coupled format, …”

“…b m (y m , x)) −1 can be separated by the (2K + 1)-term sinc-quadrature, with the exponential convergence in the number of terms (similar to [33,64]). Proposition 4.2 leads to the constructive rank-(2K + 1) approximation of grad x u M (y, x) in the weakly coupled format (2.3), and it opens the way to the design of direct QTT-truncated solver for the class of elliptic problems with parameter-dependent coefficients for d = 1.…”

“…For this application a fully separable low-rank canonical approximations are constructed in (M + 1)-dimensional tensor space [64], where the d-dimensional physical variable enters the tensor space as a single argument. S-truncated preconditioned iteration applies to solving discrete sPDE in the form Numerical results for sPDEs based on QTT and hierarchical formats are presented in [62,66].…”