Preconditioning the coupled heart and torso bidomain model with an almost linear complexity

Charles Pierre (1)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

The bidomain model is widely used in electro-cardiology to simulate spreading of excitation in the myocardium and electrocardiograms. It reads a system of two parabolic reaction diffusion equations coupled with an ODE system. Its discretization displays an ill-conditioned system matrix to be inverted at each time step: simulation based on the bidomain model therefore are associated with high computational costs. In this paper we propose a preconditioning for the bidomain model in an extended framework including a coupling with the surrounding tissues (the torso). The preconditioning is based on a formulation of the discrete problem that is shown to be symmetric positive. A block $LU$ decomposition of the system together with a heuristic approximation (referred to as the monodomain model) are the key ingredients for the preconditioning definition. Numerical results are provided for two test cases: a 2D realistic one (based on a segmented heart medical image geometry) and a 3D academical one. The analysis of the resulting computational cost (both in terms of CPU time and of iteration number) show an almost linear complexity, i.e. of type $n\log^\alpha(n)$ .

Mots clés: Preconditioning; Electro-cardiology; Bidomain and monodomain models; Hierarchical matrices; Reaction diffusion equations; Numerical simulations

Domaines: Analyse numérique [math.NA]

Charles Pierre. Preconditioning the coupled heart and torso bidomain model with an almost linear complexity. 2010. ⟨hal-00525976v1⟩