On Numerical Computation Of Two Inverse Eigenvalue Problems

Inverse eigenvalue problems arise in many applications,including structural dynamics,vibration,control design,Sturm-Liouville inverse problem,graph theory,and structural damage detection,etc.There are a variety of problems appeared in these applications can be solved by the solution of corresponding inverse eigenvalue problems.Therefore,the study of inverse eigenvalue problems is not only of theoretical significance,but also of practical value in engineering practice and scientific computing.This thesis focuses on the numerical solution for two different inverse eigenvalue problems.This thesis first studies the inverse eigenvalue problem for positive doubly stochastic matrices,which aims to construct a positive doubly stochastic matrix from the prescribed realizable spectral data.Doubly stochastic matrices play an important role in many applications such as communication theory of satellite-switched,multiple-access systems,quantum mechanics,the assignment problem,graph theory,and graph-based clustering,etc.There exist some necessary or sufficient existence conditions on the inverse eigenvalue problem for positive doubly stochastic matrices.Some constructive methods were proposed during the study of existence theory.Recently,some Riemannian optimization methods have been developed for solving the eigenvalue problem and the inverse eigenvalue problem.In particular,in 2018,Zhao,Bai and Jin gave a Riemannian inexact Newton-CG method for solving the inverse eigenvalue problem for nonnegative matrices.Sparked by this,based on the geometric properties of the set of positive doubly stochastic matrices and the real Schur decomposition,in this thesis,monotone/nonmonotone Riemannian inexact Newton-CG methods are proposed for solving the inverse eigenvalue problem for positive doubly stochastic matrices.Under some assumptions,the global and quadratic convergence of the proposed methods is established.Also,by using the computed real Schur decomposition,invariant subspaces of the constructed solution to the inverse eigenvalue problem for positive doubly stochastic matrices are obtained.Finally,some numerical examples,including an application in the digraph,are presented to illustrate that the proposed methods can solve effectively the inverse eigenvalue problem for positive doubly stochastic matrices.Next,the structural damage detection problem with incomplete mode shape data is discussed.For a vibratory structure in healthy way,it is crucial to carry on reliable and effective nondestructive damage diagnosis.There exist various damage detection methods.In particular,a kind of vibration-based methods has attracted much attention.The basic idea of this kind of methods is that the dynamic features(e.g.,natural frequencies and mode shapes)of a structure can be changed due to structural damages.Recently,the damage detection methods based on natural frequencies and mode shapes have been developed.In 2010,Li et al.presented the generalized flexibility matrix method.However,few natural frequencies and mode shapes can be measured by using structural dynamics experiments.Moreover,the measured mode shapes are often incomplete.On the other hand,there exist only a few damage elements.This poses a challenge for the structural damage detection.In this thesis,based on the incomplete mode shapes,a generalized flexibility matrix-based hard thresholding pursuit method is proposed for detecting the few damage locations and damage severity.Compared with the mode-shape expansion methods,the proposed method needs much less components of the involved mode shapes.Finally,numerical examples show that the proposed damage detection method can detect the locations of damaged elements effectively and provide the reliable prediction of damage severity.In addition,compared with the flexibility matrix method and the generalized flexibility matrix method with complete mode shapes,the proposed method needs less sensors and can also exactly find the locations of damaged elements.