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

本课题致力于研究与物理和工程应用相关的三个反二次特征问题。其目的是加深对二次模型重构的理论理解和推导数值算法，从而在满足某些特定的数学条件的同时，减少由于现有技术的限制而导致的模型的不精确性和不确定性。二次模型重建中最困难的任务是满足关联约束，这些约束可以内在地继承于某种机械结构的物理可行性，或者可以外在地受到某种设计参数的期望性质的驱动。最大的挑战，也是实践中的迫切要求，就是必须仅使用最新的计算技术获得的部分特征信息来进行重建。约束模型重构反问题对于理解和管理复杂系统是必不可少的，但在可解性、灵敏度和计算量等方面仍有许多问题没有得到解答。研究者们各自对二次模型构造问题做出了重大贡献，现在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上。因此，这项拟议的工作在工程界和数学科学界都应该具有令人信服的独立兴趣。在数学建模中，根据系统的服务或预期行为来验证、确定或估计系统参数的反问题技术至关重要。本文主要研究与物理和工程应用无关的逆模型重建问题。这些问题受到科学和工业应用的强烈刺激，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，该项目将影响使用这些应用的各种行业，包括航空航天、汽车、制造和生物医学工程。这些工业面临的最大挑战是用有限的工程和计算资源制造日益改进的产品。为了令人满意地完成模型修正任务，这些行业投入了大量的资金和精力，但由于缺乏适当的理论和计算工具，往往迫使这些行业以临时方式解决自己的问题。一个改进的分析模型可以自信地用于未来的设计，是实现这一目标的必要工具。所提出的研究不仅有坚实的数学基础，而且在利用工业数据进行建模和实验方面具有重要的数学意义，应立即受到业界的欢迎。此外，在这个项目上工作了四年的学生将接受将数学和科学计算与工程和应用科学的各个领域相结合的宝贵的跨学科培训。这样的专业知识很少找到，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.