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，研究者提出了一种一阶优化算法来求解反应-扩散方程的时间隐式格式。在本研究中，我们对这一配备二次正则化项的一阶算法进行了理论分析。我们给出了充分条件，证明所提算法及其时间连续极限能够以指数级速度收敛到期望的时间隐式数值解。我们从理论和数值上均证明该收敛速率与网格尺寸无关，这使得我们的方法适用于大规模问题。通过对各类反应-扩散方程进行的一系列数值实验，我们验证了算法的有效性。数值实验部分还讨论了最优超参数的选择，并与若干经典求根算法进行了比较。