@demofox Naïve question: is this faster and/or more efficient than solving the Laplace equation? I somehow had in my mind that there were pretty good Laplace solvers out there (notice that I might be badly wrong about it).

@j_bertolotti @demofox IIRC the trick here is that it gives you the value of a solution at a specific point without requiring to solve for the whole domain.

@lisyarus @demofox What this is exploiting is that the solution of the Laplace equation is also the steady state of the diffusion equation. So while it is true you don't need to solve for parts of the domain far away from the boundaries (i.e. the data you want to interpolate/extrapolate), you still need to "diffuse" from all those points.
If you tell me that solving the Laplace equation is a lot more onerous than doing a random walk I am ready to believe you, but it doesn't feel obvious to me.

@lisyarus @j_bertolotti oh and it can also solve poisson equations (using greens functions, which is just a tiny bit more calculation per step) which is neat, not just laplace.

@demofox @lisyarus Which brings me to the next (also very naïve) question: as the Green functions of the Poisson eq on many surfaces is well known, why not find the diffusion steady state as a sum of those instead of doing the whole Brownian random walk?

@j_bertolotti @demofox @lisyarus Walk on Spheres in fact uses the known Greens functions and Poisson kernels of simple shapes (by default the "Sphere" in Walk on Spheres, but you can also do Walk on Rectangles, etc). The benefit is that you can apply WoS to arbitrary domains where these functions are not known in closed form. As previously mentioned, you can perform point-evaluation, and you don't need to mesh/discretize the domain or boundaries.