Mathematical Sciences: Inverse Eigenvalue Problems

[9422280] PRIVATE本研究涉及特征值反问题。从规定的光谱数据中建立一个以矩阵形式描述的物理模型的理论和实践都将被研究。这一建议延续并扩展了以前的研究。特别地，将为每个反特征值问题制定一个动力系统，以确保特定任务连续发生。然后将使用计算机实验、高分辨率图形和符号操作来探索动力学，并结合分析。这项研究将导致对特征值反问题理解的根本性进展。这个项目有望找到重要的应用范围，从数值算法的新发展到困难问题的理论解决。由于特征值反问题出现在各种各样的学科中，由此产生的技术将对许多科学和工程领域产生影响。特征值反问题在控制设计、隔振、系统识别、勘探与遥感以及信号处理等许多重要应用中都有出现。在所有这些应用领域中，一个重要的共同现象是，某个系统的物理参数是根据其动力学行为的知识或期望来重建的，特别是其固有频率和/或正常模式。从这项拟议的工作中取得的进展将有助于理解几个战略领域中出现的问题，特别是在全球变化、制造业、运输和环境监测等领域。

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.