Solving Inverse PDE Problems using Grid-Free Monte Carlo Estimators

TL;DR

This paper explores using grid-free Monte Carlo estimators for inverse PDE problems, aiming to reconstruct PDE inputs efficiently without domain meshing, inspired by techniques from physically-based rendering.

Contribution

It introduces a gradient-based approach combining Monte Carlo methods with inverse PDE solving, offering a potentially more robust and efficient alternative to traditional mesh-based solvers.

Findings

Unbiased estimators for inverse PDE problems demonstrated

Potential for linear time complexity in gradient computation

Preliminary results show promise for combining rendering techniques with PDE solving

Abstract

Modeling physical phenomena like heat transport and diffusion is crucially dependent on the numerical solution of partial differential equations (PDEs). A PDE solver finds the solution given coefficients and a boundary condition, whereas an inverse PDE solver goes the opposite way and reconstructs these inputs from an existing solution. In this article, we investigate techniques for solving inverse PDE problems using a gradient-based methodology. Conventional PDE solvers based on the finite element method require a domain meshing step that can be fragile and costly. Grid-free Monte Carlo methods instead stochastically sample paths using variations of the walk on spheres algorithm to construct an unbiased estimator of the solution. The uncanny similarity of these methods to physically-based rendering algorithms has been observed by several recent works. In the area of rendering, recent progress has led to the development of efficient unbiased derivative estimators. They solve an adjoint form of the problem and exploit arithmetic invertibility to compute gradients using a constant amount of memory and linear time complexity. Could these two lines of work be combined to compute cheap parametric derivatives of a grid-free PDE solver? We investigate this question and present preliminary results.