Discovery of new methods that may yield reconstruction algorithms in inverse problems

1.包络法及其正则化的数值实现从控制方程解的一组柯西数据出发，给出了多边形腔或包裹体凸包提取公式的数值实现。为了解释数值结果，我们考虑了当数据含有误差时如何对公式进行修正，并给出了修正公式。探针法的发展我们考虑了混合型边值问题的反问题。通过研究所谓反射解的详细行为，我们发现首席调查者发现的探测法可以应用于这些问题。特别地，我们给出了De Giorgi-Nash-Moser定理在探测方法中的一个应用。我们发现了一种基于Mittag-Leffler函数的解析性和渐近性的方法，该方法从Dirichlet-to-Neumann映射中得到未知包含边界的可见部分，并将该方法应用于一个交替电流的数学模型。包络法在多层材料反问题中的应用我们考虑了从Dirichlet-to-Neumann映射中提取未知夹杂信息的问题，该夹杂嵌入在具有不同恒定电导率的背景层状材料中。为此，我们构造了背景材料控制方程的指数增长解，并研究了它们的渐近行为。利用这些解的性质，我们给出了一个求包含凸包的公式。

1. A numerical implementation of the enclosure method and its regularization We made a numerical implementation of an extraction formula of the convex hull of polygonal cavities or inclusions from a single set of the Cauchy data of a solution of the governing equation. In order to explain the numerical results we considered how to modify the formula when the data contain error and gave a modified formula.2. Development of the probe methodWe considered inverse problems for the mixed type boundary value problems. By studying the detailed behavior of the so-called reflected solutions, we found that the probe method discovered by the head investigator can be applied to those problems. In particular, we gave an application of the De Giorgi-Nash-Moser theorem to the probe method.3. A generalization of the enclosure methodWe discovered a method that is based on the analyticity and asymptotic behavior of Mittag-Leffler's function and yields the visible parts of the boundary of unknown inclusions from the Dirichlet-to-Neumann map.Moreover we applied the method to a mathematical model for alternative current.4 . An application of the enclosure method to an inverse problem for the multilayered materialWe considered the problem of extracting information about unknown inclusions embedded in a background layered material that has different constant conductivities across finitely many parallel planes, from the Dirichlet-to-Neumann map. For the purpose we constructed the exponentially growing solutions of the governing equation for the back ground material and studied their asymptotic behavior. Using the property of those solutions, we gave an extraction formula of the convex hull of the inclusions .