A row-sampling-based randomised finite element method for elliptic partial differential equations

TL;DR

This paper introduces a randomized finite element method that accelerates solutions for elliptic PDEs by using low-dimensional projections and non-uniform sampling, achieving significant computational speedups with controlled errors.

Contribution

It proposes a novel randomized FEM approach that employs low-rank approximations and non-uniform sampling to efficiently solve high-dimensional elliptic PDEs.

Findings

Achieves approximately tenfold reduction in computational time.

Maintains moderate approximation accuracy despite speedup.

Validates approach on Dirichlet and Neumann boundary problems.

Abstract

We consider a randomised implementation of the finite element method (FEM) for elliptic partial differential equations on high-dimensional models. This is motivated by applications where model predictions are essential for real-time process diagnostics. In these circumstances it is imperative to expedite prediction without a significant compromise in the model's fidelity, which in turn relies on the rapid assembly and solution of the associated system of equations typically at the many-query context. Our approach involves converting the solution of the linear, symmetric positive definite FEM system into an over-determined least squares problem, whose solution is then projected onto a low-dimensional subspace. The resulting low-dimensional system can be effectively sketched as a product of two high-dimensional matrices using a parameter-dependent non-uniform sampling distribution, utilising only a small subset of the model's parameters. Although different to the optimal sampling distributions based on the statistical leverage-scores of the rows of the matrices, we show that the distance between them shrinks for an appropriate choice of the projection subspace. For the approximate solution we bound the incurring errors due to the projection, subspace approximation and sketching and show that the overall error is dominated by the condition number of the projected stiffness matrix. Our approach is tested on simulations on the Dirichlet and Neumann problems for the steady-state diffusion equation. The results show that our approach has on average a tenfold improvement on the computational times compared to the classical deterministic framework at the expense of a moderately small approximation error.