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.
Keywords: domain decomposition, optimized schwarz methods, computational electrocardiology
@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 }, pages = {583--608}, publisher = {EDP-Sciences}, volume = {47}, number = {2}, year = {2013}, doi = {10.1051/m2an/2012040}, mrnumber = {3021699}, zbl = {1274.92021}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2012040/} }
TY - JOUR AU - Gerardo-Giorda, Luca AU - Perego, Mauro TI - Optimized Schwarz Methods for the Bidomain system in electrocardiology JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 583 EP - 608 VL - 47 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012040/ DO - 10.1051/m2an/2012040 LA - en ID - M2AN_2013__47_2_583_0 ER -
%0 Journal Article %A Gerardo-Giorda, Luca %A Perego, Mauro %T Optimized Schwarz Methods for the Bidomain system in electrocardiology %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 583-608 %V 47 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012040/ %R 10.1051/m2an/2012040 %G en %F 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 , Volume 47 (2013) no. 2, pp. 583-608. doi : 10.1051/m2an/2012040. http://www.numdam.org/articles/10.1051/m2an/2012040/
[1] LifeV software. http://www.LifeV.org.
[2] Trilinos software. http://trilinos.sandia.gov.
[3] New non-overlapping domain decomposition methods for the time-harmonic Maxwell system. SIAM J. Sci. Comput. 28 (2006) 102-122. | MR | Zbl
and ,[4] On the finite element solution of the pure Neumann problem. SIAM Rev. 47 (2005) 50-66. | MR | Zbl
and ,[5] Domain decomposition algorithms, in Acta Numerica 1994. Cambridge University Press (1994) 61-143. | MR | Zbl
and ,[6] Méthode de décomposition de domaine pour l'équation d'advection-diffusion. C. R. Acad. Sci. 313 (1991) 623-626. | MR | Zbl
, and ,[7] 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 | Zbl
and ,[8] Models of cardiac tissue electrophysiology : Progress, challenges and open questions. Progr. Bioph. Molec. Biol. 104 (2011) 22-48.
, , , , , , , , and ,[9] A guide to modelling cardiac electrical activity in anatomically detailed ventricles. Progr. Bioph. Molec. Biol. 96 (2008) 19-43.
and ,[10] A parallel solver for reaction-diffusion systems in computational electrocardiology. Math. Models Methods Appl. Sci. 14 (2004) 883-911. | MR | Zbl
and ,[11] Computational electrocardiology : mathematical and numerical modeling, in Complex Systems in Biomedicine - A. Quarteroni, edited by L. Formaggia and A. Veneziani. Springer, Milan (2006). | MR
, and ,[12] 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 | Zbl
and ,[13] An analysis for a nonoverlapping domain decomposition iterative procedure. SIAM J. Sci. Comput. 18 (1997) 1517-1525. | MR | Zbl
,[14] 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 | Zbl
and ,[15] Optimized Schwarz Methods for Maxwell's equations. SIAM J. Sci. Comput. 31 (2009) 2193-2213. | MR | Zbl
, and ,[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 | Zbl
,[17] Absorbing boundary conditions for domain decomposition. Appl. Numer. Math. 27 (1998) 341-365. | MR | Zbl
and ,[18] Spectral domain decomposition methods for the solution of acoustic and elastic wave propagation. Geophys. 61 (1996) 1160-1174.
, , and ,[19] 2d and 3d elastic wave propagation by pseudo-spectral domain decomposition method. J. Seismology 1 (1997) 237-251.
, , and ,[20] Optimized Schwarz methods. SIAM J. Numer. Anal. 44 (2006) 699-731. | MR | Zbl
,[21] 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 | Zbl
and ,[22] An optimized Schwarz method with two-sided Robin transmission conditions for the Helmholtz equation. Int. J. Numer. Meth. Fluids 55 (2007) 163-175. | MR | Zbl
, and ,[23] Optimal Schwarz waveform relaxation for the one dimensional wave equation. SIAM J. Numer. Anal. 41 (2003) 1643-1681. | MR | Zbl
, and ,[24] Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24 (2002) 38-60. | MR | Zbl
, and ,[25] 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).
, , and ,[26] A model-based block-triangular preconditioner for the Bidomain system in electrocardiology. J. Comput. Phys. 228 (2009) 3625-3639. | MR | Zbl
, , , and ,[27] Analysis and optimization of Robin-Robin partitioned procedures in Fluid-Structure Interaction problems. SIAM J. Numer. Anal. 48 (2010) 2091-2116. | MR
, and ,[28] Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology. ESAIM : M2AN 45 (2011) 309-334. | Numdam | MR | Zbl
, and ,[29] Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems. Appl. Math. Lett. 1 (1988) 299-302. | MR | Zbl
, and ,[30] Simulating the electrical behavior of cardiac tissue using the Bidomain model. Crit. Rev. Biomed. Eng. 21 (1993) 1-77.
,[31] The optimized order 2 method : Application to convection-diffusion problems. Future Gener. Comp. Syst. 18 (2001) 17-30. | Zbl
, and ,[32] Numerical solution of the bidomain equations. Phil. Trans. R. Soc. A. 367 (2009) 1931-1950. | MR | Zbl
, , , and ,[33] Mathematical models and numerical methods for the forward problem in cardiac electrophysiology. Comput. Vis. Sci. 5 (2003) 215-239. | Zbl
, , , , and ,[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 | Zbl
,[35] A model of the ventricular cardiac action potential : depolarization, repolarization and their interaction. Circ. Res. 68 (1991) 1501-1526.
and ,[36] An a posteriori error estimator for model adaptivity in electrocardiology. Comput. Methods Appl. Mech. Eng. 200 (2011) 2727-2737. | MR | Zbl
, and ,[37] A scalable Newton-Krylov-Schwarz method for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 3 (2009) 3861-3883. | MR | Zbl
, and ,[38] Factorization of the convection-diffusion operator and the Schwarz algorithm. M3AS 5 (1995) 67-93. | MR | Zbl
and ,[39] Multilevel additive Schwarz preconditioners for the Bidomain reaction-diffusion system. SIAM J. Sci. Comput. 31 (2008) 420-443. | MR | Zbl
and ,[40] 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
and ,[41] Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process. J. Comput. Appl. Math. 145 (2002) 49-70. | MR | Zbl
and ,[42] Algebraic multigrid preconditioners for the Bidomain reaction-diffusion system. Appl. Numer. Math. 59 (2009) 3033-3050. | MR | Zbl
and ,[43] 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
and ,[44] An efficient generalization of the Rush-Larsen method for solving electro-physiology membrane equations. ETNA 35 (2009) 234-256. | MR | Zbl
and ,[45] 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,.
, , and ,[46] Mathematical Modelling the Electrical Activity of the Heart. World Scientific, Singapore (2005). | MR | Zbl
, and ,[47] Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999). | MR | Zbl
and ,[48] Action potential propagation in a thick strand of cardiac muscle. Circ. Res. 68 (1991) 162-173.
,[49] Springer, Berlin (2004). | Zbl
, .[50] A hybrid multilevel Schwarz method for the Bidomain model. Comput. Methods Appl. Mech. Eng. 197 (2008) 4051-4061. | MR | Zbl
,[51] Domain Decomposition : Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press (1996). | MR | Zbl
, and .[52] 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
, and ,[53] Overlapping Schwarz methods for Maxwell's equations in three dimensions. Numer. Math. 86 (2000) 733-752. | MR | Zbl
,[54] Domain Decomposition Methods - Algorithms and Theory. Springer Ser. Comput. Math. 34 (2004). | Zbl
and ,[55] Reaction-diffusion systems for the macroscopic Bidomain model of the cardiac electric field. Nonlinear Anal. Real World Appl. 10 (2009) 849-868. | MR | Zbl
,[56] Computational techniques for solving the Bidomain equations in three dimensions. IEEE Trans. Biomed. Eng. 49 (2002) 1260-1269.
, and ,[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] Some nonoverlapping domain decomposition methods. SIAM Review 40 (1998) 857-914. | MR | Zbl
and ,[60] Iterative methods by space decomposition and subspace correction. SIAM Review 34 (1992) 581-613. | MR | Zbl
,Cited by Sources: