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Reduced-order preconditioning for bidomain simulations.

Makarand Deo1, Steffen Bauer, Gernot Plank

  • 1Department of Electrical and Computer Engineering, University of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canada. mdeo@ucalgary.ca

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|May 24, 2007
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Summary
This summary is machine-generated.

A new Arnoldi-based preconditioned conjugate gradient (A-PCG) method significantly speeds up cardiac bidomain simulations. This novel approach offers faster solutions than traditional incomplete LU preconditioning for large, sparse linear systems.

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Area of Science:

  • Computational Biology and Physiology
  • Numerical Analysis and Scientific Computing

Background:

  • Simulating cardiac bidomain equations requires solving large, sparse, linear systems (Ax = b) at each time step.
  • Efficient solvers are crucial for tractable simulations, with iterative methods like preconditioned conjugate gradient (PCG) offering memory advantages over direct methods.
  • The performance of PCG heavily relies on effective preconditioners to reduce iteration counts and accelerate convergence.

Purpose of the Study:

  • To introduce and evaluate a novel preconditioner for the preconditioned conjugate gradient (PCG) method, termed A-PCG.
  • To assess the performance of the A-PCG method in solving large-scale linear systems arising from finite element-based cardiac bidomain simulations.
  • To compare the A-PCG method's efficiency against established incomplete LU (ILU) preconditioning.

Main Methods:

  • A novel preconditioner (A-PCG) was developed based on system order reduction using the Arnoldi method.
  • The A-PCG method was applied to solve large-order systems generated from finite element formulations of cardiac bidomain simulations.
  • Performance comparison was conducted against the incomplete LU (ILU) preconditioning method.
  • A cascaded preconditioner approach combining A-PCG with successive overrelaxation (SOR) was investigated to balance speed and accuracy.

Main Results:

  • The A-PCG method demonstrated considerably faster approximate solution generation compared to ILU preconditioning, often within a single iteration.
  • The cascaded A-PCG-SOR approach effectively reduced computational demands (memory and run time) by using A-PCG for rapid approximation and SOR for refinement.
  • Memory requirements for A-PCG were found to be between direct LU decomposition and the ILU method.
  • The proposed A-PCG scheme yielded significant speedups for solving time-evolving bidomain systems.

Conclusions:

  • The A-PCG method presents a highly efficient approach for solving linear systems in cardiac bidomain simulations.
  • The cascaded preconditioner strategy offers a practical solution for optimizing computational resources in complex physiological modeling.
  • This novel preconditioning technique significantly accelerates the simulation of cardiac electrophysiology.