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

本课题致力于研究三个与物理和工程应用相关的反二次特征问题。目的是发展理论认识和推导二次模型重建的数值算法，从而在满足某些特定数学条件的同时，减少由于现有技术的限制而导致的模型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如何满足相关的约束条件，这些约束条件可以从内在上继承自某个机械结构的物理可行性，也可以由外部驱动于某个设计参数的期望性质。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，而这些信息是由最先进的计算技术提供的。约束模型重构的逆问题对于复杂系统的理解和管理至关重要，但在可解性、敏感性和计算方面仍有许多问题没有得到解决。研究者们已经对二次模型的构建问题做出了重要的贡献，现在他们打算将他们的研究和联合专业知识扩展到这些具有挑战性的逆问题。因此，这项提议的工作应该在工程和数学科学界都有引人注目的独立兴趣。在数学建模中，根据观察到的或预期的行为来验证、确定或估计系统参数的反问题技术是至关重要的。本文主要研究具有物理和工程应用针对性的逆模型重构问题。这些问题已经受到科学和工业应用的强烈推动，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在有限的工程和计算资源下制造日益改进的产品。为了令人满意地执行模型更新任务，这些行业已经投入了大量的资金和精力。然而，缺乏适当的理论和计算工具往往迫使这些行业以一种特殊的方式解决他们的问题。在实现这一目标的过程中，一个改进的分析模型是一个重要的工具，它可以用于未来的设计。所提出的研究不仅具有强大的数学基础，而且在利用工业数据进行数学建模和实验方面具有重要意义，应立即受到工业界的欢迎。此外，在这个项目中工作四年的学生将获得一个有价值的跨学科训练，将数学和科学计算与工程和应用科学的各个领域结合起来。这样的专业知识很难找到，但学术界和工业界的需求都在不断增加。

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.