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

结构化二次反特征值问题的自动结构生成、纠错和半定编程技术：理论、算法和应用

• 批准号: 1014666
• 负责人: Moody Chu
• 金额: $ 19.5万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1014666&HistoricalAwards=false
• 关键词: Automated; Structure; Generation; Error; Correction

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.

数学建模已经成为几乎所有科学学科中不可或缺的任务。然而，由于大多数信息收集设备或方法只有有限的带宽，人们无法避免所采用的模型往往不准确的事实。因此，根据系统的观察或预期行为来验证、确定或估计系统参数的逆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组的有限元离散等重要应用中，最常用的模型之一是二次铅笔的概念。“基于特征信息的约束二次模型重构”反问题对于理解和处理复杂系统是必不可少的，但对其理论和计算的基本理解仍处于相当原始的状态。这一建议旨在发展理论上的理解，并将这一概念落实到新的数值算法中，这些算法在稳健性、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种机制(随后是一个软件包)，在保持系统所要求的可行性条件的同时，自动、系统地、普遍地减少模型中的不准确性和不确定性。本研究将用创新但有前途的方法来解决二次特征值反问题的三个具体挑战，其中包括结构自动生成、一致性校正和半定规划技术。从数值算法的新发展到疑难问题的理论解决，这个项目有望得到重要的应用。在数学建模中，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逆问题技术至关重要。本文主要研究具有物理和工程应用意义的逆模型重建问题。这些问题受到科学和工业应用的强烈推动，包括桥梁、建筑物和高速公路的振动控制和稳定性分析等结构力学，声音预测编码等振动声学，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。这些行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。这些行业花费了大量的资金和精力来圆满地完成模型修正任务，但由于缺乏适当的理论和计算工具，往往迫使这些行业临时解决问题。改进的分析模型可以放心地用于未来的设计，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，而且在利用工业数据进行数学建模和实验方面也具有重要意义，应该立即受到业界的欢迎。从事这个项目的学生将接受一项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结合起来。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。