Yang QiFollow

Author ORCID Identifier

https://orcid.org/0000-0002-1710-4207

Date of Award

Spring 6-1-2022

Document Type

Thesis (Master's)

Department or Program

Computer Science

First Advisor

Wojciech Jarosz

Abstract

Poisson’s equations and Laplace’s equations are important linear partial differential equations (PDEs)
widely used in many applications. Conventional methods for solving PDEs numerically often need to
discretize the space first, making them less efficient for complex shapes. The random walk on spheres
method (WoS) is a grid-free Monte-Carlo method for solving PDEs that does not need to discrete the
space. We draw analogies between WoS and classical rendering algorithms, and find that the WoS
algorithm is conceptually identical to forward path tracing.
We show that solving the Poisson’s equation is equivalent to solving the Green’s function for every
pair of points in the domain. Inspired by similar approaches in rendering, we propose a novel WoS
reformulation that operates in the reverse direction. Besides this, using the corrector function enables
us to use control variates to estimate the Green’s function. Implementations of this algorithm show
improvement over classical WoS in solving Poisson’s equation with sparse sources. Our approach
opens exciting avenues for future algorithms for PDE estimation which, analogous to light transport,
connect WoS walks starting from sensors and sources and combine different strategies for robust
solution algorithms in all cases.

Original Citation

@article{qi22bidirectional, author = "Qi, Yang and Seyb, Dario and Bitterli, Benedikt and Jarosz, Wojciech", title = "A bidirectional formulation for {Walk} on {Spheres}", journal = "Computer Graphics Forum (Proceedings of EGSR)", year = "2022", month = jul, volume = "41", number = "4", issn = "1467-8659", doi = "10/jgzr", keywords = "Brownian motion, partial differential equations, PDEs, Monte Carlo", abstract = "Numerically solving partial differential equations (PDEs) is central to many applications in computer graphics and scientific modeling. Conventional methods for solving PDEs often need to discretize the space first, making them less efficient for complex geometry. Unlike conventional methods, the walk on spheres (WoS) algorithm recently introduced to graphics is a grid-free Monte Carlo method that can provide numerical solutions of Poisson equations without discretizing space. We draw analogies between WoS and classical rendering algorithms, and find that the WoS algorithm is conceptually equivalent to forward path tracing. Inspired by similar approaches in light transport, we propose a novel WoS reformulation that operates in the reverse direction, starting at source points and estimating the Green's function at ``sensor'' points. Implementations of this algorithm show improvement over classical WoS in solving Poisson equation with sparse sources. Our approach opens exciting avenues for future algorithms for PDE estimation which, analogous to light transport, connect WoS walks starting from sensors and sources and combine different strategies for robust solution algorithms in all cases." }

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