算子广义逆的反序律及其在算子方程迭代求解中的应用

The theory of generalized inverses of operator has wide application background in the fields of Differential Equations, Nonlinear Equations, Dynamical Systems, Statistical and Stochastic Processes. Especially, the high performance approximate solutions of the singular linear operator equations and the solutions of the singular linear operator equations in the infinite dimension space can be represented by the generalized inverses of the operator. Therefore, how to use the theory of generalized inverses of operator to study the solutions of the singular linear operator equations is a hot issue in the world. In order to solve the singular linear operator equations effectively, the project intends to study the following contents: First, study the reverse order laws of the generalized inverses of operator and provide the necessary and sufficient conditions for the reverse order laws of the generalized inverses of multiple operator products. Second, by these reverse order laws, we will design a new fast and effective iterative algorithm to deal with the high performance approximate solutions of the singular linear operator equation Tx = b in infinite dimensional space. The project have very important meaning to lecubrate the generalized inverse of operator and have inspiration and reference for the research of large-scale scientific computing.

算子广义逆理论在微分方程、非线性方程、动力系统、统计及随机过程等领域有着广泛的应用背景。特别是一般奇异线性算子方程以及无限维空间中奇异线性算子方程的高性能近似解都可以用算子的广义逆来表示。因此，如何利用算子的广义逆理论来研究奇异线性算子方程的近似迭代求解成为国内外学者普遍关心的热点问题。为了有效地求解奇异线性算子方程，本项目拟研究如下内容：一、研究算子广义逆的反序律，给出无限维空间中任意多个有界奇异线性算子乘积的广义逆其反序律成立的充分必要条件。二、利用得到的反序律理论设计出一套新的、快速有效的迭代算法来处理无限维空间中奇异线性算子方程Tx=b的高性能近似求解问题。项目的完成对深入和拓展算子广义逆的研究具有重要的意义，对研究大规模科学计算具有启发和借鉴作用。

无限维空间中的奇异线性算子方程Tx=b的高性能近似求解问题在实际当中有着广泛的应用背景。如何设计出新的、快速有效的迭代算法来求解无限维空间中奇异线性算子方程的高性能近似解，是国内外学者关心的热点问题。在本项目中我们研究了基于算子广义逆反序律理论的奇异线性算子方程迭代求解，取得了两个方面的研究进展。第一，利用无限维空间中有界线性算子的广义逆理论、投影算子广义逆的逼近理论以和线性算子的矩阵化理论，给出了无限维空间中任意多个有界奇异线性算子乘积的{1, 2}-逆，{1, 3, 4}-逆和加权M-P逆它们的反序律成立的充分必要条件。第二，利用所得的{1, 2}-逆的反序律，设计出了一种新的迭代算法来求解无限维空间中奇异线性算子方程的极小范数和最小二乘解。此外以项目为依托，项目组成员还讨论了算子乘积广义逆的反序律在反特征值问题研究和卡特生乘积图的联通性研究中的应用。