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

许多科学问题都采用逆问题的形式，其中所研究的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。此类问题的一个例子是根据当地重力场的测量来计算地球的密度。 这种反问题很少能够通过解析求解，并且通常需要数值近似来找到解。在这个研究项目中，研究人员旨在引入一种新颖的迭代算法来解决反问题，发展其理论和计算特性，并确定其在应用中的性能。 预计新算法将适用于当前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，目前在断层摄影、考古学、天体物理学和其他科学领域非常感兴趣。该研究项目探索了一种新颖的迭代算法来解决一类反问题，包括第一类 Fredholm 积分方程、拉普拉斯变换反演、统计中的混合分布估计以及求解稀疏线性方程组。研究人员计划（i）引入一种新颖的迭代算法来解决这些类型以及其他类型的逆问题，（ii）开发其理论和计算特性，以及（iii）确定其在应用中的性能。这项研究的动机是估计非参数混合模型中混合密度的统计问题。迄今为止，还没有通用算法可以产生全局一致的混合密度估计量，就强度量几乎肯定收敛的意义上来说；只能得到弱收敛结果。正在开发的算法的一个重要特征是，如果它以平滑密度函数初始化，则估计器也必然是平滑密度函数。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质。新颖的迭代算法的形式，以及它产生平滑密度估计量的事实，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普遍的全球一致性结果并展示收敛率。

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.