title: Reduced-order preconditioning for bidomain simulations.
abstract: Simulations of the bidomain equations involve solving large, sparse, linear systems of the form Ax = b. Being an initial value problems, it is solved at every time step. Therefore, efficient solvers are essential to keep simulations tractable. Iterative solvers, especially the preconditioned conjugate gradient (PCG) method, are attractive since memory demands are minimized compared to direct methods, albeit at the cost of solution speed. However, a proper preconditioner can drastically speed up the solution process by reducing the number of iterations. In this paper, a novel preconditioner for the PCG method based on system order reduction using the Arnoldi method (A-PCG) is proposed. Large order systems, generated during cardiac bidomain simulations employing a finite element method formulation, are solved with 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 was suggested. The A-PCG was applied to quickly obtain an approximate solution, and 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 those of direct LU but more than ILU method. The proposed scheme is shown to yield significant speedups when solving time evolving systems.
authors.affiliation: Department of Electrical and Computer Engineering, University of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canada. mdeo@ucalgary.ca
authors.forename: Makarand
authors.initials: M
authors.lastname: Deo
authors.forename: Steffen
authors.initials: S
authors.lastname: Bauer
authors.forename: Gernot
authors.initials: G
authors.lastname: Plank
authors.forename: Edward
authors.initials: E
authors.lastname: Vigmond
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D025802:Electric Capacitance; D004558:Electric Stimulation; D004594:Electrophysiology; D020342:Finite Element Analysis; D006329:Heart Conduction System / Q000502:physiology*; D006801:Humans; D008955:Models, Cardiovascular; D017711:Nonlinear Dynamics; D009716:Numerical Analysis, Computer-Assisted; D015203:Reproducibility of Results; D012984:Software; D016276:Ventricular Function