An improved discrete least-squares/reduced-basis method for parameterized elliptic PDEs

TL;DR

This paper introduces an enhanced method combining discrete least-squares and reduced-basis techniques to efficiently approximate solutions of parameterized elliptic PDEs, achieving significant computational savings and improved convergence.

Contribution

It presents a novel integrated approach that incorporates reduced-basis methods into DLS for elliptic PDEs, leading to faster computations and broader applicability.

Findings

Significant reduction in storage and online evaluation costs.

Improved convergence rates with quasi-optimal polynomial spaces.

Numerical experiments confirm theoretical advantages.

Abstract

It is shown that the computational efficiency of the discrete least-squares (DLS) approximation of solutions of stochastic elliptic PDEs is improved by incorporating a reduced-basis method into the DLS framework. The goal is to recover the entire solution map from the parameter space to the finite element space. To this end, first, a reduced-basis solution using a weak greedy algorithm is constructed, then a DLS approximation is determined by evaluating the reduced-basis approximation instead of the full finite element approximation. The main advantage of the new approach is that one only need apply the DLS operator to the coefficients of the reduced-basis expansion, resulting in huge savings in both the storage of the DLS coefficients and the online cost of evaluating the DLS approximation. In addition, the recently developed quasi-optimal polynomial space is also adopted in the new approach, resulting in superior convergence rates for a wider class of problems than previous analyzed. Numerical experiments are provided that illustrate the theoretical results.