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

确定初始边界的反问题 一个系统在空间和时间上的演化条件， 在不同点对系统的不完整观测 在空间领域内部， 在许多工程和科学应用中。 该项目涉及一系列理论和 这些问题的数值方面的某些类别的 抛物型偏微分方程 最低 保证唯一解决方案和精确（或， 解析）的形式，这些解决方案进行了调查。 他们的 数值确定系统小误差稳定性 参数和测量数据进行了研究，使用各种 拓扑、泛函分析和正则化 技术. 具有相应误差的数值过程 边界，基于精确的解决方案也将开发。 这项工作的另一个方面涉及理论和 本征函数法的数值分析 初始数据的展开和Whittaker Cardinal展开 初始和边界数据。 各种正则化技术 并将研究其相应的误差范围， 特征函数法 Whittaker Cardinal技术 本质上是稳定的，所以主要目的是扩展 研究人员最近的工作更一般的方程， 域，以及边界数据的恢复。

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.