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This is a review paper of the role of Carleman estimates in the theory of Multidimensional Coefficient Inverse Problems since the first inception of this idea in 1981.

This is a review paper of the role of Carleman estimates in the theory of Multidimensional Coefficient Inverse Problems since the first inception of this idea in 1981.

Coefficient Inverse Problems (CIPs) for Partial Differential Equations (PDEs) represent a very important tool for such needs of the Army as imaging of unknown targets hidden in cluttered heterogeneous backgrounds. The goal of this project is the development of globally convergent numerical methods for a wide class of CIPs. These methods are tested on mathematical models of the interest to the Army such as imaging of antipersonnel land mines and targets on battlefields covered by smogs and flames. In our definition global convergence entails: (1) a rigorous convergence analysis that does not depend on the quality of the initial guess, and (2) numerical simulations that confirm the advertised convergence property. A conventional way to solve a CIP is via the minimization of a least squares objective functional. This functional characterizes misfit between the data and the solution of that PDE with a guess for the unknown coefficient. However, it is well known to researchers working on computations of inverse problems that the phenomenon of multiple local minima of these functionals represents the major obstacle for the development of reliable numerical methods for multidimensional CIPs. This phenomenon in turn is caused by the above mentioned non-linearity and ill-posedness. Because of local minima, one should somehow guess in advance about a good approximation for the solution. Without the availability of a first good guess, however, there is no guarantee that the calculated coefficient is indeed close to the correct one. In our terminology these are locally convergent numerical methods. In other words, their convergence to the correct solution can be guaranteed only if the starting point is located in a small neighborhood of this solution. Because of local minima, conventional numerical methods for multidimensional CIPs are locally convergent ones. However, in many important applications the first good guess is unavailable.

The objective was to try to develop a useful numerical algorithm for the Drazin inverse and to analyze the numerical aspects of the applications of the Drazin inverse relating to the study of homogeneous Markov chains and systems of linear differential equations with singular coefficient matrices. It is felt that all objectives were accomplished with a measurable degree of success.

v.p p. 28 cm

55 p. 28 cm

We consider the inverse problem of determining the highly oscillatory coefficient $a^\epsilon$ in partial differential equations of the form $-\nabla\cdot (a^\epsilon\nabla u^\epsilon)+bu^\epsilon = f$ from given measurements of the solutions. Here, $\epsilon$ indicates the smallest characteristic wavelength in the problem ($0 0$, and exploration seismology, $b < 0$.