Grid-free Monte Carlo methods based on the walk on spheres (WoS) algorithm solve fundamental partial differential equations (PDEs) like the Poisson equation without discretizing the problem domain or 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. We introduce the walk on stars (WoSt) algorithm, 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 star-shaped domains. We identify such domains via the closest point on the visibility silhouette, by simply augmenting a standard bounding volume hierarchy with normal information. Overall, WoSt is an easy modification of WoS, and retains the many attractive features of grid-free Monte Carlo methods such as progressive and view-dependent evaluation, trivial parallelization, and sublinear scaling to increasing geometric detail.

翻译：基于行走球算法（WoS）的无网格蒙特卡洛方法无需离散问题域或通过有限基函数近似即可求解基本偏微分方程（如泊松方程），从而避免了求解过程中的混叠现象及网格生成的诸多挑战。然而对于具有复杂几何结构的问题，实际可用的无网格方法长期局限于基本狄利克雷边界条件。我们提出行走之星（WoSt）算法，可求解具有任意混合纽曼与狄利克雷边界条件的线性椭圆型偏微分方程。其核心创新在于：通过将WoS算法使用的球体替换为星形域，可高效模拟反射布朗运动（体现纽曼条件）。我们通过可见性轮廓上的最近点识别此类星形域，只需在标准包围体层次结构中附加法向信息即可实现。总体而言，WoSt是对WoS的简易扩展，保留了无网格蒙特卡洛方法的诸多优势特性，包括渐进式与视点相关评估、直观并行化能力，以及随几何细节增加呈亚线性扩展的特性。