Stable numerical Laplace Transform inversion technique without over- and undershoot

Williams Zanga, William Knottenbelt

Abstract:

Numerical inversion of Laplace transform is known to be equivalent to approximating a shifted version of the Dirac impulse function with a linear combination of complex exponentials. From this knowledge, we construct a general framework to approximate that function with concentrated matrix exponential distributions, characterized by low coefficient of variation. That structure generalizes the method proposed by Horvath, Talyigas and Telek; and it guarantees numerical inversions without positive or negative overshoots. Optimization is done for a specific class of inversion methods within that framework, with a semi-deterministic algorithm based upon evolution strategy and
gradient descent. This result in approximation errors evolving as O(1/n2). Finally, we propose an analytical method with error of type O(1/n) to bypass optimization.