Given $y$, I wish to find $x$ (which lies in a known certain range) such that $F(x) - y = 0$, for some monotonically increasing $F$ which I can approximate at specific values of $x$ by Monte Carlo integration. Currently I am just performing many evaluations over a grid.
Is there a nice algorithm to achieve this? It feels like there should be some cunning way to, given the current sampled values, choose a new point to sample to increase our information about where the root lies.