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

Effective integral transformations enable simplification of mathematical modeling and computational inference by mapping the original signal representation into a more manipulatable state space. Designing specialized data maps transforming (probabilistic) state representations capitalizes on contemporary computational advances and relates to well-known application scenarios, such as kernel machines, MDP transition dynamics modeling and policy optimizations, and generative modeling. A well-known classical example is the Laplace transform, which maps the subspace of differential equations to the space of algebraic equations, where the solution can be obtained by inverting the algebraic equation. The Laplace transform (LT) and 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.



The classical Laplace transform may also be viewed as a bounded and invertible linear operator, a unitary isomorphism between positive real square integrable functions and the complex Hardy space on the right half plane. A proper decay factor γ may be introduced to enforce square integrability e^-γtf(t)∈L² ([0,∞)) and the ILT can be computed exactly via the Bromwhich integral L^-1(F(s))=∫^γ+i∞ _γ-i∞ e^st F(s)ds. However, for some functions and observed discrete signals, the precise function class is often intractable. In this work, we propose a numerical LT-ILT computing framework, implemented in R, which is approximately invertible for a large class of signals, exhibits sufficient robustness, and facilitates numerical computations through appropriate parameter estimations and signal approximations.



In addition, we will present various challenges and open problems relating to random matrix theory, harmonic analysis, formulation of LT-ILT on groups, and a Clifford algebra approach to define Laplace transform to higher spacetime dimensions. Theoretical aspects, related conjectures, and empirical evaluation of the algorithm will be discussed.