有限差分近似奇异摄动问题所获线性系统 的Cholesky分解
Cholesky factorisation of linear systems coming from finite difference approximations of singularly perturbed problems
AI总结：
研究奇异摄动反应扩散方程离散化所得大线性系统求解，分析Cholesky分解在奇异摄动问题中的性能问题及参数影响范围。
摘要：
我们考虑了当两个维度的奇异摄动反应扩散方程被离散化时所得到的大线性方程组的求解。标准方法如中心有限差分法会导致系统矩阵为正定矩阵。此类系统的直接求解器基于Cholesky分解。然而，如MacLachlan和Madden（SIAM J. Sci. Comput. 35-5（2013），pp. A2225-A2254）所观察到的，这些求解器在奇异摄动问题中可能表现不佳。我们分析了因数中条目分布基于其大小的现象，解释了这一现象，并给出了扰动和离散化参数范围内预期表现不佳的界限。
英文摘要：
We consider the solution of large linear systems of equations that arise when two-dimensional singularly perturbed reaction-diffusion equations are discretized. Standard methods for these problems, such as central finite differences, lead to system matrices that are positive definite. The direct solvers of choice for such systems are based on Cholesky factorisation. However, as observed by MacLachlan and Madden (SIAM J. Sci. Comput. 35-5 (2013), pp. A2225-A2254), these solvers may exhibit poor performance for singularly perturbed problems. We provide an analysis of the distribution of entries in the factors based on their magnitude that explains this phenomenon, and give bounds on the ranges of the perturbation and discretization parameters where poor performance is to be expected.
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