Algebraic Multigrid Preconditioner for the Cardiac Bidomain Model

Abstract

The bidomain equations are considered to be one of the most complete descriptions of the electrical activity in cardiac tissue, but large scale simulations, as resulting from discretization of an entire heart, remain a computational challenge due to the elliptic portion of the problem, the part associated with solving the extracellular potential. In such cases, the use of iterative solvers and parallel computing environments are mandatory to make parameter studies feasible. The preconditioned conjugate gradient (PCG) method is a standard choice for this problem. Although robust, its efficiency greatly depends on the choice of preconditioner. On structured grids, it has been demonstrated that a geometric multigrid preconditioner performs significantly better than an incomplete LU (ILU) preconditioner. However, unstructured grids are often preferred to better represent organ boundaries and allow for coarser discretization in the bath far from cardiac surfaces. Under these circumstances, algebraic multigrid (AMG) methods are advantageous since they compute coarser levels directly from the system matrix itself, thus avoiding the complexity of explicitly generating coarser, geometric grids. In this paper, the performance of an AMG preconditioner (BoomerAMG) is compared with that of the standard ILU preconditioner and a direct solver. BoomerAMG is used in two different ways, as a preconditioner and as a standalone solver. Two 3-D simulation examples modeling the induction of arrhythmias in rabbit ventricles were used to measure performance in both sequential and parallel simulations. It is shown that the

A comparison of coupled and uncoupled solvers for the cardiac Bidomain model

“…Previous works have considered several variants of these splitting and/or uncoupling techniques. In particular, the uncoupled approach has been considered in [1,22,35,36,45,47,[51][52][53][54]. The splitting of reaction and diffusion terms has been considered in [35,53,54], while a three steps Strang splitting, with reaction -diffusion -reaction steps, has been considered in [22,47,51].…”

“…The splitting of reaction and diffusion terms has been considered in [35,53,54], while a three steps Strang splitting, with reaction -diffusion -reaction steps, has been considered in [22,47,51]. An additional approximation is made in [35,54] by substituting the mass matrix with the identity. A recap of some splitting and uncoupling approaches can be found in the review papers [5,20,28,53].…”

“…Alternatively, most previous works have considered IMEX time discretizations and/or operator splitting schemes, where the reaction and diffusion terms are treated separately, see e.g. [4,6,8,14,20,22,28,35,38,45,47,[53][54][55]. The advantage of IMEX and operator splitting schemes is that they only require the solution of a linear system for the parabolic and elliptic PDEs at each time step.…”

“…Many different preconditioners have been proposed in order to devise efficient iterative solvers for the linear systems deriving from both the splitting techniques: Symmetric Successive Over Relaxation [32], block diagonal or triangular [6,17,33,34,52], optimized Schwarz [18], multigrid [1,[33][34][35]46,54], multilevel Schwarz [29,30,40,42].…”

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

“…Preconditioning is therefore mandatory. Attempts in the recent literature have employed diagonal preconditioners, Symmetric Successive Over Relaxation [10], Block Jacobi preconditioners with incomplete LU factorization (ILU) [18], multigrid [1,13]; see [19] for an overview.…”