论文标题
抛物线方程的层次插值分解预处理
Hierarchical Interpolative Factorization Preconditioner for Parabolic Equations
论文作者
论文摘要
本说明提出了一个有效的预处理，用于求解线性和半线性抛物线方程。借助曲柄尼古尔森时间阶梯法，每个时间步骤的代数方程系统都使用共轭梯度方法求解，并通过层次插值分解进行了预处理。在抛物线方程的离散化中产生的刚度矩阵通常具有较大的状况数量，因此预处理变得至关重要，尤其是对于大时间步骤。我们建议将分层插值分解用作共轭梯度迭代的预处理。层次插值分解仅计算一次，提供了有效，准确的近似近相。结果，预处理的共轭梯度迭代在少量迭代中收敛。与其他经典的精确和近似因素化相比，例如Cholesky或Cholesky不完整，可以在线性时间内计算层次插值分解，并且其逆的应用具有线性的复杂性。数值实验证明了该方法的性能和结合梯度迭代的降低。
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.
