Continuous Analogues of Krylov Subspace Methods for Differential Operators
Continuous Analogues of Krylov Subspace Methods for Differential Operators
复制标题
微分算子的 Krylov 子空间方法的连续类似
DOI：
10.1137/18m1177810
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发表时间：
2019
影响因子：
2.9
通讯作者：
Townsend, Alex
中科院分区：
文献类型：
--
作者：
Gilles, Marc Aurèle;Townsend, Alex
共轭梯度法，最小残量法和广义最小残量法的类比推导出解决边值问题（BVP）涉及常微分方程。出现了两个挑战：在建立Krylov子空间的同时对解施加边界条件，保证了基于Krylov的方法在无界算子上的收敛性。我们的方法采用投影算子来保证边界条件得到满足，我们开发了一个算子预处理器，确保在有限次迭代后计算出近似解。所开发的Krylov方法是实用的迭代边值问题求解器，特别是当一个快速的算子函数产品是有效的。还提出了偏微分算子的一个推广。
Analogues of the conjugate gradient method, minimum residual method, and generalized minimum residual method are derived for solving boundary value problems (BVPs) involving ordinary differential equations. Two challenges arise: imposing the boundary conditions on the solution while building up a Krylov subspace and guaranteeing convergence of the Krylov-based method on unbounded operators. Our approach employs projection operators to guarantee that the boundary conditions are satisfied, and we develop an operator preconditioner that ensures that an approximate solution is computed after a finite number of iterations. The developed Krylov methods are practical iterative BVP solvers that are particularly efficient when a fast operator-function product is available. An extension to partial differential operators is also presented.
影响因子：
2.7
作者：
Aurentz, Jared L.;Trefethen, Lloyd N.
通讯作者：
Trefethen, Lloyd N.