Numerical Methods and Analysis for Computing Forward and Inverse Laplace Transform For discrete and continuous signals

Specialized data maps transforming (probabilistic) state representations mapping the original signal representation into a more manipulatable state space capitalizes on contemporary computational advances and relates to well-known application scenarios, such as kernel machines, MDP transition dynamics modeling and policy optimization, and generative modeling. In this work we focus on the classic example of Laplace transform, which maps the subspace of differential equations to the space of algebraic equations, where the differential equation solution can be obtained by inverting the algebraic equation solution. Effective integral transformations enable simplification of mathematical modeling and computational inference. The Laplace transform (LT) with its inverse (ILT) represents a family of integral transforms that have direct applications in contemporary data science, statistical inference, and probabilistic modeling. However, practical computational challenges inhibit their utilization on complex or implicit functions, noisy observations, and incomplete data.

We propose a numerical LT-ILT computing framework, empirically robust, implemented in R, and facilitates numerical computations through appropriate parameter estimations and signal approximations. We illustrate an idealized analysis relaxing the data matrix construction assumptions to bound the smallest singular values to argue algorithmic stability according to theoretical results in random matrix theory.