Numerical real inversion of the Laplace transform by reproducing kernel and multiple-precision arithmetic

Abstract

We consider numerical inversion of the Laplace transform. It is ill-posed in the sense of Hadamard. We introduce some reproducing kernel Hilbert spaces and show a new real inversion algorithm employing Tikhonov regularization [1, 2]. A regularized equations is well-posed, and its discretization is expected to have stability and convergence in an appropriate norm. A small regularization parameter is required for accurate approximation by regularization, which causes numerical instability of its computational process. We give a remark that theoretical stability is not equivalent to stability of computational processes. We use of multiple-precision arithmetic [3] to reduce the influence of rounding errors for reliable numerical computations. The proposed algorithm based on Tikhonov regularization and multiple-precision arithmetic is applicable to problems involving singularities.