Automated Structure Generation, Error Correction, and Semi-Definite Programming Techniques for Structured Quadratic Inverse Eigenvale Problems: Theory, Algorithms and Applications

Mathematical modeling has become an indispensable task in almost every discipline of sciences. However, since most of the information gathering devices or methods have only finite bandwidth, one cannot avoid the fact that the models employed often are not exact. Techniques of inverse problems that validate, determine, or estimate the parameters of the system according to its observed or expected behavior, therefore, are critically important. One of the most frequently used models in important applications including applied mechanics, electrical oscillation, vibro-acoustics, fluid mechanics, signal processing, and finite element discretization of PDEs is the notion of quadratic pencils. The inverse problem of 'constrained quadratic model reconstruction from eigeninformation' is essential for the understanding and management of complex systems, yet the fundamental understanding of either its theory or computation is still in a quite primitive state. This proposal intends to develop theoretic understanding and to implement the concept into new numerical algorithms that are effective in aspects of robustness, speed and accuracy. The ultimate goal of this project is to establish a mechanism (followed by a software package) that can automatically, systematically and universally reduce the inexactness and uncertainty within the model while maintaining feasibility conditions required by the system.

This research will take on three specific challenges for solving quadratic inverse eigenvalue problems with innovative but promising approaches among which are the automated structure generation, consistency correction, and semi-definite programming techniques. This project is expected to find important applications ranging from new development of numerical algorithms to theoretic solution of difficult problems. The resulting technology would significantly advance knowledge in the emerging field of model updating and related problems which, in turn, would have substantial impact on broad areas in scientific and engineering fields.

In mathematical modeling, techniques of inverse problems that validate, determine, or estimate the parameters of the system according to its observed or expected behavior are critically important. This research concentrates on the inverse model reconstruction problems with their pertinence to physical and engineering applications. These problems have been strongly motivated by scientific and industrial applications, including structural mechanics such as vibration control and stability analysis of bridges, buildings and highways, vibro-acoustics such as predictive coding of sound, biomedical signal and image processing, time series forecasting, information technology, and others. Thus this project will impact a wide variety of industries utilizing these applications, including aerospace, automobile, manufacturing and biomedical engineering. The greatest challenge facing these industries is to manufacture increasingly improved products with limited engineering and computing resources. A great deal of money and effort has been spent in these industries to satisfactorily perform the model updating task.

However, the lack of proper theory and computational tools often force these industries to solve their problems in an ad hoc fashion. An improved analytical model that can be used with confidence for future designs is an essential tool in achieving this objective. The proposed research has not only strong mathematical foundation but also significant mathematical modeling and experimental aspects using industrial data which should be instantly welcome by the industries. Students working on this project will receive a valuable inter-disciplinary training blending mathematics and scientific computing with various areas of engineering and applied sciences.

Such expertise is rare to find, but there is an increasing demand both in academia and industries.