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arXiv · 2610.15573

Walk-on-Ambient-Patch: Solver-Adapted Isometric Embeddings for Projected Walk on Spheres

Yunfeng Liao·Xiucheng Li
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Abstract

Projected Walk on Spheres (PWoS) solves surface PDEs through ambient sampling and closest-point projection, but tightly folded embeddings can force short steps or ambiguous projections despite simple intrinsic geometry. We introduce Walk-on-Ambient-Patch (WoAP), which treats the embedding as a solver variable. Edge-isometry constraints and tangent-point repulsion separate intrinsically distant regions, optionally in higher codimension; patch states track residual projection branches without introducing PDE boundaries. We analyze the resulting geometry--sampling tradeoff: additional dimensions provide separation freedom but reduce uniform tangent progress, while strong tangent-biased proposals incur large importance-weight moments. Folded-surface experiments demonstrate reduced projection conflicts, a separate hard-constraint pilot verifies metric preservation, and an internal fixed-RMSE comparison favors higher dimensions at stricter accuracy targets. Together, these results motivate choosing the embedding and projection state jointly with the stochastic solver.

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Yunfeng Liao, Xiucheng Li. 2026-10-11. Walk-on-Ambient-Patch: Solver-Adapted Isometric Embeddings for Projected Walk on Spheres. https://arxiv.org/abs/2610.15573

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