A Generalized Gamma Method for Numerical Inversion of the Laplace Transform in Probabilistic Problems

This paper considers the problem of numerically inverting the Laplace transform in probabilistic methods, where it is necessary to reconstruct distribution densities, reliability functions, waiting time distributions, and other probability–time characteristics from an image. It has been shown that a limitation of the classical Alfrey formula is the quadratic growth of the variance of the exponential kernel used, leading to a loss of localization with increasing time. To overcome this drawback, a generalized Alfrey Gamma method is proposed, based on a family of centered gamma kernels with controlled variance decreasing as t2/n. For the corresponding integral operator, an explicit regularized inversion formula has been derived through the Bromwich integral, and pointwise convergence has been proven for continuous originals. Using smooth test distribution densities, it has been shown that introducing the parameter n systematically improves accuracy compared to the classical case of n = 1, and in the considered settings, it brings the results closer to the Gaver–Stehfest method. The resulting approach is considered as an interpretable regularized tool for probabilistic calculations in which both the accuracy and control of smoothing are important during the reconstruction of distributions.