Symmetric Tridiagonal Inverse Quadratic Eigenvalue Problems with Partial Eigendata

In this paper we concern the inverse problem of constructing the n-by-n real symmetric tridiagonal matrices C and K so that the monic quadratic pencil Q(¸) := ¸2I + ¸C + K (where I is the identity matrix) possesses the given partial eigendata. We ¯rst provide the su±cient and necessary conditions for the existence of an exact solution to the inverse problem from the self-conjugate set of prescribed four eigenpairs. To ¯nd a physical solution for the inverse problem where the matrices C and K are weakly diagonally dominant and have positive diagonal elements and negative o®-diagonal elements, we consider the inverse problem from the partial measured noisy eigendata. We propose a regularized smoothing Newton method for solving the inverse problem. The global and quadratic convergence of our approach is established under some mild assumptions. Some numerical examples and a practical engineering application in vibrations show the e±ciency of our method

Optimization Approaches for Inverse Quadratic Eigenvalue Problems

Introduction Main Problems（Our approaches，Main results，Numerical experiments） Concluding Remark

Numerical Optimization Methods For Solving Several Inverse Quadratic Eigenvalue Problems

反二次特征值问题在结构力学、振动、控制设计、应用力学及电路理论等方面有着 广泛的应用背景。工程上往往利用有限元等技巧将结构体离散为矩阵二阶系统，并通过 测量得到的自然频率和模态来更新系统物理矩阵。 反二次特征值问题旨在构造物理矩阵使得重构的二阶系统满足给定的部分特征信 息，同时保持原始模型的结构性质：如对称性、正定性、稀疏性及内部连通性等。本文 主要探讨了几类结构化反二次特征值问题的优化算法。本文由以下四部分组成。 第一节主要介绍了反二次特征值问题的背景、已有的数值方法、及本文所涉及的几 类结构化反二次特征值问题。 第二节主要考虑基于部分特征信息基础上的参数化模型修正问题。这类...Inverse quadratic eigenvalue problems (IQEPs) arise in structural dynamics, vibra- tion, applied mechanics and circuity theory, etc. In engineering, vibrating structures are usually discretised to a matrix second-order system via the finite element technique, etc, and update the physical matrices of the original model using the measured natural frequencies and mode shapes. The IQEP aims to re...学位：理学博士院系专业：数学科学学院信息与计算数学系_计算数学学号：1902009015360

On a numerical construction of doubly stochastic matrices with prescribed eigenvalues

We study the inverse eigenvalue problem for finding doubly stochastic matrices with specified eigenvalues. By making use of a combination of Dykstra's algorithm and an alternating projection process onto a non-convex set, we derive hybrid algorithms for finding doubly stochastic matrices and symmetric doubly stochastic matrices with prescribed eigenvalues. Furthermore, we prove that the proposed algorithms converge and linear convergence is also proved. Numerical examples are presented to demonstrate the efficiency of our method.Comment: 16 page

Singular limits for 4-dimensional semilinear elliptic problems with exponential nonlinearity

Using some nonlinear domain decomposition method, we prove the existence of singular limits for solution of semilinear elliptic problems with exponential nonlinearity.Comment: 29 page