Collaborative Proposal: Quadratic Inverse Eigenvalue Problems for Model Updating in Science and Engineering: Theory and Computation

本项目致力于研究三个具有物理和工程应用价值的二次特征值反问题。目的是发展理论理解和推导二次模型重建的数值算法，以便在满足某些特定数学条件的同时，减少由于当前技术的限制而导致的模型中固有的不精确性和不确定性。二次模型重构中最困难的问题是满足相关的约束条件，这些约束条件可能是从某种机械结构的物理可行性中固有的，也可能是由某种设计参数的期望性质所驱动的，最大的挑战也是实践中的迫切要求，是重建必须只使用部分特征信息，这是由国家的最先进的计算技术。约束模型重构的逆问题对于理解和管理复杂系统是必不可少的，但其可解性、灵敏度和计算量等问题仍有待解决。 研究人员已经取得了显着的贡献，二次模型的建设problemsindividually，现在打算扩大他们的调查和joinexpertise这些具有挑战性的逆问题。因此，这一建议的工作应该是令人信服的独立的兴趣在boththe工程和数学科学community.In数学建模，技术的反问题，验证，确定，或估计系统的参数，根据itsobserved或预期的行为是至关重要的。本文主要研究具有物理意义和工程应用价值的逆模型重建问题.这些问题已经强烈地被科学和工业应用所激发，包括结构力学如桥梁、建筑物和高速公路的振动控制和稳定性分析，振动声学如声音的预测编码，生物医学信号和图像处理，时间序列预测，信息技术等。因此，该项目将影响到利用这些应用的各种行业，包括航空航天、汽车、制造业和生物医学工程。这些行业面临的最大挑战是用有限的工程和计算资源制造越来越多的改进产品。这些行业已经花费了大量的金钱和精力来满意地执行模型更新任务，然而，缺乏适当的理论和计算工具往往迫使这些行业以特设的方式解决他们的问题。一个改进的分析模型，可以用于未来的设计信心是一个重要的工具，在实现这一目标。所提出的研究不仅具有强大的数学基础，而且具有重要的数学建模和实验方面，使用IDUNOMIC数据，这应该立即受到业界的欢迎。此外，在这个项目上工作四年的学生将获得一个有价值的跨学科培训，将数学和科学计算与工程和应用科学的各个领域相结合。 这样的专业知识很难找到，但在学术界和工业界都有越来越多的需求。

This project is devoted to the study of three inverse quadraticeigenproblems with their pertinence to physical and engineeringapplications. The aim is to develop theoretic understanding andderive numerical algorithms for the quadratic model reconstructionso that the inexactness and uncertainty inherent in the model dueto the limitation of current technologies are reduced while certainspecific mathematical conditions are satisfied. The most difficulttask in the quadratic model reconstruction is to satisfy the associatedconstraints which could be inherited intrinsically from the physicalfeasibility of a certain mechanical structure or could be drivenextrinsically by the desirable property of a certain design parameter.The greatest challenge, which is also an imperative requirement inpractice, is that the reconstruction must be carried out using onlypartial eigeninformation which are available by the state-of-the-artcomputational techniques. The inverse problem of constrained modelreconstruction is essential for the understanding and management ofcomplex systems, yet many questions on the solvability, sensitivity,and computation remain unanswered. The investigators have madesignificant contributions to the quadratic model construction problemsindividually and now intend to extend their investigation and joinexpertise to these challenging inverse problems. This proposed worktherefore should be of compelling independent interest within boththe engineering and mathematical sciences communities.In mathematical modelling, techniques of inverse problems that validate,determine, or estimate the parameters of the system according to itsobserved or expected behavior are critically important. This researchconcentrates on the inverse model reconstruction problems with theirpertinence to physical and engineering applications. These problems havebeen strongly motivitated by scietific and industrial applications,including structural mechanics such as vibration control and stabilityanalysis of bridges, buildings and highways, vibro-acoustics such aspredictive coding of sound, biomedical signal and image processing,time series forecasting, information technology, and others. Thus thisproject will impact a wide variety of industries utilizing theseapplications, including aerospace, automobile, manufacturing andbiomedical engineering. The greatest challenge facing these industriesis to manufacture increasingly improved products with limited engineeringand computing resources. A great deal of money and efforts have been spentin these industries to satisactorily perform the model updating task.However, the lack of proper theory and computational tools often forcethese industries to solve their problems in an ad hoc fashion. An improvedanalytical model that can be used with confidence for future designs isan essential tool in achieving this obejective. The propsed research hasnot only strong mathematical foundation but also significant matematicalmodelling and experimental aspects using idustrial data which should beinstantly welcome by the industries. Furthermore, the students workingon this project for four years will receive a valuable interdisciplnarytraining blending mathematics and scietific computing with various areasof engineering and applied sciences. Such expertise is rare to find,but there is an increasing demand both inacademia and industries.