The aim of this work is to compare a new uncoupled solver for the cardiac Bidomain model with a usual coupled solver. The Bidomain model describes the bioelectric activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction-diffusion partial differential equation (PDE) and an elliptic linear PDE. This system models at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations (ODEs), the so-called membrane model, describing the ionic currents through the cellular membrane. A novel uncoupled solver for the Bidomain system is here introduced, based on solving twice the parabolic PDE and once the elliptic PDE at each time step, and it is compared with a usual coupled solver. Three-dimensional numerical tests have been performed in order to show that the proposed uncoupled method has the same accuracy of the coupled strategy. Parallel numerical tests on structured meshes have also shown that the uncoupled technique is as scalable as the coupled one. Moreover, the conjugate gradient method preconditioned by Multilevel Hybrid Schwarz preconditioners converges faster for the linear systems deriving from the uncoupled method than from the coupled one. Finally, in all parallel numerical tests considered, the uncoupled technique proposed is always about two or three times faster than the coupled approach.
Mots clés : operator splitting, multilevel preconditioners, parallel computing
@article{M2AN_2013__47_4_1017_0,
author = {Colli Franzone, P. and Pavarino, L. F. and Scacchi, S.},
title = {A comparison of coupled and uncoupled solvers for the cardiac {Bidomain} model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1017--1035},
publisher = {EDP-Sciences},
volume = {47},
number = {4},
year = {2013},
doi = {10.1051/m2an/2012055},
mrnumber = {3082287},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2012055/}
}
TY - JOUR AU - Colli Franzone, P. AU - Pavarino, L. F. AU - Scacchi, S. TI - A comparison of coupled and uncoupled solvers for the cardiac Bidomain model JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1017 EP - 1035 VL - 47 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012055/ DO - 10.1051/m2an/2012055 LA - en ID - M2AN_2013__47_4_1017_0 ER -
%0 Journal Article %A Colli Franzone, P. %A Pavarino, L. F. %A Scacchi, S. %T A comparison of coupled and uncoupled solvers for the cardiac Bidomain model %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1017-1035 %V 47 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012055/ %R 10.1051/m2an/2012055 %G en %F M2AN_2013__47_4_1017_0
Colli Franzone, P.; Pavarino, L. F.; Scacchi, S. A comparison of coupled and uncoupled solvers for the cardiac Bidomain model. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 4, pp. 1017-1035. doi : 10.1051/m2an/2012055. http://www.numdam.org/articles/10.1051/m2an/2012055/
[1] , and , Solving the cardiac Bidomain equations for discontinuous conductivities. IEEE Trans. Biomed. Eng. 53 (2006) 1265-1272.
[2] , , , , , , and , PETSc Users Manual.Tech. Rep. ANL-95/11 - Revision 2.1.5, Argonne National Laboratory (2002).
[3] , , , , , , and , PETSc home page. http://www.mcs.anl.gov/petsc (2001).
[4] , , , and , Mathematical modeling of electrocardiograms: a numerical study. Ann. Biomed. Eng. 38 (2010) 1071-1097.
[5] , , , , , , , , and , Models of cardiac tissue electrophysiology: Progress, challenges and open questions. Progr. Biophys. Molec. Biol. 104 (2011) 22-48.
[6] and , A parallel solver for reaction-diffusion systems in computational electrocardiology. Math. Mod. Meth. Appl. Sci. 14 (2004) 883-911. | MR | Zbl
[7] , and , Mathematical and numerical methods for reaction-diffusion models in electrocardiology, in Modeling of Physiological flows, edited by D. Ambrosi, A. Quarteroni and G. Rozza. Springer (2011) 107-142.
[8] , and , Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models. Math. Biosci. 197 (2005) 33-66. | MR | Zbl
[9] , , , and , Adaptivity in space and time for reaction-diffusion systems in Electrocardiology. SIAM J. Sci. Comput. 28 (2006) 942-962. | MR | Zbl
[10] , , and , Adaptive finite element simulation of ventricular fibrillation dynamics. Comput. Visual. Sci. 12 (2009) 201-205. | MR
[11] , and , Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72 (1996) 313-348. | MR | Zbl
[12] and , Multilevel additive methods for elliptic finite element problems. Parallel algorithms for partial differential equations (Kiel 1990) Notes Numer. Fluid Mech. 31 (1991) 58-69. | MR | Zbl
[13] and , Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15 (1994) 604-620. | MR | Zbl
[14] and , Semi-implicit time-discretization schemes for the Bidomain model. SIAM J. Numer. Anal. 46 (2008) 2443-2468. | MR | Zbl
[15] and , Decoupled time-marching schemes in computational cardiac electrophysiology and ECG numerical simulation. Math. Biosci. 226 (2010) 58-75. | MR | Zbl
[16] , , , , , , , , , , and , Cardiac cell modelling: observations from the heart of the cardiac physiome project. Prog. Biophys. Mol. Biol. 104 (2011) 2-21.
[17] , , , and , A model-based block-triangular preconditioner for the Bidomain system in electrocardiology. J. Comput. Phys. 228 (2009) 3625-3639. | MR | Zbl
[18] , and , Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology. Math. Model. Numer. Anal. 45 (2011) 309-334. | Numdam | MR | Zbl
[19] , , , , and , Laminar structure of the heart: ventricular myocyte arrangement and connective tissue architecture in the dog. Amer. J. Physiol. Heart Circ. Physiol. 269 (1995) H571-H582.
[20] , , , and , Numerical solution of the bidomain equations. Philos. Trans. R. Soc. A 367 (2009) 1931-1950. | MR | Zbl
[21] and , A model of the ventricular cardiac action potential: depolarization, repolarization, and their interaction. Circ. Res. 68 (1991) 1501-1526.
[22] , , and , An order optimal solver for the discretized bidomain equations. Numer. Linear Algebra Appl. 14 (2007) 83-98. | MR | Zbl
[23] and , MeTis: Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 4.0. http://www.cs.umn.edu/~metis/. University of Minnesota, Minneapolis, MN (2009).
[24] and , Decoupled Schwarz algorithms for implicit discretization of nonlinear Monodomain and Bidomain systems. Math. Mod. Meth. Appl. Sci. 19 (2009) 1065-1097. | MR | Zbl
[25] , and . A scalable Newton-Krylov-Schwarz method for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31 (2009) 3861-3883. | MR | Zbl
[26] and , A fully implicit parallel algorithm for simulating the non-linear electrical activity of the heart. Numer. Linear Algebra Appl. 11 (2004) 261-277. | MR | Zbl
[27] and , Homogenization of syncytial tissues. Crit. Rev. Biomed. Eng. 21 (1993) 137-199.
[28] , , , , , , and , A numerical guide to the solution of the bidomain equations of cardiac electrophysiology. Progr. Biophys. Molec. Biol. 102 (2010) 136-155.
[29] and , Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31 (2008) 420-443. | MR | Zbl
[30] and , Parallel Multilevel Schwarz and Block Preconditioners for the Bidomain Parabolic-Parabolic and Parabolic-Elliptic Formulations. SIAM J. Sci. Comput. 33 (2011) 1897-1919. | MR | Zbl
[31] , and . Multiscale modeling for the bioelectric activity of the heart. SIAM J. Math. Anal. 37 (2006) 1333-1370. | MR | Zbl
[32] and , Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process. J. Comput. Appl. Math. 145 (2002) 49-70. | MR | Zbl
[33] and , Algebraic multigrid preconditioners for the bidomain reaction-diffusion system. Appl. Numer. Math. 59 (2009) 3033-3050. | MR | Zbl
[34] and , Fast structured AMG preconditioning for the bidomain model in electrocardiology. SIAM J. Sci. Comput. 33 (2011) 721-745. | MR | Zbl
[35] G. Plank, M. Liebmann, R. Weber dos Santos, E.J. Vigmond and G. Haase, Algebraic Multigrid Preconditioner for the Cardiac Bidomain Model. IEEE Trans. Biomed. Eng. 54 (2007) 585-596.
[36] , , , and , A comparison of Monodomain and Bidomain reaction-diffusion models for action potential propagation in the human heart. IEEE Trans. Biomed. Eng. 53 (2006) 2425-2434.
[37] , The use of numerical integration in finite element methods for solving parabolic equations. In Topics in Numerical Analysis, edited by J.J.H. Miller. Academic Press (1973) 233-264. | MR | Zbl
[38] and , An advanced algorithm for solving partial differential equation in cardiac conduction. IEEE Trans. Biomed. Eng. 46 (1999) 1166-1168.
[39] and , Numerical Approximation of Partial Differential Equations. Springer (1997). | MR | Zbl
[40] , A hybrid multilevel Schwarz method for the bidomain model. Comput. Methods Appl. Mech. Eng. 197 (2008) 4051-4061. | MR | Zbl
[41] , A multilevel hybrid Newton-Krylov-Schwarz method for the Bidomain model of electrocardiology. Comput. Methods Appl. Mech. Eng. 200 (2011) 717-725. | MR | Zbl
[42] , , and , Computing cardiac recovery maps from electrograms and monophasic action potentials under heterogeneous and ischemic conditions. Math. Mod. Methods Appl. Sci. 20 (2010) 1089-1127. | MR | Zbl
[43] , and , A numerically efficient model for the simulation of defibrillation in an active bidomain sheet of myocardium. Math. Biosci. 166 (2000) 85-100. | Zbl
[44] , and , Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press (1996). | MR | Zbl
[45] , , and , Solving the coupled system improves computational efficiency of the Bidomain equations. IEEE Trans. Biomed. Eng. 56 (2009) 2404-2412.
[46] , , and , Multigrid block preconditioning for a coupled system of partial differential equations modeling the electrical activity in the heart. Comput. Methods Biomech. Biomed. Eng. 5 (2002) 397-409.
[47] , and , An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso. Math. Biosci. 194 (2005) 233-248. | MR | Zbl
[48] H. Si, http://tetgen.berlios.de/. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany.
[49] , Galerkin Finite Element Methods for Parabolic Problems. Springer (1997). | MR | Zbl
[50] and , Domain Decomposition Methods: Algorithms and Theory. Comput. Math. Springer-Verlag, Berlin 34 (2004). | Zbl
[51] and , Operator splitting and adaptive mesh refinement for the Luo-Rudy I model. J. Comput. Phys. 196 (2004) 645-679. | MR | Zbl
[52] , and , Computational techniques for solving the bidomain equations in three dimensions. IEEE Trans. Biomed. Eng. 49 (2002) 1260-1269.
[53] E.J. Vigmond, R. Weber dos Santos, A.J. Prassl, M. Deo and G. Plank, Solvers for the cardiac bidomain equations. Progr. Biophys. Molec. Biol. 96 (2008) 3-18.
[54] R. Weber dos Santos, G. Plank, S. Bauer and E.J. Vigmond, Parallel multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 51 (2004) 1960-1968.
[55] , An efficient numerical technique for the solution of the monodomain and bidomain equations. IEEE Trans. Biomed. Eng. 53 (2006) 2139-2147.
[56] , 3D current-voltage-time surfaces unveil critical repolarization differences underlying similar cardiac action potentials: A model study. Math. Biosci. 233 (2011) 98-110. | MR | Zbl
[57] , Multilevel Schwarz methods. Numer. Math. 63 (1992) 521-539. | MR | Zbl
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