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arXiv · 2401.14602

Numerical analysis of a first-order computational algorithm for reaction-diffusion equations via the primal-dual hybrid gradient method

Shu Liu·Xinzhe Zuo·Stanley Osher·Wuchen Li
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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.

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Shu Liu, Xinzhe Zuo, Stanley Osher, Wuchen Li. 2024-01-26. Numerical analysis of a first-order computational algorithm for reaction-diffusion equations via the primal-dual hybrid gradient method. https://arxiv.org/abs/2401.14602

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