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

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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.