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

Nicolás A. Barnafi, Ngoc Mai Monica Huynh, Luca F. Pavarino, Simone Scacchi

TL;DR

This work develops and analyzes robust parallel nonlinear solvers for the Bidomain equations in cardiac electrophysiology, casting the decoupled ODE-PDE time discretization into a variational potential $\Psi$ whose stationary points reproduce the discrete system. The authors prove global convergence for Quasi-Newton methods (notably BFGS) and nonlinear Conjugate Gradient (Fletcher--Reeves) under precise regularity and convexity assumptions, and they validate these results with extensive parallel numerical experiments. Across a range of scenarios, including ischemia and full activation-recovery, quasi-Newton methods consistently deliver superior convergence and scalability, while first-order methods offer advantages in matrix-free, GPU-enabled settings. The practical impact is a set of robust, scalable solvers that can outperform standard Newton-based approaches and potentially halve solve times relative to IMEX discretizations, enabling faster, more reliable cardiac simulations.

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.

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

TL;DR

This work develops and analyzes robust parallel nonlinear solvers for the Bidomain equations in cardiac electrophysiology, casting the decoupled ODE-PDE time discretization into a variational potential whose stationary points reproduce the discrete system. The authors prove global convergence for Quasi-Newton methods (notably BFGS) and nonlinear Conjugate Gradient (Fletcher--Reeves) under precise regularity and convexity assumptions, and they validate these results with extensive parallel numerical experiments. Across a range of scenarios, including ischemia and full activation-recovery, quasi-Newton methods consistently deliver superior convergence and scalability, while first-order methods offer advantages in matrix-free, GPU-enabled settings. The practical impact is a set of robust, scalable solvers that can outperform standard Newton-based approaches and potentially halve solve times relative to IMEX discretizations, enabling faster, more reliable cardiac simulations.

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.
Paper Structure (36 sections, 2 theorems, 36 equations, 12 figures, 7 tables, 2 algorithms)

This paper contains 36 sections, 2 theorems, 36 equations, 12 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. The Bidomain Equations for Cardiac Electrophysiology
  4. Time and space discretizations
  5. Finding a suitable time-discrete Bidomain potential
  6. Nonlinear Bidomain Solvers
  7. Quasi-Newton (QN) methods
  8. Nonlinear Conjugate Gradient
  9. Mesh-independence property
  10. Quasi-Newton methods
  11. Newton methods
  12. Convergence Analysis for the Bidomain Equations
  13. Convergence analysis of Quasi-Newton methods (BFGS algorithm)
  14. Property A1 (twice differentiability of the objective function)
  15. Property A2 (convexity of the level set)
  16. ...and 21 more sections

Key Result

Theorem 1

Consider the level set of the potential $\Psi$ defined in (eq:potential) $\mathcal{L}(\hat{\vectorsym u}) \coloneqq \{\vectorsym u=(u_i, u_e) \in V \times \widetilde{V}: \Psi(\vectorsym u)\leq \Psi(\hat{\vectorsym u})\}$ together with an initial guess $\vectorsym u_0 \in V\times \widetilde{V}$ and t If all the above properties hold, the BFGS algorithm converges superlinearly.

Figures (12)

  • Figure 1: Idealized left ventricle geometry.
  • Figure 2: Solver tuning, time evolution. Fixed number of processors $N_p=16$ and fixed mesh of $64\times64\times64$ elements (550k DoFs). Ten Tusscher--Panfilov ionic model. Time evolution of global CPU times (in seconds).
  • Figure 3: Robustness of (a) Newton-MG and (b) inexact-Newton solvers with respect to the problem size. Fixed number of processors $N_p=16$ and increasing number of degrees of freedom from 72k to 2M. Nonlinear iterations over the time interval $[0,1]$ ms.
  • Figure 4: Robustness of (a) QN preonly and (b) QN jac-low solvers with respect to the problem size. Fixed number of processors $N_p=16$ and increasing number of degrees of freedom from 72k to 2M. Nonlinear iterations over the time interval $[0,1]$ ms.
  • Figure 5: Robustness of NGMRES (left) NCG-PRP (right) solvers with respect to the problem size. Fixed number of processors $N_p=16$ and increasing number of degrees of freedom from 72k to 2M. Nonlinear iterations over the time interval $[0,1]$ ms.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1: BFGS convergence
  • Theorem 2: NCG convergence
  • Remark 2
  • Remark 3