Coarse-Proxy Reduced Basis Methods for Integral Equations

TL;DR

This paper presents a novel coarse-proxy reduced basis method for integral equations, leveraging low-resolution models and operator interpolation to efficiently construct reduced bases, significantly improving computational performance.

Contribution

The paper introduces a new reduced basis approach using coarse-proxy models and operator interpolation, tailored for integral equations, with easy implementation and parallelization advantages.

Findings

Significant performance improvements over high-fidelity models.

Effective error reduction with more aggressive proxy selection.

Successful application to Radiative Transport and Laplace equations.

Abstract

In this paper, we introduce a new reduced basis methodology for accelerating the computation of large parameterized systems of high-fidelity integral equations. Core to our methodology is the use of coarse-proxy models (i.e., lower resolution variants of the underlying high-fidelity equations) to identify important samples in the parameter space from which a high quality reduced basis is then constructed. Unlike the more traditional POD or greedy methods for reduced basis construction, our methodology has the benefit of being both easy to implement and embarrassingly parallel. We apply our methodology to the under-served area of integral equations, where the density of the underlying integral operators has traditionally made reduced basis methods difficult to apply. To handle this difficulty, we introduce an operator interpolation technique, based on random sub-sampling, that is aimed specifically at integral operators. To demonstrate the effectiveness of our techniques, we present two numerical case studies, based on the Radiative Transport Equation and a boundary integral formation of the Laplace Equation respectively, where our methodology provides a significant improvement in performance over the underlying high-fidelity models for a wide range of error tolerances. Moreover, we demonstrate that for these problems, as the coarse-proxy selection threshold is made more aggressive, the approximation error of our method decreases at an approximately linear rate.