Automated Structure Generation, Error Correction, and Semi-Definite Programming Techniques for Structured Quadratic Inverse Eigenvale Problems: Theory, Algorithms and Applications

数学建模已成为几乎每一门科学学科中不可缺少的任务。然而，由于大多数信息收集设备或方法只有有限的带宽，人们不能避免的事实是，所采用的模型往往是不准确的。 因此，根据系统的观测或预期行为来验证、确定或估计系统参数的反问题技术是至关重要的。在应用力学、电振荡、振动声学、流体力学、信号处理和偏微分方程的有限元离散化等重要应用中，最常用的模型之一是二次束的概念。"由特征信息重构约束二次模型"的反问题对于理解和管理复杂系统是必不可少的，但无论是理论还是计算，对它的基本理解都还处于相当原始的状态。该建议旨在发展理论上的理解，并将概念实现到新的数值算法中，这些算法在鲁棒性，速度和精度方面都是有效的。这个项目的最终目标是建立一个机制（随后是一个软件包），可以自动、系统和普遍地减少模型中的不精确性和不确定性，同时保持系统所需的可行性条件。这项研究将采用创新但有前途的方法来解决二次逆特征值问题，其中包括自动结构生成、一致性校正和半定规划技术。该项目有望在从数值算法的新发展到困难问题的理论解决等方面找到重要的应用。由此产生的技术将显着推进知识的新兴领域的模型更新和相关的问题，反过来，将有实质性的影响，在科学和工程领域的广泛领域。在数学建模，技术的逆问题，验证，确定，或估计参数的系统，根据其观察到的或预期的行为是至关重要的。本研究主要针对物理和工程应用中的逆模型重建问题进行研究。这些问题已经强烈地受到科学和工业应用的激励，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来制造日益改进的产品。这些行业已经花费了大量的金钱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特定的方式解决问题。一个改进的分析模型，可以用于未来的设计信心是实现这一目标的一个重要工具。所提出的研究不仅具有坚实的数学基础，而且具有重要的数学建模和实验方面，使用工业数据，应该立即受到业界的欢迎。从事该项目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。这种专业知识很难找到，但学术界和工业界的需求都在增加。

Mathematical modeling has become an indispensable task in almost every discipline of sciences. However, since most of the information gathering devices or methods have only finite bandwidth, one cannot avoid the fact that the models employed often are not exact. Techniques of inverse problems that validate, determine, or estimate the parameters of the system according to its observed or expected behavior, therefore, are critically important. One of the most frequently used models in important applications including applied mechanics, electrical oscillation, vibro-acoustics, fluid mechanics, signal processing, and finite element discretization of PDEs is the notion of quadratic pencils. The inverse problem of "constrained quadratic model reconstruction from eigeninformation" is essential for the understanding and management of complex systems, yet the fundamental understanding of either its theory or computation is still in a quite primitive state. This proposal intends to develop theoretic understanding and to implement the concept into new numerical algorithms that are effective in aspects of robustness, speed and accuracy. The ultimate goal of this project is to establish a mechanism (followed by a software package) that can automatically, systematically and universally reduce the inexactness and uncertainty within the model while maintaining feasibility conditions required by the system.This research will take on three specific challenges for solving quadratic inverse eigenvalue problems with innovative but promising approaches among which are the automated structure generation, consistency correction, and semi-definite programming techniques. This project is expected to find important applications ranging from new development of numerical algorithms to theoretic solution of difficult problems. The resulting technology would significantly advance knowledge in the emerging field of model updating and related problems which, in turn, would have substantial impact on broad areas in scientific and engineering fields.In mathematical modeling, 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 motivated by scientific 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 effort has been spent in these industries to satisfactorily 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 objective. The proposed research has not only strong mathematical foundation but also significant mathematical modeling and experimental aspects using industrial data which should be instantly welcome by the industries. Students working on this project will receive a valuable inter-disciplinary training blending mathematics and scientific computing with various areas of engineering and applied sciences.Such expertise is rare to find, but there is an increasing demand both in academia and industries.