I am looking for explanations of algorithms to adaptively sample a function of two variables $f(x,y)$, in a given domain $x_0\le x \le x_1$, $y_0\le y \le y_1$. Intuitively, I want to sample more densely regions were $f(x,y)$ has more variation. I know almost nothing a priori about the function, except that inside the domain $x_0\le x \le x_1$, $y_0\le y \le y_1$, there maybe regions where it is infinite (that I obviously don't want to sample, so the algorithm should detect that too), and that everywhere else, the function is continuous. These singular regions are "smooth sets".
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Looks like you probably would like to steal some ideas from this discussion about implementing something similar in Julia. It's hard to cleanly summarize since there's a lot of different details, but essentially it's an algorithm which does a simple grid and then adaptively adds points recursively in the regions where the linear approximation is worse. It's like a course grid adaptivity in finite element methods where you take more grid points and see how far your interpolation was off at the grid points, and if it was a lot, you accept the new grid points and recursively re-apply the coarsening operation. Then details ensue, but it's all open source with a related discussion so you can check it out all the gory details. Taking that from 1D to 2D likely involves new code but the same principles.
Chris Rackauckas
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the answer maybe this https://scicomp.stackexchange.com/a/13048/25606
It sounds like you want the ** multi-precision plot function from SymPy**, which is capable of plotting arbitrary black-box functions over a given range.
