Regularized Kaczmarz Solvers for Robust Inverse Laplace Transforms

Marta González-Lázaro, Eduardo Viciana, Víctor Valdivieso, Ignacio Fernández () and Francisco Manuel Arrabal-Campos ()
Additional contact information
Marta González-Lázaro: Department of Chemistry and Physics, Research Centre CIAIMBITAL, University of Almería, 04120 Almería, Spain
Eduardo Viciana: Department of Engineering, Research Centre CIAIMBITAL, University of Almería, 04120 Almería, Spain
Víctor Valdivieso: Department of Engineering, Research Centre CIAIMBITAL, University of Almería, 04120 Almería, Spain
Ignacio Fernández: Department of Chemistry and Physics, Research Centre CIAIMBITAL, University of Almería, 04120 Almería, Spain
Francisco Manuel Arrabal-Campos: Department of Engineering, Research Centre CIAIMBITAL, University of Almería, 04120 Almería, Spain

Mathematics, 2025, vol. 13, issue 13, 1-29

Abstract: Inverse Laplace transforms (ILTs) are fundamental to a wide range of scientific and engineering applications—from diffusion NMR spectroscopy to medical imaging—yet their numerical inversion remains severely ill-posed, particularly in the presence of noise or sparse data. The primary objective of this study is to develop robust and efficient numerical methods that improve the stability and accuracy of ILT reconstructions under challenging conditions. In this work, we introduce a novel family of Kaczmarz-based ILT solvers that embed advanced regularization directly into the iterative projection framework. We propose three algorithmic variants—Tikhonov–Kaczmarz, total variation (TV)–Kaczmarz, and Wasserstein–Kaczmarz—each incorporating a distinct penalty to stabilize solutions and mitigate noise amplification. The Wasserstein–Kaczmarz method, in particular, leverages optimal transport theory to impose geometric priors, yielding enhanced robustness for multi-modal or highly overlapping distributions. We benchmark these methods against established ILT solvers—including CONTIN, maximum entropy (MaxEnt), TRAIn, ITAMeD, and PALMA—using synthetic single- and multi-modal diffusion distributions contaminated with 1% controlled noise. Quantitative evaluation via mean squared error (MSE), Wasserstein distance, total variation, peak signal-to-noise ratio (PSNR), and runtime demonstrates that Wasserstein–Kaczmarz attains an optimal balance of speed (0.53 s per inversion) and accuracy (MSE = 4.7 × 10 − 8 ), while TRAIn achieves the highest fidelity (MSE = 1.5 × 10 − 8 ) at a modest computational cost. These results elucidate the inherent trade-offs between computational efficiency and reconstruction precision and establish regularized Kaczmarz solvers as versatile, high-performance tools for ill-posed inverse problems.

Keywords: inverse Laplace transform; Kaczmarz method; Wasserstein regularization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/13/2166/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/13/2166/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:13:p:2166-:d:1693369

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().