The propagation of the action potential in the heart chambers is accurately described by the Bidomain model, which is commonly accepted and used in the specialistic literature. However, its mathematical structure of a degenerate parabolic system entails high computational costs in the numerical solution of the associated linear system. Domain decomposition methods are a natural way to reduce computational costs, and Optimized Schwarz Methods have proven in the recent years their effectiveness in accelerating the convergence of such algorithms. The latter are based on interface matching conditions more efficient than the classical Dirichlet or Neumann ones. In this paper we analyze an Optimized Schwarz approach for the numerical solution of the Bidomain problem. We assess the convergence of the iterative method by means of Fourier analysis, and we investigate the parameter optimization in the interface conditions. Numerical results in 2D and 3D are given to show the effectiveness of the method.
@article{M2AN_2013__47_2_583_0,
author = {Gerardo-Giorda, Luca and Perego, Mauro},
title = {Optimized Schwarz Methods for the Bidomain system in electrocardiology},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {47},
year = {2013},
pages = {583-608},
doi = {10.1051/m2an/2012040},
mrnumber = {3021699},
zbl = {1274.92021},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2013__47_2_583_0}
}
Gerardo-Giorda, Luca; Perego, Mauro. Optimized Schwarz Methods for the Bidomain system in electrocardiology. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) pp. 583-608. doi : 10.1051/m2an/2012040. http://gdmltest.u-ga.fr/item/M2AN_2013__47_2_583_0/
[1] LifeV software. http://www.LifeV.org.
[2] Trilinos software. http://trilinos.sandia.gov.
[3] and , New non-overlapping domain decomposition methods for the time-harmonic Maxwell system. SIAM J. Sci. Comput. 28 (2006) 102-122. | MR 2219289 | Zbl 1106.78014
[4] and , On the finite element solution of the pure Neumann problem. SIAM Rev. 47 (2005) 50-66. | MR 2149101 | Zbl 1084.65111
[5] and , Domain decomposition algorithms, in Acta Numerica 1994. Cambridge University Press (1994) 61-143. | MR 1288096 | Zbl 0809.65112
[6] , and , Méthode de décomposition de domaine pour l'équation d'advection-diffusion. C. R. Acad. Sci. 313 (1991) 623-626. | MR 1133498 | Zbl 0736.76042
[7] and , Symmetrized method with optimized second-order conditions for the Helmholtz equation, in Domain decomposition methods (Boulder, CO, 1997). Amer. Math. Soc. 10 (1998) 400-407. | MR 1649636 | Zbl 0909.65105
[8] , , , , , , , , and , Models of cardiac tissue electrophysiology : Progress, challenges and open questions. Progr. Bioph. Molec. Biol. 104 (2011) 22-48.
[9] and , A guide to modelling cardiac electrical activity in anatomically detailed ventricles. Progr. Bioph. Molec. Biol. 96 (2008) 19-43.
[10] and , A parallel solver for reaction-diffusion systems in computational electrocardiology. Math. Models Methods Appl. Sci. 14 (2004) 883-911. | MR 2069498 | Zbl 1068.92024
[11] , and , Computational electrocardiology : mathematical and numerical modeling, in Complex Systems in Biomedicine - A. Quarteroni, edited by L. Formaggia and A. Veneziani. Springer, Milan (2006). | MR 2488001
[12] and , Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis, edited by A. Lorenzi and B. Ruf. Birkhauser (2002) 49-78. | MR 1944157 | Zbl 1036.35087
[13] , An analysis for a nonoverlapping domain decomposition iterative procedure. SIAM J. Sci. Comput. 18 (1997) 1517-1525. | MR 1465670 | Zbl 0892.65074
[14] and , An Optimized Schwarz Algorithm for the compressible Euler equations, in Domain Decomposition Methods in Science and Engineering XVI (Proceedings of the DD16 Conference). Springer-Verlag (2007) 173-180. | MR 2334101 | Zbl 1213.76124
[15] , and , Optimized Schwarz Methods for Maxwell's equations. SIAM J. Sci. Comput. 31 (2009) 2193-2213. | MR 2516149 | Zbl 1192.78044
[16] , Optimized Schwarz Methods with Robin conditions for the Advection-Diffusion Equation, in Domain Decomposition Methods in Science and Engineering XVI (Proceedings of the DD16 Conference). Springer-Verlag (2007) 181-188. | MR 2334102 | Zbl 1183.65030
[17] and , Absorbing boundary conditions for domain decomposition. Appl. Numer. Math. 27 (1998) 341-365. | MR 1644668 | Zbl 0935.65135
[18] , , and , Spectral domain decomposition methods for the solution of acoustic and elastic wave propagation. Geophys. 61 (1996) 1160-1174.
[19] , , and , 2d and 3d elastic wave propagation by pseudo-spectral domain decomposition method. J. Seismology 1 (1997) 237-251.
[20] , Optimized Schwarz methods. SIAM J. Numer. Anal. 44 (2006) 699-731. | MR 2218966 | Zbl 1117.65165
[21] and , Méthodes de relaxation d'ondes pour l'équation de la chaleur en dimension 1. C. R. Acad. Sci. Paris, Sér. I 336 (2003) 519-524. | MR 1975090 | Zbl 1028.65100
[22] , and , An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation. Int. J. Numer. Meth. Fluids 55 (2007) 163-175. | MR 2344706 | Zbl 1125.65114
[23] , and , Optimal Schwarz waveform relaxation for the one dimensional wave equation. SIAM J. Numer. Anal. 41 (2003) 1643-1681. | MR 2035001 | Zbl 1085.65077
[24] , and , Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24 (2002) 38-60. | MR 1924414 | Zbl 1021.65061
[25] , , and , A model adaptive strategy for computational electrocardiology. Domain Decomposition Methods in Science and Engineering XXI (Proceedings of the DD21 Conference). Springer-Verlag. To appear (2013).
[26] , , , and , A model-based block-triangular preconditioner for the Bidomain system in electrocardiology. J. Comput. Phys. 228 (2009) 3625-3639. | MR 2511070 | Zbl 1187.92053
[27] , and , Analysis and optimization of Robin-Robin partitioned procedures in Fluid-Structure Interaction problems. SIAM J. Numer. Anal. 48 (2010) 2091-2116. | MR 2740543 | Zbl pre05931231
[28] , and , Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology. ESAIM : M2AN 45 (2011) 309-334. | Numdam | MR 2804641 | Zbl 1274.92022
[29] , and , Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems. Appl. Math. Lett. 1 (1988) 299-302. | MR 963704 | Zbl 0656.65097
[30] , Simulating the electrical behavior of cardiac tissue using the Bidomain model. Crit. Rev. Biomed. Eng. 21 (1993) 1-77.
[31] , and , The optimized order 2 method : Application to convection-diffusion problems. Future Gener. Comp. Syst. 18 (2001) 17-30. | Zbl 1050.65124
[32] , , , and , Numerical solution of the bidomain equations. Phil. Trans. R. Soc. A. 367 (2009) 1931-1950. | MR 2512073 | Zbl 1185.65169
[33] , , , , and , Mathematical models and numerical methods for the forward problem in cardiac electrophysiology. Comput. Vis. Sci. 5 (2003) 215-239. | Zbl 1050.92017
[34] , On the Schwarz alternating method. III : a variant for nonoverlapping subdomains, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, held in Houston, Texas, edited by T.F. Chan, R. Glowinski, J. Périaux and O. Widlund, SIAM Philadelphia, PA (1990). | MR 1064345 | Zbl 0704.65090
[35] and , A model of the ventricular cardiac action potential : depolarization, repolarization and their interaction. Circ. Res. 68 (1991) 1501-1526.
[36] , and , An a posteriori error estimator for model adaptivity in electrocardiology. Comput. Methods Appl. Mech. Eng. 200 (2011) 2727-2737. | MR 2811911 | Zbl 1230.92026
[37] , and , A scalable Newton-Krylov-Schwarz method for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 3 (2009) 3861-3883. | MR 2556566 | Zbl 1205.65261
[38] and , Factorization of the convection-diffusion operator and the Schwarz algorithm. M3AS 5 (1995) 67-93. | MR 1314997 | Zbl 0826.65102
[39] and , Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31 (2008) 420-443. | MR 2460784 | Zbl 1185.65179
[40] 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 2831039 | Zbl 1233.65070
[41] and , Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process. J. Comput. Appl. Math. 145 (2002) 49-70. | MR 1914350 | Zbl 1006.65102
[42] and , Algebraic multigrid preconditioners for the Bidomain reaction-diffusion system. Appl. Numer. Math. 59 (2009) 3033-3050. | MR 2560833 | Zbl 1171.92017
[43] and , Non-symmetric Algebraic Multigrid Preconditioners for the Bidomain reaction-diffusion system, in Numerical Mathematics and Advanced Applications, ENUMATH 2009, Part 2 (2010) 729-736. | MR 2560833 | Zbl pre05896827
[44] and , An efficient generalization of the Rush-Larsen method for solving electro-physiology membrane equations. ETNA 35 (2009) 234-256. | MR 2582815 | Zbl 1185.92005
[45] , , and , A comparison of Monodomain and Bidomain Reaction-Diffusion models for Action Potential Propagation in the Human Heart. IEEE Trans. Biomed. Eng. 53 (2006) 2425-2435,.
[46] , and , Mathematical Modelling the Electrical Activity of the Heart. World Scientific, Singapore (2005). | MR 2174981 | Zbl 1120.92015
[47] and , Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999). | MR 1857663 | Zbl 0931.65118
[48] , Action potential propagation in a thick strand of cardiac muscle. Circ. Res. 68 (1991) 162-173.
[49] , . Springer, Berlin (2004). | Zbl 1051.92025
[50] , A hybrid multilevel Schwarz method for the Bidomain model. Comput. Methods Appl. Mech. Eng. 197 (2008) 4051-4061. | MR 2458128 | Zbl 1194.78048
[51] , and . Domain Decomposition : Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press (1996). | MR 1410757 | Zbl 0857.65126
[52] , 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 2142490 | Zbl 1063.92018
[53] , Overlapping Schwarz methods for Maxwell's equations in three dimensions. Numer. Math. 86 (2000) 733-752. | MR 1794350 | Zbl 0980.78010
[54] and , Domain Decomposition Methods - Algorithms and Theory. Springer Ser. Comput. Math. 34 (2004). | Zbl 1069.65138
[55] , Reaction-diffusion systems for the macroscopic Bidomain model of the cardiac electric field. Nonlinear Anal. Real World Appl. 10 (2009) 849-868. | MR 2474265 | Zbl 1167.35403
[56] , and , Computational techniques for solving the Bidomain equations in three dimensions. IEEE Trans. Biomed. Eng. 49 (2002) 1260-1269.
[57] E.J. Vigmond, R. Weber dos Santos, A.J. Prassl, M. Deo and G. Plank, Solvers for the caridac Bidomain equations. Prog. Biophys. Mol. Biol. 96 (2008) 3-18.
[58] R. Weber dos Santos, G. Planck, S. Bauer and E.J. Vigmond, Parallel multigrid preconditioner for the cardiac Bidomain model. IEEE Trans. Biomed. Eng. 51 (2004) 1960-1968.
[59] and , Some nonoverlapping domain decomposition methods. SIAM Review 40 (1998) 857-914. | MR 1659681 | Zbl 0913.65115
[60] , Iterative methods by space decomposition and subspace correction. SIAM Review 34 (1992) 581-613. | MR 1193013 | Zbl 0788.65037