Geometric Queries on Closed Implicit Surfaces for Walk on Stars

TL;DR

The paper addresses the lack of reliable geometric queries for WoSt on implicit domains by formulating all required queries (CPQ, ray intersection, CSPQ, and RRBQ) as constrained global optimization or constraint-satisfaction problems over closed implicit surfaces defined by zero level sets of $f_D$ and $f_R$. Leveraging interval analysis with branch-and-bound, the authors derive tight, dimension-aware bounds for Robin boundaries and present a mesh-free WoSt workflow that includes point sampling on reflecting boundaries, all implemented in Julia with IntervalArithmetic.jl. This work delivers the first complete WoSt pipeline for closed implicit domains, enabling direct PDE solving on implicit boundaries and preserving WoSt’s mesh-free advantages. Practical impact includes robust, mesh-free simulation capabilities for complex implicit geometries and potential extensions to neural SDFs and inverse problems in PDE contexts.

Abstract

Walk on stars (WoSt) is currently one of the most advanced Monte Carlo solvers for PDEs. Unfortunately, the lack of reliable geometric query approaches has hindered its applicability to boundaries defined by implicit surfaces. This work proposes a geometric query framework over closed implicit surfaces for WoSt, under the scope of walkin' Robin. Our key observation is that all WoSt queries can be formulated as constrained global optimization or constraint satisfaction problems. Based on our formulations, to solve the highly non-convex problems, we adopt a branch-and-bound approach based on interval analysis. To the best of our knowledge, our method is the first to study closest silhouette point queries and Robin radius bound queries on closed implicit surfaces. Our formulations and methods first enable mesh-free PDE solving via WoSt when boundaries are defined by closed implicit surfaces.

Figures (6)

• Figure 1: Walk on stars (WoSt) with boundaries defined by closed implicit surfaces. Our method can perform complete WoSt geometric queries on various closed implicit representations (here we demonstrate harmonic RBFs miller2024diffwos and SDF operations). WoSt based on our method yields correct solutions to the Laplace Dirichlet-Robin problems.
• Figure 2: Closest silhouette point query (CSPQ). Figures (a) and (b) illustrate the landscapes of $f_\text{R}$ and $\psi = \nabla f_\text{R}(z) \cdot (z-x)$, resp. Figure (c) visualizes the minimization process.
• Figure 3: Robin radius bound query (RRBQ). Figure (a) shows the landscape of $f_\text{R}$ and $\tilde{\mu}$ (\ref{['sec:rrbq_disc']}). Figure (b) shows the landscape of the objective, optimization process and result. In figure (c), we randomly sample points on $\partial\Omega_\text{R}$ inside the queried circle, and plot the distribution of $\rho_\mu$miller2024robin. As illustrated, $\rho_\mu \in [0, 1]$ when the radius equals to $R_\text{R}^*$, while increasing the radius slightly yields invalid ranges of $\rho_\mu$.
• Figure 4: Sanity check on the effect of increasing $\mu$ on the radius. We set $\tilde{\mu}(z)$ to a constant w.r.t.$z$, increase it from (a) to (c), and perform RRBQ. The results show the query converges to CPQ on $\partial\Omega_\text{R}$ as $\mu$ increases, which agrees with the results reported by miller2024robin.
• Figure 5: RRBQ in 3D. Figures (a) and (b) show the optimization process and results in $\mathbb{R}^3$ and on a slice, resp. Comparing (b) and (c), we see that regions where $r>\frac{\cos\theta}{\tilde{\mu}(z)}$ is not satisfied are set to $+\infty$ in \ref{['eq:new_3d_optimizer']}, then the optimizer skips these regions during the coarse stage. The sampled values (see \ref{['fig:robin_radius_query']} caption) of $\rho_\mu$ fall within $[0.4891, 0.99996]$.