Various Inverse Problems in Partial Differential Equations and Methods for their Solutions

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9971674RundellThis proposal investigates several inverse problems for partial differential equations. Our particular focus is in uniqueness results that seek the minimal amount of data required to determine unknown coefficients lying within certain predetermined classes. We are also interested in reconstruction methods based on iteration schemes and in particular to schemes that utilize higher derivatives of the coefficient to data map. Finally, for certain undetermined coefficient problems we propose to investigate the following: given information (or assumptions) on the error in the data, that is, what type of distribution it obeys and what are the significant moments, what is the distribution of the possible reconstructions of the coefficient?Many objects of physical interest cannot be studied directly. Examples include imaging the interior of the body, the determination of cracks within solid objects, and material parameters such as the conductivity of inaccessible objects. When these problems are translated into mathematical terms they take the form of partial differential equations, the Lingua Franca of the mathematical sciences. The additional unknowns in the model translate into unknown parameters in the equations and we attempt to solve for these by means of further measurements. In this proposal we deal with the practical aspects of this from a mathematical perspective: We are interested in the question of when a unique determination can be made, as well as designing algorithms for the efficient numerical recovery of the unknowns. The mathematical techniques used and to be developed are applicable to several different scientific problems. Examples are: (1) The recovery of the location and shape of interior objects from surface measurements. Applications here include the determination of tumors (which are characterized by a greater metabolic rate) and buried mines and munitions (characterized by a higher electrical conductivity). (2) The determination of the interior density profile of the sun from spectral data. This is important because it would provide an independent check of our current understanding of both the composition of a star such as the sun and the underlying physics with which we model its internal reactions.