Multilevel additive schwarz preconditioners for the bidomain reaction-diffusion system

SIAM Journal on Scientific Computing

Volumn 31, Issue 1, 2008, Pages 420-443

  Pavarino, Luca F ^a   Scacchi, Simone ^b  

^a UNIVERSITY OF MILAN   (Italy)

^b UNIVERSITY OF PAVIA   (Italy)

Author keywords

Cardiac Bidomain model; Domain decomposition preconditioners; Multilevel additive Schwarz; Parallel numerical simulations; Reaction diffusion equations

Indexed keywords

CELL MEMBRANES; COMPUTER OPERATING SYSTEMS; DIFFUSION IN LIQUIDS; DOMAIN DECOMPOSITION METHODS; LINEAR EQUATIONS; LINEAR SYSTEMS; NONLINEAR EQUATIONS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; SCALABILITY;

ADDITIVE SCHWARZ PRECONDITIONERS; BIDOMAIN; CARDIAC BIDOMAIN MODEL; DOMAIN DECOMPOSITION PRECONDITIONERS; MULTILEVEL ADDITIVE SCHWARZ; MULTILEVEL ALGORITHM; OPTIMALITY; PARALLEL NUMERICAL SIMULATION; REACTION DIFFUSION EQUATIONS; THREE DIMENSIONS;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 55049098326     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/070706148     Document Type: Article

References (54)

1. T. M. AUSTIN, M. L. TREW, AND A. J. PULLAN, Solving the cardiac Bidomain equations for discontinuous conductivities, IEEE Trans. Biomed. Engrg., 53 (2005), pp. 1265-1272.
2. T. M. AUSTIN, M. L. TREW, AND A. J. PULLAN, A comparison of multilevel solvers for the cardiac Bidomain equations, in Proceedings of IEEE-EMBS, 27th Ann. Engrg. Med. Biol. Soc., 2005, pp. 7204-7207.
3. S. BALAY, K. BUSCHELMAN, W. D. GROPP, D. KAUSHIK, M. KNEPLEY, L. CURFMAN MCINNES, B. F. SMITH, AND H. ZHANO, PETSc Users Manual, Technical report ANL-95/11 - Revision 2.1.5, Argonne National Laboratory, Argonne, IL, 2002.
4. S. BALAY, K. BUSCHELMAN, W. D. GROPP, D. KAUSHIK, M. KNEPLEY, L. CURFMAN MCINNES, B. F. SMITH, AND H. ZHANG, PETSc home page, http://www.mcs.anl.gov/petsc.
5. X.-C. CAI, Additive Schwarz algorithms for parabolic convection-diffusion equations, Numer. Math., 60 (1991), pp. 41-61.
6. X.-C. CAI, Multiplicative Schwarz methods for parabolic problems, SIAM J. Sci. Comput., 15 (1994), pp. 587-603.
7. P. COLLI FRANZONE, P. DEUFLHARD, B. ERDMANN, J. LANG, AND L. F. PAVARINO, Adaptivity in space and time for reaction-diffusion systems in electrocardiology, SIAM J. Sci. Comput., 28 (2006), pp. 942-962.
8. P. COLLI FRANZONE AND L. F. PAVARINO, A parallel solver for reaction-diffusion systems in computational electrocardiology, Math. Models Methods Appl. Sci., 14 (2004), pp. 883-911.
9. P. COLLI FRANZONE, L. F. PAVARINO, AND B. TACCARDI, Simulating patterns of excitation, repolarization and action potential duration with cardiac Bidomain and Monodomain models, Math. Biosci., 197 (2005), pp. 35-66.
10. P. COLLI FRANZONE, L. F. PAVARINO, AND G. SAVARÈ, Computational electrocardiology: Mathematical and numerical modeling, in Complex Systems in Biomedicine, A. Quarteroni et al., eds., Springer, New York, 2006, pp. 187-241.
11. P. COLLI FRANZONE AND G. SAVARÉ, Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis, A. Lorenzi and B. Ruf, eds., Birkhauser, Boston, 2002, pp. 49-78.
12. M. DRYJA AND O.B. WIDLUND, Multilevel additive methods for elliptic finite element problems, in Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth Gamm-Seminar Kiel, 1990, W. Hackbusch, ed., Vieweg-Verlag, Braunschweig, 1991, Notes Numer. Fluid Mech., 31 (1991), pp. 58-69.
13. M. DRYJA AND O. B. WIDLUND, Domain decomposition algorithms with small overlap, SIAM J. Sci. Comput., 15 (1994), pp. 604-620.
14. M. DRYJA, M. V. SARKIS, AND O. B. WIDLUND, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer. Math., 72 (1996), pp. 313-348.
15. D. B. GESELOWITZ AND W. T. MILLER, A Bidomain model for anisotropic cardiac muscle, Ann. Biomed. Eng., 11 (1983), pp. 191-206.
16. C. S. HENRIQUEZ, Simulating the electrical behavior of cardiac tissue using the Bidomain model, Crit. Rev. Biomed. Eng., 21 (1993), pp. 1-77.
17. N. HOOKE, Efficient Simulation of Action Potential Propagation in a Bidomain, Ph.D. thesis, Department of Computer Science, Duke University, Durham, NC, 1992.
18. J. P. KEENER AND J. SNEYD, Mathematical Physiology, Springer, New York, 1998.
19. I. J. LEGRICE, B. H. SMAILL, AND P. J. HUNTER, Laminar structure of the heart: A mathematical model, Amer. J. Physiol. (Heart Circ. Physiol.), 272 (1997), pp. H2466-H2476.
20. C. LUO AND Y. RUDY, A model of the ventricular cardiac action potential: Depolarization, repolarization, and their interaction, Circ. Res., 68 (1991), pp. 1501-1526.
21. K.-A. MARDAL, B. F. NIELSEN, X. CAI, AND A. TVEITO, An order optimal solver for the discretized Bidomain equations, Numer. Linear Algebra Appl., 14 (2007), pp. 83-98.
22. M. MURILLO AND X. C. CAI, A fully implicit parallel algorithm for simulating the non-linear electrical activity of the heart, Numer. Linear Algebra Appl., 11 (2004), pp. 261-277.
23. M. MUNTEANU AND L. F. PAVARINO, Implicit parallel solvers in computational electrocardiology, in Applied Analysis and Differential Equations, O. Carja and I. I. Vrabie, eds., World Scientific, River Edge, NJ, 2007, pp. 255-266.
24. M. MUNTEANU, Overlapping Additive Schwarz Methods for Nonlinear Parabolic Reactiondiffusion Problems, Ph.D. thesis, Department of Mathematics, University of Milano, Milan, Italy, 2008.
25. M. A. OLSHANSKII AND A. REUSKEN, On the convergence of a multigrid method for linear reaction-diffusion problems, Computing, 65 (2000), pp. 193-202.
26. L. F. PAVARINO AND P. COLLI FRANZONE, Parallel solution of cardiac reaction-diffusion models, in Domain Decomposition Methods in Science and Engineering, LNCSE 40, Kornhuber et al., eds., Springer, New York, 2004, pp. 669-676.
27. M. PENNACCHIO, G. SAVARÉ, AND P. COLLI FRANZONE, Multiscale modeling for the bioelectric activity of the heart, SIAM J. Math. Anal., 4 (2005), pp. 1333-1370.
28. M. PENNACCHIO AND V. SIMONCINI, Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process, J. Comput. Appl. Math., 145 (2002), pp. 49-70.
29. J. PORMANN, A Simulation System for the Bidomain Equations, Ph.D. thesis, Department of Electrical and Computer Engineering, Duke University, Durham, NC, 1999.
30. 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. Engrg., 54 (2007), pp. 585-596.
31. A. J. PULLAN, L. K. CHENG, AND L. M. BUIST, Mathematically Modeling the Electrical Activity of the Heart: From Cell to Body Surface and Back, World Scientific, River Edge, NJ, 2005.
32. Z. QU AND A. GARPINKEL, An advanced algorithm for solving partial differential equation in cardiac conduction, IEEE Trans. Biomed. Engrg., 46 (1999), pp. 1166-1168.
33. W. QUAN, S. J. EVANS, AND H. M. HASTINGS, Efficient integration of a realistic two-dimensional cardiac tissue model by domain decomposition, IEEE Trans. Biomed. Engrg., 45 (1998), pp. 372-385.
34. A. QUARTERONI AND A. VALLI, Numerical Approximation of Partial Differential Equations, Springer, Berlin, 1994.
35. Y. RUDY AND J. R. SILVA, Computational biology in the study of cardiac ion channels and cell electrophysiology, Quart. Rev. Biophys., 39 (2006), pp. 57-116.
36. H. I. SALEHEEN AND K. T. NO, A new three-dimensional finite-difference Bidomain formulation for inhomogeneous anisotropic cardiac tissues, IEEE Trans. Biomed. Engrg., 45 (1998), pp. 15-25.
37. S. SCACCHI, Multilevel Schwarz Preconditioned for the Bidomain System and Applications to Electrocardiology, Ph.D. thesis, Department of Mathematics, University of Pavia, Pavia, Italy, 2008.
38. S. SCACCHI AND L. F. PAVARINO, A hybrid multilevel Schwartz method for the Biodomain model, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 4051-4061.
39. K. SKOUIBINE AND W. KRASSOWSKA, Increasing the computational efficiency of a Bidomain model of defibrillation using a time-dependent activating function, Ann. Biomed. Eng., 28 (2000), pp. 772-780.
40. K. SKOUIBINE, N. TRAYANOVA, AND P. MOORE, A numerically efficient model for the simulation of defibrillation in an active Bidomain sheet of myocardium, Math. Biosci., 166 (2000), pp. 85-100.
41. B. F. SMITH, P. BJØRSTAD, AND W. D. GROPP, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, London, 1996.
42. N. P. SMITH, D. P. NICKERSON, E. J. CRAMPIN, AND P. J. HUNTER, Multiscale computational modelling of the heart, Acta Numer., 13 (2004), pp. 371-431.
43. J. SUNDNES, G. T. LINES, K. A. MARDAL, AND A. TVEITO, Multigrid block preconditioning for a coupled system of partial differential equations modeling the electrical activity in the heart, Comput. Methods Biomech. Biomed. Engrg., 5 (2002), pp. 397-409.
44. J. SUNDNES, G. T. LINES, AND A. TVEITO, An operator splitting method for solving the Bidornain equations coupled to a volume conductor model for the torso, Math. Biosci., 194 (2005), pp. 233-248.
45. J. SUNDNES, G. T. LINES, X. CAI, B. F. NIELSEN, K.-A. MARDAL, AND A. TVEITO, Computing the Electrical Activity of the Heart, Springer, New York, 2006.
46. J. SUNDNES, B. F. NIELSEN, K. A. MARDAL, G. T. LINES, K. A. MARDAL, AND A. TVEITO, On the computational complexity of the Bidomain and the monodomain models of electro-physiology, Ann. Biomed. Eng., 34 (2006), pp. 1088-1097.
47. A. TOSELLI AND O. B. WIDLUND, Domain Decomposition Methods: Algorithms and Theory, Comput. Math. Ser. 34. Springer, Berlin, 2004.
48. J. A. TRANGENSTEIN AND C. KIM, Operator splitting and adaptive mesh refinement for the Luo-Rudy I model, J. Comput. Phys., 196 (2004), pp. 645-679.
49. M. VENERONI, Reaction-Diffusion Systems for the Macroscopic Bidomain Model of the Cardiac Electric Field, Nonlinear Analysis: Real World Applications, 2008, to appear.
50. E. J. VIGMOND, F. AGUEL, AND N. A. TRAYANOVA, Computational techniques for solving the Bidomain equations in three dimensions, IEEE Trans. Biomed. Engrg., 49 (2002), pp. 1260-1269.
51. 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), pp. 3-18.
52. R. WEBER DOS SANTOS, G. PLANK, S. BAUER, AND E. J. VIGMOND, Parallel multigrid preconditioner for the cardiac Bidomain model, IEEE Trans. Biomed. Engrg., 31 (2004), pp. 1960-1968.
53. J. P. WHITELEY, An efficient numerical technique for the solution of the monodomain and Bidomain equations, IEEE Trans. Biomed. Engrg., 53 (2006), pp. 2139-2147.
54. X. ZHANG, Multilevel Schwarz methods, Numer. Math., 63 (1992), pp. 521-539.