Regularized Kaczmarz Solvers for Robust Inverse Laplace Transforms

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. 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. To promote reproducibility and further development, all code is freely available at https://github.com/fmarrabal/kaczmarz-ilt-solver.