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

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.