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.

在几乎所有科学学科中，数学建模已成为必不可少的任务。但是，由于大多数信息收集设备或方法仅具有有限的带宽，因此无法避免这样的事实，即使用的模型通常不准确。 因此，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。重要应用中最常用的模型之一，包括应用力学，电振荡，振荡声学，流体力学，信号处理以及PDES的有限元离散化是二次铅笔的概念。 “从征征形成的约束二次模型重建”的反问题对于对复杂系统的理解和管理至关重要，但是对其理论或计算的基本理解仍然处于相当原始的状态。该建议旨在发展理论理解并将概念实施到在鲁棒性，速度和准确性方面有效的新数值算法中。该项目的最终目标是建立一种机制（随后是软件套件），该机制可以自动，系统和普遍地降低模型内的不确定性和不确定性，同时保持系统所需的可行性条件。这项研究将面临三个特定的挑战，以解决三个特定的挑战，以解决求解典型的特征性且有望的连续性和一致性的，并且是一致的，并且是一致的，并且是一致的，并且是一致的，并且一致的一致及时，并且是一致的一致性，并且是一致的，并且是一致的，并且是一致的，并且是一致的，并且是一致的，并且是一致的，并且一致地界定了一致的差异，并且技术。预计该项目将找到从数值算法的新开发到困难问题的理论解决方案的重要应用。最终的技术将大大提高模型更新和相关问题的新知识，这反过来又将对科学和工程领域的广泛领域产生重大影响。在数学建模中，根据观察到的或预期的行为验证或估算该系统的参数的逆问题技术非常重要。这项研究集中在与物理和工程应用有关的逆模型重建问题上。这些问题是由科学和工业应用的强烈动机，包括桥梁，建筑物和高速公路的振动控制和稳定分析，诸如声音，生物医学信号和图像处理的预测编码，时间序列预测，信息技术等的结构力学分析。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使用有限的工程和计算资源生产越来越多的改进产品。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。但是，缺乏适当的理论和计算工具通常会迫使这些行业以临时的方式解决他们的问题。改进的分析模型可以充满信心地用于未来的设计，是实现这一目标的重要工具。拟议的研究不仅具有强大的数学基础，而且还具有重要的数学建模和实验方面，并使用工业数据应立即受到行业的欢迎。从事该项目的学生将获得有价值的跨学科培训融合数学和科学计算与各个工程和应用科学领域的融合。