Mathematical Sciences: Inversion and Observability of Parabolic Initial Boundary Value Problems

从系统内部各点的不完全观测到空间域，确定系统在空间和时间上演化的初始边界条件的反问题，在许多工程和科学应用中具有相当重要的意义。这个项目处理的选择理论和数值方面的这些问题的某些类别的抛物型偏微分方程。研究了保证解的唯一解和解的精确（或解析）形式所需的最小信息。它们在数值确定的系统参数和测量数据中的小误差下的稳定性使用各种拓扑，泛函分析和正则化技术进行了研究。数值程序与相应的误差范围，基于精确的解决方案也将发展。这项工作的另一个方面涉及涉及初始数据的特征函数展开和初始数据和边界数据的惠特克基数展开的方法的理论和数值分析。本文将研究特征函数法的各种正则化技术及其相应的误差范围。惠特克枢机技术本身是稳定的，因此主要目的是将研究人员最近的工作扩展到更一般的方程、域和边界数据的恢复。

The inverse problem of determining the initial boundary conditions of a system evolving in space and time from incomplete observations of the system at various points interior to the spatial domain, is of considerable importance in many engineering and scientific applications. This project deals with a selection of theoretical and numerical aspects of these problems for certain classes of parabolic partial differential equations. The minimum information needed to guarantee unique solutions and exact (or, analytic) forms of these solutions is investigated. Their stability under small errors in numerically determined system parameters and measured data are investigated using a variety of topological, functional analytic, and regularization techniques. Numerical procedures with corresponding error bounds, based on the exact solutions will also be developed. Another aspect of this work concerns the theoretical and numerical analysis of methods involving eigenfunction expansions of initial data and Whittaker Cardinal expansions of initial and boundary data. Various regularization techniques and their corresponding error bounds will be investigated for the eigenfunction method. The Whittaker Cardinal technique is inherently stable, so the main aim there is to extend the recent work of the investigators to more general equations, domains, and to the recovery of boundary data.