We introduce the walk-on-boundary (WoB) method for solving boundary value problems to computer graphics. WoB is a grid-free Monte Carlo solver for certain classes of second order partial differential equations. A similar Monte Carlo solver, the walk-on-spheres (WoS) method, has been recently popularized in computer graphics due to its advantages over traditional spatial discretization-based alternatives. We show that WoB's intrinsic properties yield further advantages beyond those of WoS. Unlike WoS, WoB naturally supports various boundary conditions (Dirichlet, Neumann, Robin, and mixed) for both interior and exterior domains. WoB builds upon boundary integral formulations, and it is mathematically more similar to light transport simulation in rendering than the random walk formulation of WoS. This similarity between WoB and rendering allows us to implement WoB on top of Monte Carlo ray tracing, and to incorporate advanced rendering techniques (e.g., bidirectional estimators with multiple importance sampling, the virtual point lights method, and Markov chain Monte Carlo) into WoB. WoB does not suffer from the intrinsic bias of WoS near the boundary and can estimate solutions precisely on the boundary. Our numerical results highlight the advantages of WoB over WoS as an attractive alternative to solve boundary value problems based on Monte Carlo.

翻译：我们引入了边界行走法（WoB）以解决计算机图形学中的边界值问题。WoB是一类针对二阶偏微分方程的无网格蒙特卡洛求解器。类似的蒙特卡洛求解器——球面行走法（WoS）——因其相较于传统空间离散化方法的优势，近年来在计算机图形学领域广受欢迎。我们证明，WoB的固有特性使其具备超越WoS的更多优势。与WoS不同，WoB自然支持内域和外域的各种边界条件（狄利克雷、诺伊曼、罗宾及混合边界）。WoB基于边界积分公式构建，其在数学上更类似于渲染中的光传输模拟，而非WoS的随机游走公式。WoB与渲染的相似性使我们能够在蒙特卡洛光线追踪基础上实现WoB，并将先进渲染技术（例如采用多重重要性采样的双向估计器、虚拟点光源方法及马尔可夫链蒙特卡洛）融入WoB中。WoB不存在WoS在边界附近的内在偏差，能够精确估计边界上的解。我们的数值结果凸显了WoB作为基于蒙特卡洛的边界值问题求解方案，相较于WoS的显著优势。