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Grid-Free Monte Carlo for Time-Dependent Diffusion — arXiv:2609.12306

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.12306
License (as archived): CC0-1.0
Archived: 2026-10-16T03:49:08.728Z
SHA-256 of the archived text: f1984e804718ddb210211d3f36ab0272178a6d4794811c93a780d56788637775
Stable URL of this page: https://metraipolis.com/documento/dc_052d2970bedb

How to cite

Grid-Free Monte Carlo for Time-Dependent Diffusion — arXiv:2609.12306 — arXiv (en), https://arxiv.org/abs/2609.12306 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_052d2970bedb (Metraipolis archive dc_052d2970bedb, sha256 f1984e804718ddb210211d3f36ab0272178a6d4794811c93a780d56788637775)

BibTeX

@misc{dc052d2970bedb,
  title = {Grid-Free Monte Carlo for Time-Dependent Diffusion — arXiv:2609.12306},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.12306},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_052d2970bedb; sha256 f1984e804718ddb210211d3f36ab0272178a6d4794811c93a780d56788637775},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.12306. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Many scientific applications require modeling how diffusive systems evolve over time, not merely their eventual steady states. While conventional steady-state analysis of partial differential equations (PDEs) on complex geometries is already hindered by costly volumetric meshing, transient analysis further requires sequential time stepping and careful step size selection. Grid-free Monte Carlo solvers such as walk on spheres (WoS) and walk on stars (WoSt) avoid this meshing bottleneck but remain largely limited to steady-state problems. We generalize WoS, for pure Dirichlet problems, and WoSt, for mixed Dirichlet--Neumann problems, to heat equations with initial conditions and time-dependent source and boundary data. We equip each random walk with a finite time budget and sample an exit time at every spatial step. If the exit time exceeds the remaining budget, the walk samples an interior point and evaluates the initial condition; otherwise, it continues with a reduced budget, accumulating source and boundary contributions. Our main technical contribution is a suite of kernel sampling and variance reduction techniques, including a low-bias, tabulation-free exit time sampler and efficient rejection samplers. Unlike grid-based transient solvers, our method directly estimates the solution at any requested time without volumetric meshing or sequential time marching. It also retains the parallel, progressive, and output-sensitive evaluation of WoS and WoSt while eliminating time step selection and temporal discretization bias entirely. Finally, we show how sharing walks enables efficient estimates at multiple target times.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-14 in math.NA).
Categorie arXiv: Graphics (cs.GR) ; Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2609.12306 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T03:16:25.007Z dalla pagina https://arxiv.org/catchup/math.NA/2026-09-14?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).