Date of Award

Fall 2024

Document Type

Thesis (Master's)

Department or Program

Computer Science

First Advisor

Wojciech Jarosz

Second Advisor

Adithya Pediredla

Third Advisor

Rohan Sawhney

Abstract

Walk on Spheres (WoS) is a grid-free Monte Carlo method for solving elliptic partial differential equations (PDEs).
Rather than discretizing the domain, WoS leverages the mean-value principle to obtain Monte Carlo estimates by recursively averaging the solution over the largest contained sphere, terminating upon reaching the boundary.
Unfortunately, WoS requires many independent estimates to achieve noise-free results.

We propose an acceleration technique for WoS, inspired by irradiance caching methods, that computes the solution at a sparse set of locations, and extrapolates these cached values to local neighborhoods. A key insight is that WoS can be extended to compute not only the solution at the sphere center but also a Fourier expansion of the solution function in the surrounding neighborhood.
Each cache point can therefore provide an accurate local estimate around, significantly reducing computation while maintaining quality.
We develop both biased and unbiased forms of our caching algorithm with stratified and geometric cache point distributions, showing that it can achieve up to an order of magnitude error reduction compared to the pointwise WoS estimator for 2D and 3D Laplace's and Poisson's equations under various boundary conditions.

Additionally, we explore a guided WoS algorithm based on zero-variance theory to reduce variance within a single forward walk. Traditional WoS samples each step uniformly over the largest sphere, disregarding the recursive solution term. By importance sampling this term using an approximate finite difference solution, the guided WoS estimator achieves further variance reduction.
This opens exciting avenues for future error reduction algorithms that combine these two orthogonal techniques: obtaining robust forward walks estimation by the guided WoS, followed by neighborhood extrapolation via Fourier method to further reduce error.

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