Robust parallel nonlinear solvers for implicit time discretizations of the Bidomain equations

TL;DR

This paper analyzes and compares nonlinear solvers for the Bidomain equations in cardiac modeling, demonstrating the superior convergence and scalability of Quasi-Newton methods and highlighting the viability of first-order methods for GPU implementations.

Contribution

It provides a rigorous proof of global convergence for Quasi-Newton and conjugate-gradient methods applied to the Bidomain system, and compares their performance and scalability.

Findings

Quasi-Newton methods converge faster than Newton-Krylov methods.

Quasi-Newton methods are robust and scalable for parallel computing.

First-order methods are competitive, especially for GPU-based matrix-free implementations.

Abstract

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.