Optimization Approaches for Inverse Quadratic Eigenvalue Problems


Zheng-Jian Bai

School of Mathematical Sciences
Xiamen University
Xiamen 561005, China
Email: zjbai@xmu.edu.cn

Abstract :  The inverse quadratic eigenvalue problem (IQEP) arises in the field of structural mechanics and vibrating structure. It aims to find three matrices, known as the mass, the damping, and the stiffness matrices, respectively such that they satisfy the measured data and preserve the exploitable structural properties such as symmetry, definiteness, sparsity and bandedness, etc., of the original model. The difficulty of this problem lies in the fact that in applications the mass, damping, and stiffness matrices should satisfy the requirement(s) of definiteness and/or bandedness.
In this talk, we first consider the IQEP where the mass matrix should be positive definite, the damping matrix symmetric, and the stiffness matrix positive semidefinite. Based on an equivalent dual optimization version of the IQEP, we present a quadratically convergent Newton-type method. Our numerical experiments confirm the high efficiency of the proposed method.

Next, we discuss the inverse problem for a discrete damped mass-spring system where the mass, damping, and stiffness matrices are all symmetric and tridiagonal. It is shown that the model can be constructed from two real eigenvalues and three real eigenvectors or a complex conjugate eigenpair and a real eigenvector. However, for large model order, the construction from these data may be sensitive to perturbations. To reduce the sensitivity, we fit an least-squares optimization problem to the overdetermined noised data. In addition, The physical realizability of the required  model is obtained by solving a positiveness-constrained least-squares optimization problem to the overdetermined corrupted eigendata.

Lecture Slide