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.

翻译：在arXiv:2305.03945 [math.NA]中，提出了一种一阶优化算法用于求解反应扩散方程的时间隐式格式。本研究对该配备二次正则化项的一阶算法进行了理论分析。我们给出了该算法及其时间连续极限指数快速收敛到预期时间隐式数值解的充分条件。通过理论和数值两方面证明，该收敛速率与网格尺寸无关，这使得所提方法适用于大规模问题。通过对多种类型反应扩散方程进行的一系列数值实验，验证了该算法的有效性。数值部分还讨论了最优超参数的选择以及与若干经典求根算法的比较。