{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15573",
  "author_total": 2,
  "authors": [
    {
      "name": "Yunfeng Liao"
    },
    {
      "name": "Xiucheng Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5m1e2d6e90fs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15573",
  "fulltext_url": null,
  "id": "c5m1e2d6e90fs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15573",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.GR"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1704|Computer Graphics and Visualization Techniques"
    ]
  ],
  "openalex": "W7221774336",
  "pdf_url": "https://arxiv.org/pdf/2610.15573",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5m1e2d6e90fs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15573",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15573"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Walk-on-Ambient-Patch: Solver-Adapted Isometric Embeddings for Projected Walk on Spheres",
  "title_text": "Walk-on-Ambient-Patch: Solver-Adapted Isometric Embeddings for Projected Walk on Spheres",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/c5m1e2d6e90fs",
  "venue": "arXiv",
  "versions": []
}