Differential Walk on Spheres

TL;DR

A Monte Carlo method for computing derivatives of the solution to a partial differential equation (PDE) with respect to problem parameters (such as domain geometry or boundary conditions) is introduced, supporting large topological changes.

Abstract

We introduce a Monte Carlo method for computing derivatives of the solution to a partial differential equation (PDE) with respect to problem parameters (such as domain geometry or boundary conditions). Derivatives can be evaluated at arbitrary points, without performing a global solve or constructing a volumetric grid or mesh. The method is hence well suited to inverse problems with complex geometry, such as PDE-constrained shape optimization. Like other walk on spheres (WoS) algorithms, our method is trivial to parallelize, and is agnostic to boundary representation (meshes, splines, implicit surfaces, etc.), supporting large topological changes. We focus in particular on screened Poisson equations, which model diverse problems from scientific and geometric computing. As in differentiable rendering, we jointly estimate derivatives with respect to all parameters -- hence, cost does not grow significantly with parameter count. In practice, even noisy derivative estimates exhibit fast, stable convergence for stochastic gradient-based optimization, as we show through examples from thermal design, shape from diffusion, and computer graphics.

Figures (13)

• Figure 1: Left: Walk on spheres (\ref{['alg:wos']}) recursively jumps to a random point on the largest sphere around the current walk location. The walk is terminated when it reaches the $\varepsilon$-shell, where the Dirichlet condition is evaluated at the closest point on the boundary. Right: Differential WoS (\ref{['alg:wos_deriv']}) also takes a walk to the $\varepsilon$-shell, but additionally launches a primary walk from an offset point close to the boundary to estimate the differential boundary condition.
• Figure 2: We can compute derivatives on a wide range of boundary representations, including those not directly handled by conventional solvers: implicit surfaces (left) and splines (center). For a solution to a Poisson equation $u\lparen*\rparen{x, \pi}$ (middle row), here we show the derivatives $\dot{u}\lparen*\rparen{x, \pi}$ (bottom row) with respect to positional parameters $\pi$ of the boundary $\partial \Omega\lparen*\rparen{\pi}$.
• Figure 3: Left: The WoS estimator for the normal derivative by Sawhney:2020:MCG in \ref{['eqn:dudn_mc_estimate']} is undefined for points on the boundary, $x\in\partial \Omega\lparen*\rparen{\pi}$, so BVC evaluate it at an offset point $x - \ell\cdot n$. Right: We instead use the backward-difference approximation of the normal derivative in \ref{['eqn:dudn_fd_estimate']} at the offset point.
• Figure 4: A Monte Carlo estimate of the product $\langle u\dot{u} \rangle$ requires estimating both the solution $u$ and derivative $\dot{u}$. A single sampled walk simultaneously provides estimates of $u$ and $\dot{u}$, but the product estimates $u \dot{u}$ are correlated and introduce bias. Rather than resort to uncorrelated estimates, which uses a single estimate from each walk, U-statistics shares complementary estimates across walks, which reduces variance without introducing bias.
• Figure 5: Finite-difference (FD) approximations of derivatives (middle-right) require only primary walks, but scale poorly with parameter count. Differential WoS (far-right) instead computes derivatives for all parameters $\pi$ with a single differential walk, leading to less noisy results at equal time.