Error attribution in gradient random walk methods for parabolic equations
Abstract
Gradient random walk (GRW) methods represent a solution's spatial derivative by weighted particles and recover the solution by cumulative summation. We examine their accuracy for the one-dimensional heat, scalar cubic reaction-diffusion, and Burgers' equations. Thirty-seed ensembles, paired reconstructions, and deterministic controls identify the sources of two apparent accuracy limits. For the heat equation, comparing cumulative bin sums at bin centers introduces a first-order alignment error, which the exact reflected-particle distribution predicts and bin-edge comparison removes. An identity for the expected squared error predicts the remaining floor and the particle count for a target root-mean-square accuracy. With unequal weights, a count chosen in advance gives an error within 1% of its prediction. Reaction-front position, speed, and shape errors decrease under refinement. For Burgers' equation, exact transformed boundary data reduce the deterministic Cole-Hopf error about 490-fold, while fixed-endpoint particle runs approach their deterministic reference.
Related
- Sign-switched antithetic coupling in gradient random-walk methods for convection-diffusion equations
- Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term
- Gradient self-improvement for mixed local and nonlocal parabolic equations
- Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations
- Asymptotically Hölder gradient regularity for degenerate or singular parabolic normalized $ p$-Laplace equations
BibTeX
@misc{abkin2026error,
title = {{Error attribution in gradient random walk methods for parabolic equations}},
author = {Stephen Abkin and Prabir Daripa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22592},
eprint = {2608.22592},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22592},
}
RIS
TY - UNPB
TI - Error attribution in gradient random walk methods for parabolic equations
AU - Stephen Abkin
AU - Prabir Daripa
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2608.22592
UR - https://arxiv.org/abs/2608.22592
AB - Gradient random walk (GRW) methods represent a solution's spatial derivative by weighted particles and recover the solution by cumulative summation. We examine their accuracy for the one-dimensional heat, scalar cubic reaction-diffusion, and Burgers' equations. Thirty-seed ensembles, paired reconstructions, and deterministic controls identify the sources of two apparent accuracy limits. For the heat equation, comparing cumulative bin sums at bin centers introduces a first-order alignment error, which the exact reflected-particle distribution predicts and bin-edge comparison removes. An identity for the expected squared error predicts the remaining floor and the particle count for a target root-mean-square accuracy. With unequal weights, a count chosen in advance gives an error within 1% of its prediction. Reaction-front position, speed, and shape errors decrease under refinement. For Burgers' equation, exact transformed boundary data reduce the deterministic Cole-Hopf error about 490-fold, while fixed-endpoint particle runs approach their deterministic reference.
ER -