Scalable parallel preconditioners for an open source cardiac electrophysiology simulation package

The numerical solution of the bidomain equations is a fundamental tool in the field of computational cardiac electrophysiology. The multi-scale nature of the processes involved and the increasing complexity of the tissue and ionic models of interest make the use of High Performance Computing techniques mandatory. The solution of the linear system arising from the Finite Element discretisation of the bidomain equations is the main bottleneck in terms of performance and parallel scalability. In this work we present and evaluate parallel implementations of two problem-specific block preconditioners, namely BD and LDU, in the framework of the open source cardiac electrophysiology simulation package Chaste. Both rely upon providing a reliable and practical method for calculating the action of the inverse of certain matrix subblocks. Our results show that: a) total execution time can be reduced by using different approximation methods for different matrix subblocks, b) there exists a good correlation between the coefficient Delta t/h(2) and the cases where BD and LDU outperform generic Algebraic Multigrid (AMG) preconditioning, and c) when applied to simulations with current state-of-the-art cardiac geometries, BD and LDU are over 3 times faster than AMG preconditioning and show similar scaling.