Mathematical Sciences:Reconstructions Methods for Inverse Problems in Multiple Dimensions

点击翻译按钮获取中文摘要

Rundell 9501030 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. However, since we have additional unknowns in the model, these introduce unknown parameters in the equations that have to be additionally solved for 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 recovery of unknown coefficients from second order partial differential operators in multi-dimensions will be investigated. Our main goal is to prove uniqueness results and to develop constructive recovery methods for several particular cases. In practice, each of these problems reduce to uniquely recovering a coefficient in an appropriately chosen finite dimensional space from a finite number of data measurements. This reduces the inverse problem to the inversion of the data-to-coefficient map in finite dimensions. In addition to developing reconstruction methods, the stability (or sensitivity) of these methods to perturbations of the data, as well as their computational cost, will be studied. ***