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

许多科学问题都以逆问题的形式存在，即所研究的系统的未知输入会产生可观察到的有噪声的输出，而目标是从输出中估计输入。此类问题的一个例子是通过测量局部引力场来计算地球的密度。只有很少的情况下，这样的反问题可以解析解决，一般需要数值近似才能找到解。在本研究项目中，研究人员旨在引入一种新的求解逆问题的迭代算法，发展其理论和计算性质，并建立其应用性能。预计新的算法将适用于目前应用数学和数值数学中正在研究的一系列问题，例如解决线性方程的稀疏系统，目前在断层扫描、考古学、天体物理学和其他科学领域都很感兴趣。本研究计划探索求解一类反问题的新迭代算法，包括第一类Fredholm积分方程、拉普拉斯变换反演、统计学中的混合分布估计和求解线性方程的稀疏系统。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逆问题，也许还有其他类型的逆问题，（ii）发展其理论和计算特性，（iii）建立其应用性能。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，还没有产生混合密度的全局一致估计的一般算法，在对强度量几乎肯定收敛的意义上；只有弱收敛结果。正在开发的算法的一个重要特征是，如果它在平滑密度函数初始化，那么估计器也必然是平滑密度函数。数值分析人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，以及它产生平滑密度估计的事实，表明这个开放问题是可以解决的；研究人员的目的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.