First results about the Tikhonov regularization methods for numerically inverting a Laplace transform function.

The real inversion of a Laplace transform function is an ill-posed problem; so, in the design of a numerical method for inverting a Laplace transform function when the function values are known on a subset of R only, it is of fundamental importance to take into account the dependence of the solution on the perturbations, even if small, that affect the data. The regularization methods allow one to transform that problem into a well-posed one, whose solution is a good approximation of that of the original problem; a crucial role, of course, is played by the choice of the so-called regularization parameter. "In this paper we discuss some results obtained by applying the cross-validation technique for the automatic estimate of the regularization parameter.''