A comparison of matrix-free isogeometric Galerkin and collocation methods for Karhunen--Lo\`eve expansion

TL;DR

This paper compares matrix-free isogeometric Galerkin and collocation methods for computing the Karhunen--Loève expansion, highlighting their relative performance on smooth versus rough covariance kernels.

Contribution

It introduces a new matrix-free isogeometric collocation method and provides a comprehensive comparison with the Galerkin approach for the K--L expansion.

Findings

Galerk in method outperforms for smooth kernels

Collocation method performs better for rough kernels

Benchmark results demonstrate method efficiencies

Abstract

Numerical computation of the Karhunen--Lo\`eve expansion is computationally challenging in terms of both memory requirements and computing time. We compare two state-of-the-art methods that claim to efficiently solve for the K--L expansion: (1) the matrix-free isogeometric Galerkin method using interpolation based quadrature proposed by the authors in [1] and (2) our new matrix-free implementation of the isogeometric collocation method proposed in [2]. Two three-dimensional benchmark problems indicate that the Galerkin method performs significantly better for smooth covariance kernels, while the collocation method performs slightly better for rough covariance kernels.