Spectral analysis of the anti-reflective algebra
详细信息 查看全文
- 作者：A. Aricò ; M. Donatelli ; Serra Capizzano
- 关键词：Inverse problem ; Vibration system ; Ill-posedness ; Regularization
- 刊名：Linear Algebra and its Applications
- 出版年：2008
- 出版时间：15 January 2008
- 年：2008
- 卷：428
- 期：2-3
- 页码：657-675
- 全文大小：244 K
文摘
In this paper we consider the inverse problem for a discrete damped mass–spring system where the mass, damping, and stiffness matrices are all symmetric tridiagonal. We first show that the model can be constructed from two real eigenvalues and three real eigenvectors or two complex conjugate eigenpairs and a real eigenvector. Then, we study the general under-determined and over-determined problems. In particular, we provide the sufficient and necessary conditions on the given two real or complex conjugate eigenpairs so that the under-determined problem has a physical solution. However, for large model order, the construction from these data may be sensitive to perturbations. To reduce the sensitivity, we propose the minimum norm solution over the under-determined noisy data and the least squares solution to the over-determined measured data. We also discuss the physical realizability of the required model by the positivity-constrained regularization method for the ill-posed under-determined problem and the least squares optimization problems with positivity-constraints for the ill-posed over-determined problem. Finally, we give simple numerical examples to illustrate the effectiveness of our methods.