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

许多科学问题都以反问题的形式出现，在反问题中，被研究系统的未知输入会产生观察到的有噪声的输出，目标是从输出中估计输入。这类问题的一个例子是根据当地引力场的测量来计算地球的密度。只有极少数情况下，这种逆问题才能得到解析解，而且通常需要数值近似才能找到解。在这项研究项目中，研究人员的目标是引入一种新的求解反问题的迭代算法，发展其理论和计算性质，并建立其在应用中的性能。预计新算法将适用于目前正在研究的一系列应用数学和数值数学问题，例如求解稀疏线性方程组，目前在层析成像、考古学、天体物理学和其他科学领域非常感兴趣。本研究项目探索一种新的迭代算法来求解一类反问题，包括第一类Fredholm型积分方程、拉普拉斯变换逆、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，也许还有其他类型的反问题，(Ii)发展它的理论和计算性质，(Iii)建立它的应用性能。这项研究的一个动机是估计非参数混合模型中的混合密度的统计问题。到目前为止，还没有通用的算法来产生混合密度的全局一致估计，在关于强度量的几乎必然收敛的意义上；只有弱收敛结果可用。正在开发的算法的一个重要特征是，如果它是以光滑密度函数初始化的，则估计器必然也是光滑密度函数。数值分析员为解决这些反问题而设计的其他算法不具有这种闭包性质。新迭代算法的形式，以及它产生光滑密度估计的事实，表明这个开放问题是可以解决的；研究人员的目标是建立一个一般的全局一致性结果，并证明收敛速度。

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.