Numerical analysis of a first-order computational algorithm for reaction-diffusion equations via the primal-dual hybrid gradient method
Abstract
In arXiv:2305.03945 [math.NA], a first-order optimization algorithm has been introduced to solve time-implicit schemes of reaction-diffusion equations. In this research, we conduct theoretical studies on this first-order algorithm equipped with a quadratic regularization term. We provide sufficient conditions under which the proposed algorithm and its time-continuous limit converge exponentially fast to a desired time-implicit numerical solution. We show both theoretically and numerically that the convergence rate is independent of the grid size, which makes our method suitable for large-scale problems. The efficiency of our algorithm has been verified via a series of numerical examples conducted on various types of reaction-diffusion equations. The choice of optimal hyperparameters as well as comparisons with some classical root-finding algorithms are also discussed in the numerical section.
Cite
@article{arxiv.2401.14602,
title = {Numerical analysis of a first-order computational algorithm for reaction-diffusion equations via the primal-dual hybrid gradient method},
author = {Shu Liu and Xinzhe Zuo and Stanley Osher and Wuchen Li},
journal= {arXiv preprint arXiv:2401.14602},
year = {2025}
}
Comments
Revised version, comments and suggestions are welcome