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Walking on Spheres and Talking to Neighbors: Variance Reduction for Laplace's Equation

Michael Czekanski, Benjamin Faber, Margaret Fairborn, Adelle Wright, David Bindel

TL;DR

This work advances mesh-free solutions to Laplace's equation by enhancing Walk on Spheres with a caching-based information-reuse strategy. By reusing exit points from walks started at nearby points and reweighting via the Poisson kernel, the authors derive variance bounds for equal-weight and variance-weighted estimators and show asymptotic runtime improvements with a deterministic cache of size $O(\delta^{-d})$. Numerical experiments on a square domain and a Julia-set boundary illustrate substantial variance reduction and improved $L^2$ accuracy without increasing the number of walks. The approach generalizes Monte Carlo methods for elliptic PDEs and opens pathways to extensions to Poisson problems, spatially varying coefficients, and higher-dimensional domains.

Abstract

Walk on Spheres algorithms leverage properties of Brownian Motion to create Monte Carlo estimates of solutions to a class of elliptic partial differential equations. We propose a new caching strategy which leverages the continuity of paths of Brownian Motion. In the case of Laplace's equation with Dirichlet boundary conditions, our algorithm has improved asymptotic runtime compared to previous approaches. Until recently, estimates were constructed pointwise and did not use the relationship between solutions at nearby points within a domain. Instead, our results are achieved by passing information from a cache of fixed size. We also provide bounds on the performance of our algorithm and demonstrate its performance on example problems of increasing complexity.

Walking on Spheres and Talking to Neighbors: Variance Reduction for Laplace's Equation

TL;DR

This work advances mesh-free solutions to Laplace's equation by enhancing Walk on Spheres with a caching-based information-reuse strategy. By reusing exit points from walks started at nearby points and reweighting via the Poisson kernel, the authors derive variance bounds for equal-weight and variance-weighted estimators and show asymptotic runtime improvements with a deterministic cache of size . Numerical experiments on a square domain and a Julia-set boundary illustrate substantial variance reduction and improved accuracy without increasing the number of walks. The approach generalizes Monte Carlo methods for elliptic PDEs and opens pathways to extensions to Poisson problems, spatially varying coefficients, and higher-dimensional domains.

Abstract

Walk on Spheres algorithms leverage properties of Brownian Motion to create Monte Carlo estimates of solutions to a class of elliptic partial differential equations. We propose a new caching strategy which leverages the continuity of paths of Brownian Motion. In the case of Laplace's equation with Dirichlet boundary conditions, our algorithm has improved asymptotic runtime compared to previous approaches. Until recently, estimates were constructed pointwise and did not use the relationship between solutions at nearby points within a domain. Instead, our results are achieved by passing information from a cache of fixed size. We also provide bounds on the performance of our algorithm and demonstrate its performance on example problems of increasing complexity.
Paper Structure (19 sections, 7 theorems, 30 equations, 6 figures, 3 algorithms)

This paper contains 19 sections, 7 theorems, 30 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Laplace's Equation
  3. Mesh-based methods
  4. Mesh-free methods
  5. Walk on Spheres
  6. Walk on Spheres Algorithm
  7. An example
  8. Implementation
  9. Information Reuse
  10. Variance Bounds
  11. Equal Weighting
  12. Variance Weighting
  13. Caching
  14. Runtime
  15. Numerical Experiments
  16. ...and 4 more sections

Key Result

Theorem 1

kakutani1944143muller1956some Given a domain $\Omega$ that is sufficiently regular and a continuous function $f$ on $\partial \Omega$, the solution to eq:laplace is where $\tau = \inf\{t > 0 : B_t \notin \Omega \}$, $B_t$ is Brownian Motion, and $\mathbb{E}_x$ denotes the expectation taken under the measure implied by conditioning on $B_0 = x$. Note that $\tau$ is random under realizations of pat

Figures (6)

  • Figure 1: Estimating $u(0,0)$ on $[-1,1]^2$ via \ref{['alg:wos']}. For clarity only the first 4 steps of each walk are shown.
  • Figure 1: Estimating $u(0,0)$ on $[-1,1]^2$ via \ref{['alg:wos']}. For clarity only the first 4 steps of each walk are shown.
  • Figure 1: Performance of \ref{['alg:wos', 'alg:eql', 'alg:var']} on \ref{['eq:exp1']} with 10 walks per point.
  • Figure 2: Average performance of original Walk on Spheres when solving \ref{['eq:exp1']} as the number of walks per point grows over 30 independent runs.
  • Figure 3: Average performance of original Walk on Spheres when solving \ref{['eq:exp1']} with 10 and 100 walks per point over 30 runs. Information reuse schemes use a fixed 25 element cache and run the same number of total walks as original Walk on Spheres.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 1: durrett2005probability
  • Corollary 2
  • Proof 1
  • Lemma 1
  • Proof 2
  • Lemma 1
  • Proof 3
  • Lemma 1
  • Proof 4
  • ...and 2 more