Numerical Methods For The Inverse Quadratic Eigenvalue Problems And Its Applications

Theory of algebra inverse eigenvalue problem is one of the main method to study the structural dynamics modification. Recently, with the partially prescribed eigendata, how to preserve the positive semi-definiteness and the sparsity of the structured matrix simultaneously in the model updating problems(MUP) has become an important research topic. This paper shows that the proximal point method(PPA) and the alternating direction method(ADM) can be used to solve the structured inverse quadratic eigenvalue problems(SQIEP) with the positive semi-definiteness and the sparsity requirements, and discuss the applications for the finite element model updating problems of the damped vibrating systems and the undamped gyroscopic systems. It provides the mathematical theory and efficient numerical methods for QIEP and MUP. The main contribution is as follows:With the mass matrix exact and the partially prescribed eigendata, it is showed that the ADM can be used to solve the monic inverse quadratic eigenvalue problem(MQIEP) which preserves some properties simultaneously, including symmetry, positive semi-definiteness and sparsity. With the special structure offered by the constraint set, the conditions for the solvability of MQIEP are discussed.By combining the ADM and the customized PPA, a general alternating direction method of multiplier(ADMM) for the MQIEP is presented. Finally, the convergence analysis is provided and the numerical examples show that the presented method works well.With the partially prescribed eigendata, the SQIEP considered in this paper concerns updating a symmetric and damped second-order finite element model so that the updated model reproduces a given set of measured eigendata, and preserves the symmetry, the positive semi-definiteness and the sparsity of the original model. By exploiting the special structure offered by the constraint set, the optimization problem for SQIEP is formulated in such a way that a proximal point like method can be used to solve the equivalent problem. And the convergence analysis is provided. Finally, the numerical examples for the damped vibrating systems show that the proposed method works well.With the partiall prescribed eigendata, the structured inverse quadratic eigenvalue problem for the undamped gyroscopic system(GQIEP) is considered, which preserves the symmetry, the anti-symmetry, the positive semi-definiteness and the sparsity of the original model simultaneously. By exploiting the special structure offered by the constraint set, a customized proximal point algorithm for GQIEP and its convergence analysis are presented. Numerical examples show that the proposed method works well.