In this work, we study the convergence and performance of nonlinear solvers for the Bidomain equations after decoupling the ordinary and partial differential equations of the cardiac system. Firstly, we provide a rigorous proof of the global convergence of Quasi-Newton methods, such as BFGS, and nonlinear Conjugate-Gradient methods, such as Fletcher--Reeves, for the Bidomain system, by analyzing an auxiliary variational problem under physically reasonable hypotheses. Secondly, we compare several nonlinear Bidomain solvers in terms of execution time, robustness with respect to the data and parallel scalability. Our findings indicate that Quasi-Newton methods are the best choice for nonlinear Bidomain systems, since they exhibit faster convergence rates compared to standard Newton-Krylov methods, while maintaining robustness and scalability. Furthermore, first-order methods also demonstrate competitiveness and serve as a viable alternative, particularly for matrix-free implementations that are well-suited for GPU computing.

翻译：本文研究了在解耦心脏系统常微分与偏微分方程后，Bidomain方程非线性求解器的收敛性与计算性能。首先，通过分析在物理合理假设下的辅助变分问题，我们严格证明了拟牛顿方法（如BFGS）和非线性共轭梯度方法（如Fletcher–Reeves）在Bidomain系统中的全局收敛性。其次，我们从执行时间、数据鲁棒性和并行可扩展性三个维度比较了多种非线性Bidomain求解器。结果表明，拟牛顿方法是Bidomain非线性系统的最优选择——其收敛速度优于标准Newton-Krylov方法，同时保持强鲁棒性和可扩展性。此外，一阶方法也展现出竞争力，可作为可行替代方案，尤其适用于面向GPU计算的矩阵自由实现模式。