Mathematical Sciences:Reconstructions Methods for Inverse Problems in Multiple Dimensions

许多有物理意义的物体不能直接研究。例如，人体内部成像，确定固体物体内部的裂缝，以及材料参数，如不可接近物体的导电性。当这些问题被翻译成数学术语时，它们采用偏微分方程的形式，这是数学科学的通用语言。然而，由于我们在模型中有额外的未知数，这些在方程中引入了未知参数，这些参数必须通过进一步的测量来额外求解。在这个建议中，我们从数学的角度来处理这个问题的实际方面：我们感兴趣的问题是什么时候可以做出一个唯一的决定，以及设计算法来有效地恢复未知的数值。我们将研究从多维的二阶偏微分算子中恢复未知系数的方法。我们的主要目标是证明唯一性结果，并为几个特殊情况开发建设性的恢复方法。在实践中，这些问题中的每一个都归结为在适当选择的有限维空间中从有限数量的数据测量中唯一地恢复一个系数。这将反问题简化为有限维数据到系数映射的反演。除了开发重建方法外，还将研究这些方法对数据扰动的稳定性（或灵敏度）以及它们的计算成本。＊＊＊

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. ***