Parallel Newton--Krylov BDDC and FETI-DP Deluxe Solvers for Implicit Time discretizations of the Cardiac Bidomain Equations

Huynh, Ngoc Mai Monica; Pavarino, Luca F.; Scacchi, Simone

doi:10.1137/20m1353848

Cited by 9 publications

(1 citation statement)

References 32 publications

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“…Numerical methods for solving the EMI equations have been analyzed in [23, 24, 25, 26, 27]. Additionally, the properties and efficiencies of these solvers have been evaluated in [28, 29, 30, 31, 7, 23, 32], and special methods for mesh generation for EMI models have been addressed in [33].…”

Section: Introductionmentioning

confidence: 99%

Evaluating computational efforts and physiological resolution of mathematical models of cardiac tissue

Jæger,

Trotter,

Cai

et al. 2024

Preprint

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Computational techniques have significantly advanced our understanding of cardiac electrophysiology, yet they have predominantly concentrated on averaged models that do not represent the intricate dynamics near individual cardiomyocytes. Recently, accurate models representing individual cells have gained popularity, enabling analysis of the electrophysiology at the micrometer level. Here, we evaluate five mathematical models to determine their computational efficiency and physiological fidelity. Our findings reveal that cell-based models introduced in recent literature offer both efficiency and precision for simulating small tissue samples (comprising thousands of cardiomyocytes). Conversely, the traditional bidomain model and its simplified counterpart, the monodomain model, are more appropriate for larger tissue masses (encompassing millions to billions of cardiomyocytes). For simulations requiring detailed parameter variations along individual cell membranes, the EMI model emerges as the only viable choice. This model distinctively accounts for the extracellular (E), membrane (M), and intracellular (I) spaces, providing a comprehensive framework for detailed studies. Nonetheless, the EMI model's applicability to large-scale tissues is limited by its substantial computational demands for subcellular resolution.

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Section: Introductionmentioning

confidence: 99%

Evaluating computational efforts and physiological resolution of mathematical models of cardiac tissue

Jæger,

Trotter,

Cai

et al. 2024

Preprint

View full text Add to dashboard Cite

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Newton–Krylov-BDDC deluxe solvers for non-symmetric fully implicit time discretizations of the bidomain model

Huynh

2022

Numer. Math.

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A novel theoretical convergence rate estimate for a Balancing Domain Decomposition by Constraints algorithm is proven for the solution of the cardiac bidomain model, describing the propagation of the electric impulse in the cardiac tissue. The non-linear system arises from a fully implicit time discretization and a monolithic solution approach. The preconditioned non-symmetric operator is constructed from the linearized system arising within the Newton–Krylov approach for the solution of the non-linear problem; we theoretically analyze and prove a convergence rate bound for the Generalised Minimal Residual iterations’ residual. The theory is confirmed by extensive parallel numerical tests, widening the class of robust and efficient solvers for implicit time discretizations of the bidomain model.

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Scalable and Robust Dual-Primal Newton–Krylov Deluxe Solvers for Cardiac Electrophysiology with Biophysical Ionic Models

2022

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The focus of this work is to provide an extensive numerical study of the parallel efficiency and robustness of a staggered dual-primal Newton–Krylov deluxe solver for implicit time discretizations of the Bidomain model. This model describes the propagation of the electrical impulse in the cardiac tissue, by means of a system of parabolic reaction-diffusion partial differential equations. This system is coupled to a system of ordinary differential equations, modeling the ionic currents dynamics. A staggered approach is employed for the solution of a fully implicit time discretization of the problem, where the two systems are solved successively. The arising nonlinear algebraic system is solved with a Newton–Krylov approach, preconditioned by a dual-primal Domain Decomposition algorithm in order to improve convergence. The theoretical analysis and numerical validation of this strategy has been carried out in Huynh et al. (SIAM J. Sci. Comput. 44, B224–B249, 2022) considering only simple ionic models. This paper extends this study to include more complex biophysical ionic models, as well as the presence of ischemic regions, described mathematically by heterogeneous diffusion coefficients with possible discontinuities between subregions. The results of several numerical experiments show robustness and scalability of the proposed parallel solver.

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Parallel Newton--Krylov BDDC and FETI-DP Deluxe Solvers for Implicit Time discretizations of the Cardiac Bidomain Equations

Cited by 9 publications

References 32 publications

Evaluating computational efforts and physiological resolution of mathematical models of cardiac tissue

Evaluating computational efforts and physiological resolution of mathematical models of cardiac tissue

Newton–Krylov-BDDC deluxe solvers for non-symmetric fully implicit time discretizations of the bidomain model

Scalable and Robust Dual-Primal Newton–Krylov Deluxe Solvers for Cardiac Electrophysiology with Biophysical Ionic Models

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