Error analysis of a Collocation method for numerically inverting a Laplace transform in case of real samples

摘要

In [S. Cuomo, L. D’Amore, A. Murli, M.R. Rizzardi, Computation of the inverse Laplace transform based on a collocation method which uses only real values, J. Comput. Appl. Math., 198 (1) (2007) 98–115] the authors proposed a Collocation method (C-method) for real inversion of Laplace transforms (Lt), based on the truncated Laguerre expansion of the inverse function:fN(x)=eσx∑k=0N-1cke-bxLk(2bx),where σ, b are parameters and ck, k∈N, are the MacLaurin coefficients of a function depending on the Lt. The computational kernel of a C-method is the solution of a Vandermonde linear system, where the right hand side is obtained evaluating the Lt on the real axis. The Bjorck Pereira algorithm has been used for solving the Vandermonde linear system, providing a computable componentwise error bound on the solution.For an inversion problem on discrete data F is known on a pre-assigned set of points (we refer to these points as samples of F) only and the major challenge is to deal with a significative loss of information. A natural approach to overcome this intrinsic difficulty is to construct a suitable fitting model that approximates the given data. In this case, we show that such approach leads to a C-method with perturbed right hand side, and then we use again the Bjorck Pereira algorithm.Starting from the error introduced by the fitting model, we study its propagation in order to determine the maximum attainable accuracy on fN. Moreover we derive a computable error bound that allows to get the suitable value of the parameter N that gives the maximum attainable accuracy.