结构特征值反问题在结构动力学、应用物理、地球物理学、振动控制、有限元模型修正、经济学以及统计学等领域中有重要的应用背景。可计算性是结构特征值反问题的基本问题之一。申请人拟结合基于矩阵流形的黎曼几何、非线性和非光滑最优化算法以及数值代数等工具，为包含半正定（二次）矩阵特征值反问题、非负矩阵特征值反问题、随机和双随机矩阵特征值反问题以及特征值反问题的最小二乘问题在内的若干重要结构特征值反问题提出一套稳定、收敛、高效的黎曼几何非线性优化算法，以突破已有算法在规模化（即矩阵阶数超过1000）计算方面的瓶颈。从而为结构特征值反问题的规模化计算打开新的研究途径并提高其实用性。

Structured inverse eigenvalue problems arise in structured dynamics, applied physics, geophysics, vibration control, finite element model updating, economics and statistics, etc.. The computability is one of fundamental questions in inverse eigenvalue problems. The applicant intends to combine Riemannian geometry over matrix manifolds, nonlinear and nonsmooth optimization methods with numerical linear algebra and propose some stable, convergent and effective numerical Riemannian optimization methods for several structured inverse eigenvalue problems including semidefinite inverse (quadratic) eigenvalue problems, nonnegative inverse eigenvalue problems, the inverse eigenvalue problem for stochastic and doubly stochastic matrices, and least-square inverse eigenvalue problems. We hope to break through the bottleneck of existing numerical algorithms in large-scale (i.e., the number of rows of a square matrix is large than 1000) computations and provide a new way for large-scale computation of inverse eigenvalue problems and improve their practical applications.