Weber dos Santos Rodrigo, Plank Gernot, Bauer Steffen, Vigmond Edward J
Department of Biosignals, Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, D-10587 Berlin, Germany.
IEEE Trans Biomed Eng. 2004 Nov;51(11):1960-8. doi: 10.1109/TBME.2004.834275.
The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue but are computationally expensive, limiting the size of the problem which can be modeled. The purpose of this study is to determine more efficient ways to solve the elliptic portion of the bidomain equations, the most computationally expensive part of the computation. Specifically, we assessed the performance of a parallel multigrid (MG) preconditioner for a conjugate gradient solver. We employed an operator splitting technique, dividing the computation in a parabolic equation, an elliptical equation, and a nonlinear system of ordinary differential equations at each time step. The elliptic equation was solved by the preconditioned conjugate gradient method, and the traditional block incomplete LU parallel preconditioner (ILU) was compared to MG. Execution time was minimized for each preconditioner by adjusting the fill-in factor for ILU, and by choosing the optimal number of levels for MG. The parallel implementation was based on the PETSc library and we report results for up to 16 nodes on a distributed cluster, for two and three dimensional simulations. A direct solver was also available to compare results for single processor runs. MG was found to solve the system in one third of the time required by ILU but required about 40% more memory. Thus, MG offered an attractive tradeoff between memory usage and speed, since its performance lay between those of the classic iterative methods (slow and low memory consumption) and direct methods (fast and high memory consumption). Results suggest the MG preconditioner is well suited for quickly and accurately solving the bidomain equations.
双域方程被广泛用于模拟心脏组织中的电活动，但计算成本高昂，限制了可建模问题的规模。本研究的目的是确定更有效的方法来求解双域方程的椭圆部分，这是计算中计算成本最高的部分。具体而言，我们评估了共轭梯度求解器的并行多重网格（MG）预处理器的性能。我们采用了算子分裂技术，在每个时间步将计算划分为抛物方程、椭圆方程和常微分方程的非线性系统。椭圆方程通过预处理共轭梯度法求解，并将传统的块不完全LU并行预处理器（ILU）与MG进行比较。通过调整ILU的填充因子以及选择MG的最佳层数，使每个预处理器的执行时间最小化。并行实现基于PETSc库，我们报告了在分布式集群上多达16个节点的二维和三维模拟结果。还提供了直接求解器以比较单处理器运行的结果。结果发现，MG求解系统所需的时间是ILU的三分之一，但内存需求大约多40%。因此，MG在内存使用和速度之间提供了一个有吸引力的权衡，因为其性能介于经典迭代方法（慢且内存消耗低）和直接方法（快且内存消耗高）之间。结果表明，MG预处理器非常适合快速准确地求解双域方程。
