Mathematical Sciences: Inverse Eigenvalue Problems

9422280楚私本研究涉及到特征值逆问题。从规定的光谱数据构建物理模型的理论和实践都将被研究，该模型以矩阵的形式以数学形式描述。这一建议延续并扩展了先前的研究。具体地说，将为每个逆特征值问题建立一个动力系统，以确保特定任务持续进行。然后将使用计算机实验、高分辨率图形和符号处理，并结合分析来探索动力学。这项研究应该会在理解逆特征值问题方面取得根本性的进步。这个项目有望找到重要的应用，从数值算法的新发展到困难问题的理论解决方案。由于逆本征值问题产生于各种各样的学科，由此产生的技术将对许多科学和工程领域产生影响。特征值逆问题在控制设计、隔振、系统辨识、探测与遥感、信号处理等领域有着广泛的应用。在所有这些应用领域中的一个重要的普遍现象是，某个系统的物理参数是根据对其动力学行为的知识或期望来重构的，特别是其固有频率和/或简正模式。从这项拟议工作中取得的进展将有助于理解在若干战略领域出现的问题，特别是在全球变化、制造、运输和环境监测领域。

9422280 Chu PRIVATE This research is concerned with inverse eigenvalue problems. Both the theory and the practice of constructing a physical model, described mathematically in the form of a matrix, from prescribed spectral data will be investigated. This proposal continues and extends previous studies. In particular, a dynamical system for each inverse eigenvalue problem will be formulated to ensure that a specific task is taking place continuously. The dynamics will then be explored using computer experiments, high resolution graphics and symbolic manipulation, in conjunction with analysis. This study should lead to fundamental advances in the understanding of inverse eigenvalue problems. This project is expected to find important applications ranging from new development of numerical algorithms to theoretical solutions of difficult problems. Since inverse eigenvalue problems arise from a wide variety of disciplines, the resulting technology would have impact on a number of scientific and engineering fields. Inverse eigenvalue problems arise in many important applications, including control design, vibration isolation, system identification, exploration and remote sensing, and signal processing. A significant common phenomenon in all these areas of application is that the physical parameter of a certain system is to be reconstructed from knowledge or expectation of its dynamical behavior, in particular its natural frequencies and/or normal modes. The advances obtained from this proposed work will contribute to the understanding of issues that arise in several strategic areas, particularly in the areas of global change, manufacturing, transportation, and environmental monitoring.