https://rubatotree.github.io/blog/tags/monte-carlo-pde/Recent content in Monte Carlo PDE on Umbrella Coffeehttps://rubatotree.github.io/blog/images/og-default.pnghttps://rubatotree.github.io/blog/images/og-default.pngHugoen-usSun, 07 Jun 2026 00:00:00 +0800https://rubatotree.github.io/blog/posts/embodied-notes-2/Sun, 07 Jun 2026 00:00:00 +0800https://rubatotree.github.io/blog/posts/embodied-notes-2/<p class="typst-parbreak"></p> <div style="display: grid; place-items: start center;"> <figure> <div style="display: grid; place-items: start center;"><img src="https://rubatotree.github.io/blog/images/embodied-notes-2/teaser.png" alt loading="lazy"></div> <div style="display: grid; place-items: start center;"> <figcaption>图 1 玩音乐的最终归宿是转行具身智能。<a id="loc-1" href="#loc-3" role="doc-biblioref">[1]</a></figcaption> </div> </figure> </div> <p class="typst-parbreak"></p> <p>TODO</p> <div class="toc" style="display: none"><details><summary>Table of Contents</summary><div><nav role="doc-toc"><ol style="list-style-type: none"><li><p></p><ul><li><a href="#loc-2">References</a></li></ul><p></p></li></ol></nav></div></details></div> <section role="doc-bibliography"> <h2 id="loc-2">References</h2> <ul style="list-style-type: none"> <li id="loc-3"><span class="prefix"><a href="#loc-1" role="doc-backlink">[1]</a></span> 山下清悟, “超かぐや姫！.” [Online]. Available: <a href="https://www.netflix.com/title/81756595" target="_blank" rel="noopener noreferrer"><span style="color: #59a4ff;"><span style="text-decoration: underline">https://www.netflix.com/title/81756595</span></span></a></li> <li><span class="prefix">[2]</span> J. Park and H. Kang, “RenderMem: Rendering as Spatial Memory Retrieval.” [Online]. Available: <a href="https://arxiv.org/abs/2603.14669" target="_blank" rel="noopener noreferrer"><span style="color: #59a4ff;"><span style="text-decoration: underline">https://arxiv.org/abs/2603.14669</span></span></a></li> <li><span class="prefix">[3]</span> R. I. C. Muchacho and F. T. Pokorny, “Walk on Spheres for PDE-based Path Planning.” [Online]. Available: <a href="https://arxiv.org/abs/2406.01713" target="_blank" rel="noopener noreferrer"><span style="color: #59a4ff;"><span style="text-decoration: underline">https://arxiv.org/abs/2406.01713</span></span></a></li> <li><span class="prefix">[4]</span> C. Jambon, M. S. Nabizadeh, and M. Konaković Luković, “Walk on Decomposed Subdomains: A Hybrid Monte Carlo–Deterministic Solver for Elliptic PDEs,” <em>ACM Trans. Graph.</em>, vol. 45, no. 4, July 2026, doi: <a href="https://doi.org/10.1145/3811340" target="_blank" rel="noopener noreferrer"><span style="color: #59a4ff;"><span style="text-decoration: underline">10.1145/3811340</span></span></a>.</li> <li><span class="prefix">[5]</span> Heskey0, “具身智能 - 9 个方向讲透 2025-2026 灵巧手智能.” [Online]. Available: <a href="https://zhuanlan.zhihu.com/p/2046746760459171551" target="_blank" rel="noopener noreferrer"><span style="color: #59a4ff;"><span style="text-decoration: underline">https://zhuanlan.zhihu.com/p/2046746760459171551</span></span></a></li> </ul> </section>https://rubatotree.github.io/blog/posts/sig26-paper-notes-3/Fri, 29 May 2026 00:00:00 +0800https://rubatotree.github.io/blog/posts/sig26-paper-notes-3/<h2 id="loc-1">1. Monte Carlo PDE</h2> <h3 id="loc-2">Probe-based Walk on Spheres for Efficient Path Reusing [<a href="https://t7imal.github.io/projects/2026wop/" target="_blank" rel="noopener noreferrer"><span style="color: #59a4ff;"><span style="text-decoration: underline">Project</span></span></a>]</h3> <img src="https://rubatotree.github.io/blog/images/sig26-paper-notes/WoP-teaser.png" alt loading="lazy"> <p><em>我自己的文章。</em></p> <p>利用路径信息重用高效优化 Walk on Spheres 系列算法。</p> <p>考虑到 Walk on Spheres 的采样过程，每一步都要在球面上均匀采样一个点作为该点解值的一个无偏估计。由 Off-center 形式的平均值公式可以得到实际上在球内的偏心点按 Poisson Kernel 分布在球面上采样一个点得到的结果也是解值的无偏估计。因此 WoS 均匀采样得到的解值可以为球内每个点所重用，在算法上就是得到一条完整路径后将求解值 Splat 回路径上的每个球内。这个算法被我们称为 Naive Path Reuse，不清楚现在 sig 上那篇 Talking to Neighbors 有没有扩展成这样的形式。</p> <p class="typst-parbreak"></p> <div style="display: grid; place-items: start center;"> <figure> <div style="display: grid; place-items: start center;"><img src="https://rubatotree.github.io/blog/images/sig26-paper-notes/wop_path_reuse.png" alt loading="lazy"></div> <div style="display: grid; place-items: start center;"> <figcaption>图 1 我们的 Naive Path Reuse 算法</figcaption> </div> </figure> </div> <p class="typst-parbreak"></p> <p>然后考虑到这样 Splat 的开销过于大（球内每个点都需要查询并访存一次），正好 25 年 11 月又出来了 Harmonic Caching 的文章，发现后者正好可以将 Splat 的任务转化为求解几个 Fourier 系数的任务。因此和 HC 类似地在求解域预放置一些探针球，游走时取一个探针球做 Poisson Kernel 采样找到边界，若不存在探针球就 Fallback 到传统的 WoSt 算法，可以验证这个行为仍然是满足布朗运动的。找到 Dirichlet 边界后就将求解到的结果送回探针球中贡献 Fourier 系数。然后就非常快了。</p>