There are many implementations available of algorithms that numerically invert the Laplace transform in one dimension (e.g. Talbot or Euler algorithms) in languages such as Mathematica, Matlab and Python.
I am looking for an implementation (in any language) in case of multiple dimensions, or something that I could code up myself. Please let me know if you are aware of this.
Apperantly it is not so hard. You can just take e.g. the Talbot algorithm, and then write a double for-loop, where each for loop inverts one dimension.