A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems

TL;DR

This paper introduces a tensor approximation method based on ideal minimal residual formulations for efficiently solving high-dimensional problems, with convergence guarantees and goal-oriented strategies.

Contribution

It presents a novel perturbation of minimal residual methods and a greedy algorithm for low-rank tensor approximations without prior solution information.

Findings

Method achieves quasi-optimal low-rank approximations.

Convergence of the greedy algorithm is proven under certain conditions.

Applied successfully to stochastic PDEs discretized in tensor spaces.

Abstract

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with residual norm corresponding to the error in a specified solution norm. We introduce and analyze an iterative algorithm that is able to provide a controlled approximation of the optimal approximation of the solution in a given low-rank subset, without any a priori information on this solution. We also introduce a weak greedy algorithm which uses this perturbed minimal residual method for the computation of successive greedy corrections in small tensor subsets. We prove its convergence under some conditions on the parameters of the algorithm. The residual norm can be designed such that the resulting low-rank approximations are quasi-optimal with respect to particular norms of interest, thus yielding to goal-oriented order reduction strategies for the approximation of high-dimensional problems. The proposed numerical method is applied to the solution of a stochastic partial differential equation which is discretized using standard Galerkin methods in tensor product spaces.