Reduced Collocation Methods: Reduced Basis Methods in the Collocation Framework

TL;DR

This paper introduces two novel reduced basis methods tailored for the collocation framework, offering efficient and accurate alternatives to traditional Galerkin approaches, especially for non-affine problems.

Contribution

It presents the first reduced basis strategies specifically designed for collocation methods, including the Least Squares Reduced Collocation Method and Empirical Reduced Collocation Method.

Findings

High efficiency and accuracy demonstrated in numerical tests.

Methods match or surpass traditional Galerkin reduced basis methods.

Elimination of costly online procedures for non-affine problems.

Abstract

In this paper, we present the first reduced basis method well-suited for the collocation framework. Two fundamentally different algorithms are presented: the so-called Least Squares Reduced Collocation Method (LSRCM) and Empirical Reduced Collocation Method (ERCM). This work provides a reduced basis strategy to practitioners who {prefer} a collocation, rather than Galerkin, approach. Furthermore, the empirical reduced collocation method eliminates a potentially costly online procedure that is needed for non-affine problems with Galerkin approach. Numerical results demonstrate the high efficiency and accuracy of the reduced collocation methods, which match or exceed that of the traditional reduced basis method in the Galerkin framework.