Grid-free Monte Carlo methods based on the \emph{walk on spheres (WoS)} algorithm solve fundamental partial differential equations (PDEs) like the Poisson equation without discretizing the problem domain, nor approximating functions in a finite basis. Such methods hence avoid aliasing in the solution, and evade the many challenges of mesh generation. Yet for problems with complex geometry, practical grid-free methods have been largely limited to basic Dirichlet boundary conditions. This paper introduces the \emph{walk on stars (WoSt)} method, which solves linear elliptic PDEs with arbitrary mixed Neumann and Dirichlet boundary conditions. The key insight is that one can efficiently simulate reflecting Brownian motion (which models Neumann conditions) by replacing the balls used by WoS with \emph{star-shaped} domains; we identify such domains by locating the closest visible point on the geometric silhouette. Overall, WoSt retains many attractive features of other grid-free Monte Carlo methods, such as progressive evaluation, trivial parallel implementation, and logarithmic scaling relative to geometric complexity.

翻译：基于“漫步球体（WoS）”算法的无网格蒙特卡洛方法可求解泊松方程等基本偏微分方程（PDEs），无需离散问题域或在有限基函数中近似函数。此类方法因此避免了求解过程中的混叠现象，并规避了网格生成面临的诸多挑战。然而，对于具有复杂几何结构的问题，实用化的无网格方法长期以来主要局限于基本的狄利克雷边界条件。本文提出“漫步繁星（WoSt）”方法，可求解具有任意混合诺伊曼与狄利克雷边界条件的线性椭圆型偏微分方程。核心创新在于：通过将WoS方法使用的球体替换为“星形”域，可高效模拟反射布朗运动（即诺伊曼条件的建模方式）；我们通过定位几何轮廓上的最近可见点来识别此类星形域。总体而言，WoSt方法保留了其他无网格蒙特卡洛方法的诸多优势特性，包括渐进式求解、简易并行实现以及随几何复杂度呈对数增长的计算规模。