一种求解边界值问题的实用边界行走（WoB）方法
A Practical Walk-on-Boundary Method for Boundary Value Problems
AI总结：
本文向计算机图形学引入了边界行走（WoB）方法，这是一种无网格Monte Carlo求解器，支持多种边界条件，并利用与光传输模拟的相似性结合高级渲染技术，解决了WoS在边界附近的固有偏差问题。
摘要：
我们向计算机图形学引入了用于求解边界值问题的边界行走（walk-on-boundary, WoB）方法。WoB是一种无网格的Monte Carlo求解器，适用于特定类别的二阶偏微分方程。类似的Monte Carlo求解器——球面行走（walk-on-spheres, WoS）方法，近期因其在传统基于空间离散化的替代方案上的优势，而在计算机图形学领域得到推广。我们表明，WoB的内在特性可带来超越WoS的进一步优势。与WoS不同，WoB自然支持内部和外部域的各种边界条件（Dirichlet、Neumann、Robin及混合条件）。WoB建立在边界积分公式之上，在数学上，它比WoS的随机游走公式更类似于渲染中的光传输模拟。WoB与渲染之间的这种相似性使我们能够在Monte Carlo光线追踪之上实现WoB，并将高级渲染技术（例如，带有多重重要性采样的双向估计器、虚拟点光源方法以及Markov链Monte Carlo）纳入WoB。WoB不存在WoS在边界附近固有的偏差，并且能够在边界上精确估计解。我们的数值结果突显了WoB相对于WoS的优势，使其成为基于Monte Carlo求解边界值问题的一种极具吸引力的替代方案。
英文摘要：
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.
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