Shape Reconstruction and Other Inverse Problems

该项目旨在进一步发展/DIV技术在线性和非线性逆/DIV问题的数值解。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，在该项目中感兴趣的一个领域是大类的反问题，涉及/DIV时刻。例如，重建给定矩的/DIV平面区域的形状的问题是具有挑战性的数学和/DIV数值问题，其应用于计算机断层摄影、/DIV地球物理勘探和热成像，仅举几例。一般来说，（不适定）反问题/DIV的解决方案需要使用先进的计算技术。例如，/DIV线性和非线性最小二乘问题的有效解，/DIV和这些问题的正则化概念在许多/DIV应用中很重要。人们普遍认为，选择一个/DIV正则化功能和正则化参数都是/DIV中心的一个稳定的数值反演方法的成功。/DIV项目关注的是不适定反问题的数值稳定解/DIV，特别是涉及/DIV矩的反问题。一个具体的问题是从其矩恢复二维/DIV形状，涉及/DIV（1）用于从矩重建形状的/DIV正则化反演方案的公式化和求解，以及（2）将这些/DIV算法应用于从重力场的/DIV测量重建重力异常

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