Collaborative Research: New stochastically-motivated solutions to classical inverse problems

合作研究：经典反问题的新随机驱动解决方案

• 批准号: 1612891
• 负责人: Stephen Walker
• 金额: $ 12.56万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612891&HistoricalAwards=false
• 关键词: Collaborative; Research; New; stochastically; motivated

Numerous scientific questions assume the form of inverse problems, in which an unknown input to a system under study gives rise to an observed noisy output, and the goal is to estimate the input from the output. An example of such a problem is calculating the density of the Earth from measurements of the local gravitational field. Only rarely can such inverse problems be solved analytically, and in general numerical approximations are required to find solutions. In this research project, the investigators aim to introduce a novel iterative algorithm for solving inverse problems, develop its theoretical and computational properties, and establish its performance in applications. It is anticipated that the new algorithm will be adaptable to a range of problems currently under investigation in applied and numerical mathematics, for example in solving a sparse system of linear equations, currently of great interest in areas including tomography, archaeology, astrophysics, and other sciences.This research project explores a novel iterative algorithm for the solution of a class inverse problems that includes Fredholm integral equations of the first kind, Laplace transform inversion, mixing distribution estimation in statistics, and solving sparse systems of linear equations. The investigators plan to (i) introduce a novel iterative algorithm for solving inverse problems of these types, and perhaps others, (ii) develop its theoretical and computational properties, and (iii) establish its performance in applications. A motivation for the research is the statistical problem of estimating a mixing density in a nonparametric mixture model. To date, there are no general algorithms that produce globally consistent estimators of the mixing density, in the sense of almost sure convergence with respect to a strong metric; only weak convergence results are available. An important feature of the algorithm under development is that, if it is initialized at a smooth density function, then the estimator is necessarily also a smooth density function. Other algorithms designed by numerical analysts for solving these inverse problems do not have this closure property. The form of the novel iterative algorithm, along with the fact that it yields smooth density estimators, suggests that this open problem can be solved; the investigators aim to establish a general global consistency result and demonstrate rates of convergence.

许多科学问题都采用逆问题的形式，其中所研究系统的未知输入会产生观察到的噪声输出，目标是根据输出估计输入。这样的问题的一个例子是计算密度的地球从测量当地的重力场。 只有很少可以这样的反问题得到解决的分析，并在一般情况下，需要数值逼近找到解决方案。在这个研究项目中，研究人员的目标是引入一种新的迭代算法来解决反问题，发展其理论和计算特性，并建立其在应用中的性能。 预计新算法将适用于目前在应用数学和数值数学中正在研究的一系列问题，例如求解稀疏线性方程组，目前在包括断层扫描，考古学，天体物理学，本研究计画探讨一种新的迭代演算法，用以求解一类包含Fredholm积分方程的反问题，第一类是拉普拉斯变换反演、统计学中的混合分布估计和求解稀疏线性方程组。研究人员计划（i）引入一种新的迭代算法来解决这些类型的反问题，也许还有其他问题，（ii）开发其理论和计算特性，以及（iii）建立其应用性能。一个动机的研究是在非参数混合模型的混合密度估计的统计问题。迄今为止，有没有一般的算法，产生全局一致的估计的混合密度，在这个意义上几乎肯定收敛相对于一个强度量，只有弱收敛的结果。正在开发的算法的一个重要特征是，如果它是在一个光滑的密度函数初始化，那么估计也必然是一个光滑的密度函数。数值分析师为解决这些反问题而设计的其他算法不具有这种封闭性。新的迭代算法的形式，沿着的事实，它产生光滑的密度估计，表明这个开放的问题可以解决，调查人员的目标是建立一个普遍的全球一致性的结果，并证明收敛速度。