  A popular method to compute first-passage probabilities in continuous-time
Markov chains is by numerically inverting their Laplace transforms. Past
decades, the scientific computing community has developed excellent numerical
methods for solving problems governed by partial differential equations (PDEs),
making the availability of a Laplace transform not necessary here for
computational purposes. In this study we demonstrate that numerical PDE problem
solvers are suitable for computing first passage times, and can be very
efficient for this purpose. By doing extensive computational experiments, we
show that modern PDE problem solvers can outperform numerical Laplace transform
inversion, even if a transform is available. When the Laplace transform is
explicit (e.g. does not require the computation of an eigensystem), numerical
transform inversion remains the primary method of choice.
