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Shape Reconstruction and Other Inverse Problems

形状重建和其他反问题

基本信息

批准号：
9971010
负责人：
Gene Golub
金额：
$ 44.68万
依托单位：
Stanford University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1999
资助国家：
美国
起止时间：
1999-09-01 至 2003-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971010&HistoricalAwards=false
关键词：
Shape Reconstruction Other Inverse Problems

项目摘要

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

该项目旨在进一步发展/DIV技术在线性和非线性逆/DIV问题的数值解中的地位。这是一个丰富的研究领域，应用于科学和工程中的许多问题。特别地，该项目中一个感兴趣的领域是涉及/DIV矩的一大类逆问题。例如，给定一个/DIV平面区域的矩重建问题是一个具有挑战性的数学和/DIV数值问题，应用于计算机断层扫描，/DIV地球物理勘探和热成像，仅举几例。一般来说，求解（不适定）逆问题需要使用先进的计算技术。例如，线性和非线性最小二乘问题的有效解，以及这些问题的正则化概念在许多应用中都很重要。人们普遍认为，正则化泛函和正则化参数的选择对稳定数值反演方法的成功与否至关重要。/DIV项目主要研究不适定逆问题的数值稳定解，特别是涉及/DIV矩的问题。一个具体的问题是从矩中恢复二维/DIV形状，涉及/DIV(1)矩重建形状的正则化反演方案的制定和求解，以及(2)将这些/DIV算法应用于重力场/DIV测量数据的重力异常重建