Non-Symmetric Algebraic Multigrid Preconditioners for the Bidomain Reaction–Diffusion system

Abstract

We deal with the efficient solution of the so-called bidomain system which is possibly the most complete model for the cardiac bioelectric activity. We study the performance of a non-symmetric structured algebraic multigrid preconditioner on the formulation generally used of the bidomain model, i.e. the one characterized by a parabolic equation coupled with an elliptic one. Our numerical results show that, for this formulation, the non-symmetric preconditioner provides the best overall performance. The Bidomain ModelThe excitation process in the myocardium is a complex phenomenon characterized by rapid ionic fluxes through the cellular membrane separating the intracellular and the interstitial fluid in the myocardium [7]. The bidomain is the most complete model for the cardiac bioelectric activity, and it consists of a non-linear Reaction-Diffusion (R-D) system of equations for the intraand extracellular potential u i and u e , coupled through the transmembrane potential v := u i − u e [9]. The nonlinearity arises through the current-voltage relationship across the membrane which is described by a set of nonlinear ODEs (see [7]). The anisotropic properties of the media are modeled by the intra-and extracellular conductivity tensors M i = M i (x) and M e = M e (x) that satisfy a uniform ellipticity condition, see [10].

Electrophysiology

“…As in the earlier work of Pennacchio and Simoncini [66], we solve (4.36) iteratively using multigrid-preconditioned Krylov subspace methods such as GMRES [79]. The monodomain model is similar to the bidomain model except that the monodomain model does not separately account for the intracellular and extracellular currents, but instead describes the state of the tissue in terms of only the transmembrane voltage v and the spatially distributed gating variables w. One way to obtain the monodomain model from the bidomain model is to assume that the extracellular compartment is well approximated as being isopotential.…”

Optimized Schwarz Methods for the Bidomain system in electrocardiology

“…The degenerate parabolic nature of this system, however, entails a very ill conditioning for the linear system associated to the Bidomain discretization. In the recent years many efforts have been devoted to the set up of efficient solvers and preconditioners to reduce the high computational costs associated to its numerical solution [9,10,33,37,41,52,56,57], possibly based on multigrid approaches [39,42,43,50,58] or suitable approximations of the equations [26]. Among these works, most are based on a proper decomposition of the computational domain in order to set up parallel preconditioners, or on suitable multigrid schemes still coupled with parallel architectures.…”