一种适用于具有序关系的埃尔米特正定线性系统的多重网格方法提案
A Proposal of Multigrid Methods for Hermitian Positive Definite Linear Systems enjoying an order relation
AI总结：
针对满足序关系的埃尔米特正定线性系统，提出一种基于相同光滑子和投影子的多重网格方法，并证明其两网格最优性，应用于多级结构线性系统实现线性时间求解。
摘要：
给定一个用于系数矩阵为$A_n$的线性系统的多重网格过程，我们讨论了当应用于系数矩阵为$B_n$的适当相关代数问题时，使用相同光滑子和相同投影子的相关多重网格过程的最优性：我们假设$A_n$和$B_n$都是正定的，且存在与$n$无关的正数$\vartheta$使得$A_n\le \vartheta B_n$。在此背景下，我们证明了两网格方法的最优性。我们应用这一基本策略来设计一种多重网格解法，用于修改多级结构（Toeplitz、循环、Hartley、正弦（$ au$类）和余弦代数）线性系统，其中系数矩阵是多级带状的且埃尔米特正定。通过这种方式，可以高效地以线性时间（相对于代数问题的规模）求解由具有各种边界条件的积分微分方程逼近产生的多个线性系统。给出并讨论了一些数值实验，涉及两网格和多网格过程。
英文摘要：
Given a multigrid procedure for linear systems with coefficient matrices $A_n$, we discuss the optimality of a related multigrid procedure with the same smoother and the same projector, when applied to properly related algebraic problems with coefficient matrices $B_n$: we assume that both $A_n$ and $B_n$ are positive definite with $A_n\le \vartheta B_n$, for some positive $\vartheta$ independent of $n$. In this context we prove the Two-Grid method optimality. We apply this elementary strategy for designing a multigrid solution for modifications of multilevel structured (Toeplitz, circulants, Hartley, sine ($τ$ class) and cosine algebras) linear systems, in which the coefficient matrix is banded in a multilevel sense and Hermitian positive definite. In such a way, several linear systems arising from the approximation of integro-differential equations with various boundary conditions can be efficiently solved in linear time (with respect to the size of the algebraic problem). Some numerical experiments are presented and discussed, both with respect to Two-Grid and multigrid procedures.
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