  In this paper we study the inverse Laplace transform. We first derive a new
global logarithmic stability estimate that shows that the inversion is severely
ill-posed. Then we propose a regularization method to compute the inverse
Laplace transform using the concept of mollification. Taking into account the
exponential instability we derive a criterion for selection of the
regularization parameter. We show that by taking the optimal value of this
parameter we improve significantly the convergence of the method. Finally,
making use of the holomorphic extension of the Laplace transform, we suggest a
new PDEs based numerical method for the computation of the solution. The
effectiveness of the proposed regularization method is demonstrated through
several numerical examples.
