Polynomial (chaos) approximation of maximum eigenvalue functions: efficiency and limitations

TL;DR

This paper investigates polynomial chaos approximations of the spectral abscissa function, emphasizing the impact of smoothness properties on approximation accuracy and limitations, especially in cases with non-differentiability.

Contribution

It highlights the critical role of smoothness in polynomial approximations of eigenvalue functions and analyzes limitations when the spectral abscissa lacks smoothness.

Findings

Smoothness significantly affects approximation errors.

Non-differentiability occurs with multiple or higher multiplicity eigenvalues.

Numerical experiments demonstrate the limitations of polynomial chaos methods.

Abstract

This paper is concerned with polynomial approximations of the spectral abscissa function (the supremum of the real parts of the eigenvalues) of a parameterized eigenvalue problem, which are closely related to polynomial chaos approximations if the parameters correspond to realizations of random variables. Unlike in existing works, we highlight the major role of the smoothness properties of the spectral abscissa function. Even if the matrices of the eigenvalue problem are analytic functions of the parameters, the spectral abscissa function may not be everywhere differentiable, even not everywhere Lipschitz continuous, which is related to multiple rightmost eigenvalues or rightmost eigenvalues with multiplicity higher than one. The presented analysis demonstrates that the smoothness properties heavily affect the approximation errors of the Galerkin and collocation-based polynomial approximations, and the numerical errors of the evaluation of coefficients with integration methods. A documentation of the experiments, conducted on the benchmark problems through the software Chebfun, is publicly available.