Here's a little ELI5 about the Laplace and inverse Laplace transform, and why the inverse transfrom is fiendishly difficult, and therefore why this result is ex

Here's a little ELI5 about the Laplace and inverse Laplace transform, and why the inverse transfrom is fiendishly difficult, and therefore why this result is extraordinarily important. Imagine you win the Megabucks lottery. The win is one hundred million dollars. You go to claim your money, but you are told you can choose between the full amount given in monthly payments over 20 years, or a lump sum. But the lump sum is not the full $100MM, it is the present value of the monthly payments discounted at a rate of 5%. To discount an amount received 10 years from now at the 5% rate, you simply divide by 1.05 ^ 10, which is very close to exp(0.05 x 10). If you actually calculate this present value using the exponential function, you say that you use "continuously compounded rates". So, for any stream of future cashflows one can calculate the present value by multiplying the cashflows with appropriate discount factors (of the type exp(-r t)) and adding them up. For different discounting rates r you obtain different present values. This present value as a function of r is the Laplace transform of the cashflow stream as a function of t. The inverse Laplace transform is solving the riddle: if I tell you the present value (PV) of some cashflows for any (positive) discount rate you want, can you calculate the cashflows? Why is this a difficult problem? Because it is "ill-conditioned". Imagine the following two cashflow streams: in the first you get $1MM every year for the next 10 years and another $1MM one hundred years from now. In the second you also get $1MM annually for the first ten years but the last $1MM is 101 years from now. For a zero discount rate the value of both cashstreams is $11MM. For a 5% they are both around $9MM and different by about $300, which is about 0.003%. For any discount rate the PV's will be very very close. In some cases "in real life" this closeness could be below machine precision level. If someone gives you 2 sets of inputs where their Laplace transforms are different by less than the machine precision levels for all values of the discount rate, then there is no hope to tell them apart knowing only their trasforms only, at least not if you don't use some multiple precision libraries. That should give you an intuition why the inverse Laplace transform is nasty. All hope is not lost though. First of all, in a typical application the Laplace transform of a function is known in closed (analytical) form, so you can actually use multiple precision libraries if you so wish. I have seen cases where people were using precision of 2000 digits in Mathematica for this. It's slow as hell, but it gets the job done. Separately, you are free to calculate the Laplace transform at any "discount rate", including complex values. If you are smart about how to choose these values, you can come up with good recipes for the Laplace transform. For hundreds of years now, the general wisdom was that various inverse numerical Laplace transform algorithms have strengths and weaknesses, but no single one is universally good. Maybe this one will be, and if so it will be indeed revolutionary.