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首页> 外文学位 >Parallel algorithms and software for time-dependent systems of nonlinear partial differential equations with an application in computational biology.
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Parallel algorithms and software for time-dependent systems of nonlinear partial differential equations with an application in computational biology.

机译：非线性偏微分方程时间相关系统的并行算法和软件及其在计算生物学中的应用。

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摘要

In this thesis we develop a fully implicit, parallel nonlinear domain decomposition method and software for solving the two dimensional bidomain equations which model the excitation process of the heart. The bidomain model consists of a coupled system of time-dependent nonlinear partial and ordinary differential equations, including both parabolic and hyperbolic types. To solve the system of equations, we use a nonlinearly implicit backward Euler discretization scheme for the time variable, and the resulting large sparse nonlinear algebraic system is solved using a Newton-Krylov-Schwarz algorithm at each time step. Within each Newton iteration, the Jacobian system of linear equations is solved inexactly using the restarted GMRES method with a restricted additive Schwarz preconditioner. In order to reduce the storage and execution time, an incomplete factorization technique is applied to each of the subdomain systems of equations.; The proposed numerical algorithms are implemented on various distributed memory parallel computers using PETSc (Portable, Extensible Toolkit for Scientific Computation) of Argonne National Laboratory. Computational experiments show that our methods and software are robust with respect to physical and mesh parameters, and the nested iterative method is scalable when using large number of processors. In addition, our nonlinearly implicit algorithm allows the use of time steps much larger than other existing methods.
机译：在本文中，我们开发了一种完全隐式，并行的非线性域分解方法和软件，用于求解对心脏的兴奋过程建模的二维双域方程。双域模型由时间相关的非线性偏微分方程和常微分方程的耦合系统组成，包括抛物线型和双曲线型。为了求解方程组，我们对时间变量使用非线性隐式向后欧拉离散化方案，然后在每个时间步使用Newton-Krylov-Schwarz算法求解所得的大型稀疏非线性代数系统。在每个Newton迭代中，使用重新启动的GMRES方法和受限加性Schwarz预调节器来精确求解线性方程组的Jacobian系统。为了减少存储和执行时间，将不完全分解技术应用于方程的每个子域系统。使用Argonne国家实验室的PETSc（用于科学计算的便携式可扩展工具包），在各种分布式内存并行计算机上实现了提出的数值算法。计算实验表明，我们的方法和软件在物理参数和网格参数方面均很健壮，并且嵌套迭代方法在使用大量处理器时可扩展。此外，我们的非线性隐式算法允许使用比其他现有方法大得多的时间步长。

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