Quasi-Monte Carlo based meshfree methods for parabolic equations on parameterized Riemannian manifolds
Abstract
We develop and analyze a randomized path sampling framework for the pointwise approximation of linear parabolic equations on parameterized Riemannian manifolds, treating both closed manifolds and manifolds with Dirichlet boundary. Based on the Feynman-Kac representation, we construct a parameter-coordinate retraction-based discretization, referred to as the p-Ret scheme, for the manifold diffusion generated by the uniformly elliptic operator, and use it to formulate the Monte Carlo (MC) and randomized quasi-Monte Carlo (RQMC) methods. We derive the local weak consistency of the p-Ret transition and first-order weak convergence of the temporal discretization on closed manifolds. For fixed h, we show that RQMC sampling yields asymptotically faster variance decay than standard MC. On closed manifolds, the variance is o(n−1) and, under additional regularity, satisfies the sharper bound O(n−2+ε) for any ε>0. For Dirichlet problems, despite the discontinuities induced by numerical first exit, nested-uniformly scrambled Sobol' sampling attains O(n−1−1/s+ε) variance for any ε>0 under suitable regularity assumptions, where s denotes the nominal path-space dimension. More notably, we further develop an Array-RQMC strategy based on stepwise low-dimensional randomized point sets and a Hilbert space-filling curve that maps high-dimensional particle states to a one-dimensional ordering, and establish a variance rate of o(n−1) under suitable assumptions. Numerical results demonstrate that both RQMC methods reduce variance and root mean square error (RMSE) relative to standard MC and remain effective for manifolds with intrinsic dimension up to $50$.
Related
- The Parabolic Harnack Inequality on Weighted Riemannian Manifolds
- On Polynomial Complex Hessian Equations I: Non-Linear Non-Spectral Elliptic Hessian Equations on Riemannian Manifolds
- Differential Harnack Estimates and Frequency Monotonicity for Filtration Equations on Riemannian Manifolds
- Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds
- Interior gradient estimates for mean curvature equations on Riemannian manifolds
BibTeX
@misc{he2026quasimonte,
title = {{Quasi-Monte Carlo based meshfree methods for parabolic equations on parameterized Riemannian manifolds}},
author = {Zhijian He and Zhiyuan Hui and Changtao Sheng and Bihao Su and Tao Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23341},
eprint = {2610.23341},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23341},
}
RIS
TY - UNPB
TI - Quasi-Monte Carlo based meshfree methods for parabolic equations on parameterized Riemannian manifolds
AU - Zhijian He
AU - Zhiyuan Hui
AU - Changtao Sheng
AU - Bihao Su
AU - Tao Zhou
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23341
UR - https://arxiv.org/abs/2610.23341
AB - We develop and analyze a randomized path sampling framework for the pointwise approximation of linear parabolic equations on parameterized Riemannian manifolds, treating both closed manifolds and manifolds with Dirichlet boundary. Based on the Feynman-Kac representation, we construct a parameter-coordinate retraction-based discretization, referred to as the p-Ret scheme, for the manifold diffusion generated by the uniformly elliptic operator, and use it to formulate the Monte Carlo (MC) and randomized quasi-Monte Carlo (RQMC) methods. We derive the local weak consistency of the p-Ret transition and first-order weak convergence of the temporal discretization on closed manifolds. For fixed h, we show that RQMC sampling yields asymptotically faster variance decay than standard MC. On closed manifolds, the variance is o(n^−1) and, under additional regularity, satisfies the sharper bound O(n^−2+ε) for any ε>0. For Dirichlet problems, despite the discontinuities induced by numerical first exit, nested-uniformly scrambled Sobol' sampling attains O(n^−1−1/s+ε) variance for any ε>0 under suitable regularity assumptions, where s denotes the nominal path-space dimension. More notably, we further develop an Array-RQMC strategy based on stepwise low-dimensional randomized point sets and a Hilbert space-filling curve that maps high-dimensional particle states to a one-dimensional ordering, and establish a variance rate of o(n^−1) under suitable assumptions. Numerical results demonstrate that both RQMC methods reduce variance and root mean square error (RMSE) relative to standard MC and remain effective for manifolds with intrinsic dimension up to $50$.
ER -