Walking on Heat Stars for Parabolic Heat Equations with Neumann Boundary Conditions

2026-06-15

AI Summary

This work addresses the challenge of mesh-free numerical solution of time-dependent parabolic heat equations under Neumann boundary conditions by introducing the “Walk on Heat Stars” method. Built upon a non-cylindrical boundary integral framework and heat-ball geometric modeling, the approach establishes, for the first time, a unified exact sampling mechanism tailored to parabolic equations. It decomposes the double-layer heat kernel into a product of Gamma and uniform distributions, enabling joint exact sampling of spatial displacements and backward time steps. Key innovations include a non-recursive gradient estimator, a spatiotemporally heteroscedastic denoising strategy, and logarithmic time-coordinate parameterization. Numerical experiments demonstrate that the method achieves the theoretical Monte Carlo convergence rate across diverse geometries and frequencies, and successfully solves heat dissipation and cooling problems with both pure and mixed Neumann boundary conditions.

Abstract

Monte Carlo methods have proven highly effective for elliptic partial differential equations through algorithms such as Walk on Spheres and Walk on Stars, which evaluate solutions at individual points without volumetric meshing or global linear solves. Extending these methods to the transient regime has remained an open challenge: parabolic equations couple space and time through an anisotropic scaling, requiring joint sampling of spatial displacements and backward time steps whose distribution was not previously available in a unified, exact form. We present Walk on Heat Stars, a grid-free Monte Carlo solver that closes this gap by extending the boundary integral framework of Walk on Stars to the parabolic setting. Our method introduces a non-cylindrical boundary integral formulation that accommodates the time-varying domains induced by heat-ball sampling. The heat ball geometry is parameterized by a logarithmic time coordinate and a spatial direction, revealing that the double-layer kernel factorizes into independent Gamma and uniform components. This factorization enables exact directional importance sampling of the recursive next walk position, the Neumann flux contribution, and the volumetric source term. We further derive a decoupled gradient estimator that expresses spatial derivatives as weighted boundary integrals of the solution, requiring no recursion on the gradient, and adapt a heteroscedastic regression-based denoiser to the space-time domain for variance reduction. We validate our method on analytical solutions across a range of geometries and spatial frequencies, confirm convergence at the expected Monte Carlo rate, and demonstrate practical applicability on heat sink and cooling scenes with mixed or pure Neumann boundary conditions.

Problem

• parabolic heat equations
• Monte Carlo methods
• Neumann boundary conditions
• space-time coupling
• grid-free solvers

Innovation

• Walk on Heat Stars
• parabolic PDEs
• Monte Carlo methods
• boundary integral formulation
• importance sampling