Spectral Methods for Parameterized Matrix Equations

TL;DR

This paper introduces spectral methods for efficiently approximating solutions to parameter-dependent linear matrix equations, providing error estimates and numerical validation.

Contribution

It develops interpolatory pseudospectral and residual-minimizing Galerkin methods for parameterized matrix equations, with theoretical error analysis and practical residual estimates.

Findings

Methods achieve accurate approximations in numerical examples.

Error estimates relate to analyticity of solutions.

Residual minimization improves approximation quality.

Abstract

We apply polynomial approximation methods -- known in the numerical PDEs context as spectral methods -- to approximate the vector-valued function that satisfies a linear system of equations where the matrix and the right hand side depend on a parameter. We derive both an interpolatory pseudospectral method and a residual-minimizing Galerkin method, and we show how each can be interpreted as solving a truncated infinite system of equations; the difference between the two methods lies in where the truncation occurs. Using classical theory, we derive asymptotic error estimates related to the region of analyticity of the solution, and we present a practical residual error estimate. We verify the results with two numerical examples.