Iteratively Regularized Broyden-Type Algorithms for Nonlinear Inverse Problems

Georgia State University Research Foundation, Inc., Atlanta GA

Investigators

Abstract

The goal of this project is to tackle major computational challenges faced by scientists and engineers in their quest to improve the accuracy and efficiency of numerical algorithms for solving large-scale inverse problems. This is a scenario where direct measurements of the unknown quantities are not feasible, and one needs to identify "cause from effect" by using (generally nonlinear) mathematical and statistical models. The resulting problems are notoriously ill-posed (or unstable), in a sense that even small measurement errors in the input data may give rise to a substantial noise propagation in the recovered solution, to the extent that this solution gets entirely destroyed. For this reason, special techniques called "regularization" must be combined with high-speed optimization procedures, so that reliable information on the unknown effect could be obtained from the available data. The key areas of application include imaging and sensing technology, machine learning, gravitational sounding, ocean acoustics, and data sciences. This project aims at the development of iteratively regularized Broyden-type numerical algorithms for solving nonlinear ill-posed inverse problems in either finite or infinite dimensional spaces. A family of new regularization methods will be designed to solve large-scale unstable least squares problems, where the Jacobian of a discretized nonlinear operator is difficult or even impossible to compute. To overcome this obstacle, PIs consider a family of Gauss-Newton and Levenberg-Marquardt algorithms with the Frechet derivative operator recalculated recursively by using Broyden-type single rank updates. To balance accuracy and stability, the pseudo-inverse for the derivative-free Jacobian is regularized in a problem-specific manner at every step of the iteration process. A variety of filters will be investigated, yielding greater flexibility in the use of qualitative and quantitative a priori information available for each particular applied problem. The proposed iteratively regularized methods will be studied in both deterministic and stochastic settings. For stochastic processes, the minimization functionals are evaluated subject to stochastic errors due to inexact computations to lower per-iteration cost, and/or unavoidable environmental noise and fluctuations. In the framework of the proposed research, PIs will conduct comprehensive convergence analysis of the new algorithms, including convergence rates and optimal policies for the selection of regularization parameters and step sizes. In addition to the theoretical investigation, a significant component of this project is to evaluate the proposed algorithms using extensive numerical experiments on real-world nonlinear inverse problems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.