Mathematical Sciences:Reconstructions Methods for Inverse Problems in Multiple Dimensions

朗德尔9501030许多有物理意义的物体不能直接研究。例如，成像身体内部，确定固体物体内的裂缝，以及材料参数，如无法接触到的物体的导电性。当这些问题被转化为数学术语时，它们就会以偏微分方程式的形式出现，这是数学科学中的语言。然而，由于我们在模型中有额外的未知数，这些未知数在方程中引入了未知参数，必须通过进一步的测量来额外求解。在这项提案中，我们从数学的角度处理这方面的实际问题：我们感兴趣的是何时可以作出独特的确定以及设计有效的未知数字恢复算法的问题。%研究从多维二阶偏微分算子中恢复未知系数。我们的主要目标是证明结果的唯一性，并为几个特定的案例开发建设性的恢复方法。在实践中，这些问题中的每一个都简化为从有限数量的数据测量中唯一地恢复适当选择的有限维空间中的系数。这将逆问题简化为有限维中数据到系数映射的逆问题。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(或敏感性)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. ***