最小二乘参数化奇异值反问题的预处理迭代算法

Preconditioned iterative method for least squares parameterized inverse singular value problems

提出一种最小二乘参数化奇异值反问题模型。首先,该最小二乘模型为混合优化问题,等价转化为流形上的光滑最小二乘问题,并运用黎曼非精确高斯牛顿法求解等价问题;其次,设计了黎曼中心预处理子,加速了黎曼高斯牛顿方程的求解,并适用于大规模问题的求解。数值实验表明,预处理的黎曼非精确高斯牛顿法可以稳定有效地求解最小二乘参数化奇异值反问题。

In order to tackle the practical issue encountered in parameterized inverse singular value problems,a more general least-squares model is considered,and the corresponding hybrid optimization problem is transformed into an equivalent smooth nonlinear least-squares problem on a manifold,which is solved by using Riemannian inexact Gauss-Newton method.A centered preconditioner is designed to accelerate the convergence of the solution of sub-problems,so it is suitable for solving large-scale problems.Finally,both effectiveness and efficiency are verified by numerical examples.