Fast Least-Squares Pad\'e approximation of problems with normal operators and meromorphic structure

TL;DR

This paper introduces a simplified and efficient Least-Squares Padé approximation method for meromorphic functions from parametric PDEs with normal operators, providing theoretical guarantees and improved numerical accuracy.

Contribution

It develops a faster, more robust version of the Least-Squares Padé approximation for normal operators, with new theoretical convergence results and practical numerical validation.

Findings

Exponential decay of pole approximation error

Convergence in measure of the approximant to the target function

Numerical results show improved accuracy over previous methods

Abstract

In this work, we consider the approximation of Hilbert space-valued meromorphic functions that arise as solution maps of parametric PDEs whose operator is the shift of an operator with normal and compact resolvent, e.g. the Helmholtz equation. In this restrictive setting, we propose a simplified version of the Least-Squares Pad\'e approximation technique introduced in [6] following [11]. In particular, the estimation of the poles of the target function reduces to a low-dimensional eigenproblem for a Gramian matrix, allowing for a robust and efficient numerical implementation (hence the "fast" in the name). Moreover, we prove several theoretical results that improve and extend those in [6], including the exponential decay of the error in the approximation of the poles, and the convergence in measure of the approximant to the target function. The latter result extends the classical one for scalar Pad\'e approximation to our functional framework. We provide numerical results that confirm the improved accuracy of the proposed method with respect to the one introduced in [6] for differential operators with normal and compact resolvent.