A Reduced Radial Basis Function Method for Partial Differential Equations on irregular domains

TL;DR

This paper introduces a novel Reduced Radial Basis Function Method (R2BFM) for efficiently solving parametric partial differential equations on irregular domains, combining stable RBF solvers with model reduction techniques.

Contribution

It presents the first R2BFM that integrates a stable RBF solver with a greedy algorithm for center selection and a collocation-based model reduction, significantly reducing computational complexity.

Findings

Efficient and accurate solutions for 2D and 3D PDEs on irregular domains.

Reduced-order models with dimensions much smaller than the original RBF centers.

Demonstrated effectiveness through numerical tests.

Abstract

We propose and test the first Reduced Radial Basis Function Method (R$^2$BFM) for solving parametric partial differential equations on irregular domains. The two major ingredients are a stable Radial Basis Function (RBF) solver that has an optimized set of centers chosen through a reduced-basis-type greedy algorithm, and a collocation-based model reduction approach that systematically generates a reduced-order approximation whose dimension is orders of magnitude smaller than the total number of RBF centers. The resulting algorithm is efficient and accurate as demonstrated through two- and three-dimensional test problems.