标题： 星上行走的闭合隐式曲面几何查询 显示英文标题

标题： Geometric Queries on Closed Implicit Surfaces for Walk on Stars

摘要： 基于闭合隐式曲面的Walk on stars（WoSt）目前是求解偏微分方程（PDEs）最先进的蒙特卡洛求解器之一。不幸的是，缺乏可靠的几何查询方法阻碍了其在由隐式曲面定义的边界上的应用。本工作提出了在Walkin' Robin框架下针对闭合隐式曲面的几何查询框架。我们的关键观察是，所有WoSt查询都可以表述为约束全局优化或约束满足问题。基于我们的公式，为了解决高度非凸的问题，我们采用了一种基于区间分析的分支定界方法。据我们所知，我们的方法是首次研究闭合隐式曲面上最近轮廓点查询和Robin半径边界查询的方法。我们的公式和方法首次实现了当边界由闭合隐式曲面定义时，通过WoSt进行无网格的PDE求解。 显示英文摘要

摘要： Walk on stars (WoSt) is currently one of the most advanced Monte Carlo solvers for PDEs. Unfortunately, the lack of reliable geometric query approaches has hindered its applicability to boundaries defined by implicit surfaces. This work proposes a geometric query framework over closed implicit surfaces for WoSt, under the scope of walkin' Robin. Our key observation is that all WoSt queries can be formulated as constrained global optimization or constraint satisfaction problems. Based on our formulations, to solve the highly non-convex problems, we adopt a branch-and-bound approach based on interval analysis. To the best of our knowledge, our method is the first to study closest silhouette point queries and Robin radius bound queries on closed implicit surfaces. Our formulations and methods first enable mesh-free PDE solving via WoSt when boundaries are defined by closed implicit surfaces.