On moment-discretization and least-squares solutions of linear integral equations of the first kind
On moment-discretization and least-squares solutions of linear integral equations of the first kind
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关于第一类线性积分方程的矩离散化和最小二乘解
DOI：
10.1016/0022-247x(76)90115-3
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发表时间：
1976
期刊：
影响因子：
--
通讯作者：
M. Nashed
中科院分区：
文献类型：
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作者：
M. Nashed
令K (s, t) 为[0, 1]×[0, 1] 上的连续函数，令K 为核K (s, t) 在空间L 2 [0, 1] 上导出的线性积分算子。本文关注的是 L 2-范数中最小化“K x− y”问题的矩离散化，其中 y 是给定的连续函数。这与矩离散方程的最小二乘解问题形成对比：∝ 0 1 K (s i, t) x (t) dt= y (s i), i= 1, 2, h., n。建立了“矩离散化”和“最小二乘”运算之间的简单交换性结果。这提出了一种近似 K† y（其中 K† 是 K 的广义逆）的过程，无需求助于正规方程 K*Kx=K*y，该过程可以与简单的数值求积公式加搭配或相关的数值和正则化方法结合使用，用于第一类线性积分方程的最小二乘解。
Let K (s, t) be a continuous function on [0, 1]×[0, 1], and let K be the linear integral operator induced by the kernel K (s, t) on the space L 2 [0, 1]. This note is concerned with moment-discretization of the problem of minimizing‖ K x− y‖ in the L 2-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation:∝ 0 1 K (s i, t) x (t) dt= y (s i), i= 1, 2, h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating K† y (where K† is the generalized inverse of K), without recourse to the normal equation K∗ Kx= K∗ y, that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind.