Walk on Multiscale Media: Compiling Editable and Composable Transport Medium Prefabs for Monte Carlo PDE Solvers
Abstract
Monte Carlo PDE solvers enable mesh-free pointwise queries, but repeatedly traversing heterogeneous media is costly, particularly in scenes with reusable or dynamically changing materials. We introduce Walk on Multiscale Media (WoMM), which compiles fine-scale transport into reusable, editable and composable transport prefabs. A probabilistic interpretation of multiscale finite-element bases and condensed operators yields coarse transitions retaining fine-scale material effects. Each prefab combines port-level operators, entrance-dependent exit distributions, and Poisson-marked trajectory records, enabling incremental updates for overlapping absorption fields, gradient-field steering through path reweighting and transient diffusion. Integrated with conventional Walk-on-Spheres in a hybrid continuous--discrete solver, these prefabs accelerate repeated pointwise PDE queries and support interactive material edits without repeatedly solving the fine-scale problem.
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BibTeX
@misc{liao2026walk,
title = {{Walk on Multiscale Media: Compiling Editable and Composable Transport Medium Prefabs for Monte Carlo PDE Solvers}},
author = {Yunfeng Liao and Xiucheng Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35952},
eprint = {2610.35952},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35952},
}
RIS
TY - UNPB
TI - Walk on Multiscale Media: Compiling Editable and Composable Transport Medium Prefabs for Monte Carlo PDE Solvers
AU - Yunfeng Liao
AU - Xiucheng Li
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.35952
UR - https://arxiv.org/abs/2610.35952
AB - Monte Carlo PDE solvers enable mesh-free pointwise queries, but repeatedly traversing heterogeneous media is costly, particularly in scenes with reusable or dynamically changing materials. We introduce Walk on Multiscale Media (WoMM), which compiles fine-scale transport into reusable, editable and composable transport prefabs. A probabilistic interpretation of multiscale finite-element bases and condensed operators yields coarse transitions retaining fine-scale material effects. Each prefab combines port-level operators, entrance-dependent exit distributions, and Poisson-marked trajectory records, enabling incremental updates for overlapping absorption fields, gradient-field steering through path reweighting and transient diffusion. Integrated with conventional Walk-on-Spheres in a hybrid continuous--discrete solver, these prefabs accelerate repeated pointwise PDE queries and support interactive material edits without repeatedly solving the fine-scale problem.
ER -