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

合作提案：科学与工程模型更新的二次逆特征值问题：理论与计算

• 批准号: 0505880
• 负责人: Moody Chu
• 金额: $ 20.59万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505880&HistoricalAwards=false
• 关键词: Collaborative; Proposal; Quadratic; Inverse; Eigenvalue

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.

该项目致力于研究三个反二次特征根问题及其与物理和工程应用的相关性。其目的是发展二次模型重构的理论理解并推导数值算法，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。二次模型重建中最困难的任务是满足相关约束，这些约束可能从本质上继承于某种机械结构的物理可行性，或者可能由某个设计参数的理想属性外在驱动。最大的挑战，也是实践中的迫切要求，是必须仅使用可通过机械结构获得的部分特征信息来进行重建。 最先进的计算技术。约束模型重构的反问题对于理解和管理复杂系统至关重要，但许多关于可解性、敏感性和计算性的问题仍未得到解答。 研究人员分别对二次模型构建问题做出了重大贡献，现在打算扩大他们的研究范围，并结合专业知识来解决这些具有挑战性的反问题。因此，这项拟议的工作应该在工程和数学科学界引起令人信服的独立兴趣。在数学建模中，根据观察到的或预期的行为验证、确定或估计系统参数的反问题技术至关重要。这项研究集中于与物理和工程应用相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，该项目将影响利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。这些行业面临的最大挑战是利用有限的工程和计算资源制造日益改进的产品。这些行业花费了大量的资金和精力来令人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临时的方式解决他们的问题。可以放心地用于未来设计的改进的分析模型是实现这一目标的重要工具。所提出的研究不仅具有坚实的数学基础，而且还具有使用工业数据进行数学建模和实验的重要意义，应该立即受到业界的欢迎。此外，从事该项目四年的学生将接受宝贵的跨学科培训，将数学和科学计算与工程和应用科学的各个领域相结合。 这种专业知识很少见，但学术界和工业界的需求都在不断增加。