This project is devoted to the study of three inverse quadratic eigenproblems with their pertinence to physical and engineering applications. The aim is to develop theoretic understanding and derive numerical algorithms for the quadratic model reconstruction so that the inexactness and uncertainty inherent in the model due to the limitation of current technologies are reduced while certain specific mathematical conditions are satisfied. The most difficult task in the quadratic model reconstruction is to satisfy the associated constraints which could be inherited intrinsically from the physical feasibility of a certain mechanical structure or could be driven extrinsically by the desirable property of a certain design parameter. The greatest challenge, which is also an imperative requirement in practice, is that the reconstruction must be carried out using only partial eigeninformation which are available by the state-of-the-art computational techniques. The inverse problem of constrained model reconstruction is essential for the understanding and management of complex systems, yet many questions on the solvability, sensitivity, and computation remain unanswered. The investigators have made significant contributions to the quadratic model construction problems individually and now intend to extend their investigation and join expertise to these challenging inverse problems. This proposed work therefore should be of compelling independent interest within both the engineering and mathematical sciences communities.
In mathematical modelling, 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 motivitated by scietific 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 efforts have been spent in these industries to satisactorily 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 obejective. The propsed research has not only strong mathematical foundation but also significant matematical modelling and experimental aspects using idustrial data which should be instantly welcome by the industries. Furthermore, the students working on this project for four years will receive a valuable interdisciplnary training blending mathematics and scietific computing with various areas of engineering and applied sciences. Such expertise is rare to find, but there is an increasing demand both inacademia and industries.
