Numerical inverse Laplace transformation using concentrated matrix exponential distributions

Abstract

This paper investigates the performance of the numerical inverse Laplace transformation (ILT) method based on concentrated matrix exponential (CME) distributions, referred to as the CME method.The CME method does not generate overshoot and undershoot (i.e., avoids Gibbs oscillation), preserves monotonicity of functions, its accuracy is gradually improving with the order, and it is numerically stable even for order 1000 when using machine precision arithmetic, while other methods get unstable already for order 100 using the same arithmetic.For ILT based tail approximation, the paper recommends an abscissa shifting approach which improves the accuracy of most ILT methods and proposes a heuristic procedure to approximate the numerical accuracy of some ILT methods.