My mind is blown several times Man. Suggest me a few books to read, so that I can understand the concepts and get the hints of connecting dots relatively easily.
Thanks! My PhD thesis ( rohansawhney.io/RohanSawhneyPhDThesis.pdf ) goes over the concepts in more detail, and covers others variants of walk on spheres we've developed as well :)
Love to see this progress! I wonder whether this could be done in the time domain as well? I know, it tends to be the case with Monte-Carlo methods, that they are timeless. They try to sample some steady state, right? But if you make the entire spacio-temporal volume your "steady state", it ought to be possible regardless. Normally you couldn't properly save, and keep access to all that data so you could completely skip out on frames. But perhaps there is a way around that somehow? Maybe something inspired by NeRFs. A sort of data structure that's iteratively updated with gradient descend anyways, so you could probably reformulate that as a PDE that can be handled with a method like this I'd think? Then you could effectively get rid of all explicit grids (globally spatial, locally surface- or volume-spatial, and temporal) and go gridless continuous end to end perhaps. It seems to me the application of refining subsurface scattering is a nice example of how that might be helpful.
Hi! It's possible to handle temporal problems such as the heat equation with walk on spheres, I recommend looking at this paper: epubs.siam.org/doi/10.1137/0114031 Since the heat equation is an initial value problem, the high-level idea is to also sample an "exit time" at each step of the random walk from a known distribution (in addition to a uniform exit location on the sphere), and to keep a counter of the total time elapsed during the walk. If the counter value exceeds the predetermined amount/time for which one wants to flow the heat, then the initial value at the current position of the random walk inside the domain is added to the Monte Carlo estimate.
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Excellent Rohan
My mind is blown several times Man. Suggest me a few books to read, so that I can understand the concepts and get the hints of connecting dots relatively easily.
Thanks! My PhD thesis ( rohansawhney.io/RohanSawhneyPhDThesis.pdf ) goes over the concepts in more detail, and covers others variants of walk on spheres we've developed as well :)
A full video! brilliant, i loved this paper IMMENSELY (my thesis is suffering from how much time ive spent thinking bout this paper)
Love to see this progress!
I wonder whether this could be done in the time domain as well?
I know, it tends to be the case with Monte-Carlo methods, that they are timeless. They try to sample some steady state, right?
But if you make the entire spacio-temporal volume your "steady state", it ought to be possible regardless.
Normally you couldn't properly save, and keep access to all that data so you could completely skip out on frames. But perhaps there is a way around that somehow? Maybe something inspired by NeRFs. A sort of data structure that's iteratively updated with gradient descend anyways, so you could probably reformulate that as a PDE that can be handled with a method like this I'd think?
Then you could effectively get rid of all explicit grids (globally spatial, locally surface- or volume-spatial, and temporal) and go gridless continuous end to end perhaps.
It seems to me the application of refining subsurface scattering is a nice example of how that might be helpful.
Hi! It's possible to handle temporal problems such as the heat equation with walk on spheres, I recommend looking at this paper: epubs.siam.org/doi/10.1137/0114031
Since the heat equation is an initial value problem, the high-level idea is to also sample an "exit time" at each step of the random walk from a known distribution (in addition to a uniform exit location on the sphere), and to keep a counter of the total time elapsed during the walk. If the counter value exceeds the predetermined amount/time for which one wants to flow the heat, then the initial value at the current position of the random walk inside the domain is added to the Monte Carlo estimate.
@@rohansawhney1583 thanks I'll check it out! sounds interesting
very nice presentation thx