On the convergence of the Gaver-Stehfest algorithm

TL;DR

This paper rigorously investigates the Gaver-Stehfest algorithm for numerical Laplace transform inversion, proving its convergence for specific classes of functions, which was previously unknown despite its widespread practical use.

Contribution

It provides the first rigorous proof of convergence for the Gaver-Stehfest algorithm under certain conditions, filling a significant gap in the theoretical understanding of this popular method.

Findings

Proves convergence for functions of bounded variation.

Establishes convergence for functions satisfying an analogue of Dini's criterion.

Addresses a longstanding open question about the method's theoretical validity.

Abstract

The Gaver-Stehfest algorithm for numerical inversion of Laplace transform was developed in the late 1960s. Due to its simplicity and good performance it is becoming increasingly more popular in such diverse areas as Geophysics, Operations Research and Economics, Financial and Actuarial Mathematics, Computational Physics and Chemistry. Despite the large number of applications and numerical studies, this method has never been rigorously investigated. In particular, it is not known whether the Gaver-Stehfest approximations converge and what is the rate of convergence. In this paper we answer the first of these two questions: We prove that the Gaver-Stehfest approximations converge for functions of bounded variation and functions satisfying an analogue of Dini criterion.