Regularization of the inverse Laplace transform by Mollification

2023/04/17 by Pierre Maréchal, Maréchal, Pierre, Faouzi Triki +3
Decision Sciences · Mathematics · #35R30 #65N21 #Analysis of PDEs (math.AP) #FOS: Mathematics #G.1.0 #G.1.9 #Mathematical functions and polynomials #Numerical methods in inverse problems #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2304.08057

openalex publication_date 2023/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the inverse Laplace transform. We first derive a new global logarithmic stability estimate that shows that the inversion is severely ill-posed. Then we propose a regularization method to compute the inverse Laplace transform using the concept of mollification. Taking into account the exponential instability we derive a criterion for selection of the regularization parameter. We show that by taking the optimal value of this parameter we improve significantly the convergence of the method. Finally, making use of the holomorphic extension of the Laplace transform, we suggest a new PDEs based numerical method for the computation of the solution. The effectiveness of the proposed regularization method is demonstrated through several numerical examples.

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