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Hierarchical Interpolative Factorization Preconditioner for Parabolic Equations
Journal of Scientific Computing ( IF 2.5 ) Pub Date : 2020-11-10 , DOI: 10.1007/s10915-020-01343-5
Jordi Feliu-Fabà , Lexing Ying

This note proposes an efficient preconditioner for solving linear and semi-linear parabolic equations. With the Crank–Nicholson time stepping method, the algebraic system of equations at each time step is solved with the conjugate gradient method, preconditioned with hierarchical interpolative factorization. Stiffness matrices arising in the discretization of parabolic equations typically have large condition numbers, and therefore preconditioning becomes essential, especially for large time steps. We propose to use the hierarchical interpolative factorization as the preconditioning for the conjugate gradient iteration. Computed only once, the hierarchical interpolative factorization offers an efficient and accurate approximate inverse of the linear system. As a result, the preconditioned conjugate gradient iteration converges in a small number of iterations. Compared to other classical exact and approximate factorizations such as Cholesky or incomplete Cholesky, the hierarchical interpolative factorization can be computed in linear time and the application of its inverse has linear complexity. Numerical experiments demonstrate the performance of the method and the reduction of conjugate gradient iterations.



中文翻译:

抛物方程的层次插值因式分解预处理器

本说明提出了一种用于求解线性和半线性抛物方程的有效预处理器。使用Crank–Nicholson时间步长方法,可以使用共轭梯度方法对每个时间步长的代数方程组进行求解,并通过分层插值因式分解进行预处理。在抛物线方程离散化过程中产生的刚度矩阵通常具有较大的条件数,因此预处理必不可少,尤其是对于较大的时间步长而言。我们建议使用分层插值分解作为共轭梯度迭代的预处理。层次插值因式分解仅计算一次,因此可以提供线性系统的有效且精确的近似逆。结果是,预处理的共轭梯度迭代以少量迭代收敛。与其他经典精确和近似因式分解(例如Cholesky或不完全Cholesky)相比,分层插值因式分解可以在线性时间内计算,并且其逆运算的应用具有线性复杂度。数值实验证明了该方法的性能以及共轭梯度迭代的减少。

更新日期:2020-11-12
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