Multilevel tensor approximation of PDEs with random data

TL;DR

This paper presents a multilevel low-rank tensor approach for efficiently solving high-dimensional random PDEs, enabling accurate statistical analysis with reduced computational cost.

Contribution

It introduces a novel multilevel framework combined with adaptive low-rank tensor techniques for high-dimensional stochastic PDEs, improving efficiency and accuracy.

Findings

Efficient approximation of random PDE solutions in low-rank tensor format.

Reduced computational cost compared to traditional methods.

Effective error balancing across multiple discretization levels.

Abstract

In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a deterministic parameter-dependent problem on a high-dimensional parameter domain. Given a hierarchy of finite element discretizations for the spatial approximation, we make use of a multilevel framework in which we consider the differences of the solution on two consecutive finite element levels in the collocation points. We then address the approximation of these high-dimensional differences by adaptive low-rank tensor techniques. This allows to equilibrate the error on all levels by exploiting analytic and algebraic properties of the solution at the same time. We arrive at an explicit representation in a low-rank tensor format of the approximate solution on the entire parameter domain, which can be used for, e.g., the direct and cheap computation of statistics. Numerical results are provided in order to illustrate the approach.