Regularized Kaczmarz Solvers for Robust Inverse Laplace Transforms

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Detalhes bibliográficos
Publicado no:	Mathematics vol. 13, no. 13 (2025), p. 2166-2195
Autor principal:	González-Lázaro, Marta
Outros Autores:	Viciana Eduardo, Valdivieso Víctor, Fernández Ignacio, Arrabal-Campos, Francisco Manuel
Publicado em:	MDPI AG
Assuntos:	Sparsity Accuracy Mathematical analysis Iterative methods Transport theory Laplace transforms Signal processing Medical imaging Solvers Maximum entropy Numerical methods Regularization Inverse problems NMR spectroscopy Magnetic resonance imaging Ill posed problems Process controls Computational efficiency Computing costs Regularization methods Probability Molecular weight Engineering Algorithms Robustness (mathematics) Signal to noise ratio
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Descrição
Resumo:	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 = <inline-formula>4.7×10−8</inline-formula>), while TRAIn achieves the highest fidelity (MSE = <inline-formula>1.5×10−8</inline-formula>) 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.
ISSN:	2227-7390
DOI:	10.3390/math13132166
Fonte:	Engineering Database