Bidomain simulations of cardiac systems often involve solving large, sparse, linear systems of the form Ax=b. These simulations are computationally very expensive in terms of run time and memory requirements. Therefore, efficient solvers are essential to keep simulations tractable. In this paper, an efficient preconditioner for the conjugate gradient (CG) method based on system order reduction using the Arnoldi method (A-PCG) is explained. Large order systems generated during cardiac bidomain simulations using a finite element method formulation, are solved using the A-PCG method. Its performance is compared with incomplete LU (ILU) preconditioning. Results indicate that the A-PCG estimates an approximate solution considerably faster than the ILU, often within a single iteration. To reduce the computational demands in terms of memory and run time, the use of a cascaded preconditioner is suggested. The A-PCG can be applied to quickly obtain an approximate solution, subsequently a cheap iterative method such as successive overrelaxation (SOR) is applied to further refine the solution to arrive at a desired accuracy. The memory requirements are less than direct LU but more than ILU method. The proposed scheme is shown to yield significant speedups when solving time evolving systems.
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