Numerical methods for computing the discrete and continuous Laplace transforms

Abstract The Laplace transform (LT) and its inverse (ILT) play vital roles in contemporary data science, probability modelling, statistical inference and spacekime analytics. For some complex functions, noisy data or incomplete observations, there are challenges with computing LT and ILT. This article reports on new numerical algorithms for computing the forward and inverse LT. The technique is applicable to both continuous functions and discrete signals, and yields computationally efficient and analytically robust results. We review a Meijer-G symbolic approach to compute the LT and test the LT and ILT on discrete data and on analytic functions with known exact transforms. We also report empirical evidence of the asymptotic behaviour of the expectation of the smallest singular value of the LT matrix, $\mathbb{E}(\sigma _{n}(\mathbf{A})) \sim \frac{1}{n^\gamma }$ for some $\gamma \in (0,2)$. Our analysis of the smallest singular values of the random matrix emerging in the ILT algorithm indicates that under certain assumptions, the matrix can be bounded. Random phase sampling independent of the radial strategy may violate isotropicities of the random matrix. This LT/ILT technique is applicable to analytical functions as well as observational signals. The article also proposes a Clifford algebra approach for generalizing the LT to higher dimensional space-time processes.