Grid-free Monte Carlo methods such as \emph{walk on spheres} can be used to solve elliptic partial differential equations without mesh generation or global solves. However, such methods independently estimate the solution at every point, and hence do not take advantage of the high spatial regularity of solutions to elliptic problems. We propose a fast caching strategy which first estimates solution values and derivatives at randomly sampled points along the boundary of the domain (or a local region of interest). These cached values then provide cheap, output-sensitive evaluation of the solution (or its gradient) at interior points, via a boundary integral formulation. Unlike classic boundary integral methods, our caching scheme introduces zero statistical bias and does not require a dense global solve. Moreover we can handle imperfect geometry (e.g., with self-intersections) and detailed boundary/source terms without repairing or resampling the boundary representation. Overall, our scheme is similar in spirit to \emph{virtual point light} methods from photorealistic rendering: it suppresses the typical salt-and-pepper noise characteristic of independent Monte Carlo estimates, while still retaining the many advantages of Monte Carlo solvers: progressive evaluation, trivial parallelization, geometric robustness, \etc{}\ We validate our approach using test problems from visual and geometric computing.

翻译：无网格蒙特卡洛方法（如球面游走）可在无需网格生成或全局求解的情况下求解椭圆型偏微分方程。然而，此类方法独立估计每个点的解，因此未能利用椭圆问题解的高度空间正则性。我们提出一种快速缓存策略：首先沿域边界（或局部感兴趣区域）随机采样点处估计解值及其导数，随后通过边界积分公式为内部点提供廉价且输出敏感的解（或其梯度）评估。与传统边界积分方法不同，我们的缓存方案引入零统计偏差，且无需密集的全局求解。此外，我们可处理不完美几何（如自交情况）及精细边界/源项，而无须修复或重采样边界表示。总体而言，本方案在精神上与真实感渲染中的虚拟点光源方法类似：它抑制了独立蒙特卡洛估计特有的椒盐噪声特征，同时保留了蒙特卡洛求解器的诸多优势——渐进评估、易并行化、几何鲁棒性等。我们通过视觉与几何计算领域的测试问题验证了该方法。