Preconditioning the bidomain model with almost linear complexity

TL;DR

This paper introduces an efficient preconditioning method for the bidomain model in electro-cardiology, significantly reducing computational costs and achieving near-linear complexity in simulations of cardiac tissue.

Contribution

It proposes a novel preconditioning approach based on a symmetric positive semi-definite formulation and a block LU decomposition with a heuristic approximation, improving computational efficiency.

Findings

Achieves almost linear computational complexity in problem size.

Reduces CPU time and iteration count in numerical simulations.

Validates method on 2D and 3D cardiac tissue models.

Abstract

The bidomain model is widely used in electro-cardiology to simulate spreading of excitation in the myocardium and electrocardiograms. It consists of a system of two parabolic reaction diffusion equations coupled with an ODE system. Its discretisation displays an ill-conditioned system matrix to be inverted at each time step: simulations based on the bidomain model therefore are associated with high computational costs. In this paper we propose a preconditioning for the bidomain model either for an isolated heart or 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 semi-definite. A block $LU$ decomposition of the system together with a heuristic approximation (referred to as the monodomain approximation) are the key ingredients for the preconditioning definition. Numerical results are provided for two test cases: a 2D test case on a realistic slice of the thorax based on a segmented heart medical image geometry, a 3D test case involving a small cubic slab of tissue with orthotropic anisotropy. The analysis of the resulting computational cost (both in terms of CPU time and of iteration number) shows an almost linear complexity with the problem size, i.e. of type $n\log^\alpha(n)$ (for some constant $\alpha$) which is optimal complexity for such problems.