Shape Reconstruction and Other Inverse Problems

本项目致力于在线性和非线性逆/DIV问题的数值解中进一步发展/DIV技术。这是一个丰富的研究/DIV领域，应用于科学和/DIV工程中的许多问题。特别是，该项目感兴趣的一个领域是涉及/DIV矩的一大类反问题。例如，在给定其矩的情况下重建平面区域的形状的问题是一个具有挑战性的数学和/DIV数值问题，在计算机层析成像、/DIV地球物理勘探和热成像等方面都有应用。例如，/DIV是线性和非线性最小二乘问题的有效解，/DIV和这类问题的正则化概念在许多/DIV应用中是重要的。人们普遍认为，a/DIV正则化泛函和正则化参数的选择都是稳定的数值反演方法成功的关键。/DIV项目涉及病态逆问题的数值稳定解/DIV，特别是涉及/DIV矩的逆问题。一个具体的问题是从它的矩恢复2维/DIV形状，涉及/DIV(1)从矩/DIV重建形状的正则化反演方案的公式和解，以及(2)将这些/DIV算法应用于根据/DIV测量的重力场重建重力异常

This project seeks to further the state of the/DIV art in the numerical solution of linear and nonlinear inverse/DIV problems. This is a rich area of research/DIV with applications to a great many problems in science and/DIV engineering. In particular, one area of interest in the project is the large class of inverse problems involving/DIV moments. For instance, the problem of reconstructing the shape of a/DIV planar region given its moments is a challenging mathematical and/DIV numerical problem, with applications in computed tomography,/DIV geophysical prospecting, and thermal imaging, to name just a few./DIVGenerally, the solution of (ill-posed) inverse problems/DIV requires the use of advanced computational techniques. For instance,/DIV the efficient solution of linear and nonlinear least squares problems,/DIV and the concept of regularization of such problems are important in many /DIVapplications. It is widely recognized that the choice of a/DIV regularizing functional and the regularization parameters are both/DIV central to the success of a stable numerical inversion approach. The/DIV project is concerned with the numerically stable solution/DIV of ill-posed inverse problems, particularly those involving/DIV moments. A specific problem is that of recovering a 2-dimensional/DIV shape from its moments, involving /DIV(1) formulation and solution of regularized inversion schemes for the/DIV reconstruction of shape from moments, and (2) applying these/DIV algorithms to the reconstruction of gravitational anomalies from/DIV measurements of gravitational fields