Parametric Integration by Magic Point Empirical Interpolation

TL;DR

This paper establishes error bounds and convergence rates for the magic point empirical interpolation method, demonstrating its effectiveness in parametric integration, especially for Fourier transforms, with broad applications across various scientific fields.

Contribution

It provides new analyticity criteria and error bounds for the magic point empirical interpolation method, extending its theoretical understanding and practical applicability.

Findings

Exponential convergence observed in numerical experiments

Method is well-suited for Fourier transform applications

Effective in diverse fields like physics, finance, and signal processing

Abstract

We derive analyticity criteria for explicit error bounds and an exponential rate of convergence of the magic point empirical interpolation method introduced by Barrault et al. (2004). Furthermore, we investigate its application to parametric integration. We find that the method is well-suited to Fourier transforms and has a wide range of applications in such diverse fields as probability and statistics, signal and image processing, physics, chemistry and mathematical finance. To illustrate the method, we apply it to the evaluation of probability densities by parametric Fourier inversion. Our numerical experiments display convergence of exponential order, even in cases where the theoretical results do not apply.