Positivity, Inverse Problems, and Operator Theory

正性、反问题和算子理论

• 批准号: 0350911
• 负责人: Mihai Putinar
• 金额: --
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0350911&HistoricalAwards=false
• 关键词: Positivity; Inverse; Problems; Operator; Theory

The project is centered on the study of mathematical aspects of inverse problems, from the point of view of functional and complex mathematical analysis. Statistics, spectral analysis, X-ray crystallography, tomography offer models and classical examples of such questions. More specifically, most of the modern inverse problems are related to (power, Fourier, wavelet) moments of mass distributions. In their turn, moment problems form a wellestablished field of research with a fascinating history of abouttwo centuries. The project focuses on specific moment problems arising in shape and volume reconstruction from distant measurements (for instance of electric or gravitational fields). An important component of the project is the study of the best approximation and the error bounds related to reconstruction algorithms.The success of previous researches of the principal investigator on similar two dimensional questions guarantee positive results in his attempt to expand his horizon to three or more dimensions. He will collaborate with several experts in pure and applied mathematics, as well as with graduate students andyoung research assistants. Their combined efforts and results will be significant for present studies in functional analysis, function theory and numerical mathematics, as well as for some specific branches of appliedmathematics and engineering.

该项目以研究反问题的数学方面为中心，从函数和复杂数学分析的角度出发。统计学、光谱分析、X射线结晶学、层析照相术提供了这些问题的模型和经典例子。更具体地说，大多数现代反问题都与(幂、傅立叶、小波)质量分布矩有关。转而，瞬间问题形成了一个有着大约两个世纪迷人历史的成熟研究领域。该项目侧重于根据远距离测量(例如电场或引力场)重建形状和体积时出现的具体力矩问题。该项目的一个重要组成部分是与重建算法相关的最佳逼近和误差界的研究。主要研究人员以前在类似二维问题上的研究的成功为他将视野扩展到三维或更多维的尝试提供了积极的结果。他将与几位纯数学和应用数学专家合作，以及与研究生和年轻的研究助理合作。他们的共同努力和结果将对目前泛函分析、函数论和数值数学的研究以及应用数学和工程的一些具体分支具有重要意义。