On Low-Rank Regularization for Ill-Posed Nonlinear Parameter Estimation

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This research has been inspired by numerous challenges in studying the transmission dynamics of infectious diseases. The investigator will focus on estimating parameters of data-enabled mathematical models of infectious disease aiming to generate forecasts of future incidence cases. The project will develop regularized computational algorithms for this data estimation which presents many challenges and includes uncertainties. The models and optimization methods to be used are important in anticipating the resources needed for disease management. Data on past and present infectious diseases will be studied, and the investigator will collaborate with the university's School of Public Health. Apart from applications in epidemiology, this project will have a broad impact on scientific disciplines including signal and image processing, biomedical imaging, gravitational sounding, chaos theory, ocean acoustics, and others. The project includes graduate student training through involvement in the research.The project aims to develop computational algorithms for estimating parameters using optimization. From an optimization standpoint, parameter estimation and forecasting from data comes down to solving an ill-posed minimization problem constrained by a system of ordinary or partial differential equations. For uncertainty quantification, multiple runs of the inversion algorithm must be carried out, preferably in real time. To address this challenge, the project will construct a family of trust-region optimization algorithms with low-rank updates for the Jacobian operator that will reduce the computational cost of a quasi-Newton step and, at the same time, incorporate an extra layer of stability in the iterative process. In case of nonlinear least squares with non-zero residuals, low-rank updates for stable Hessian evaluation will be investigated. Theoretical and numerical analysis of the new methods will be first carried out for normally solvable ill-posed operator equations and then extended to essentially ill-posed problems. The successful completion of this project will advance the understanding of ill-posed inverse problems and facilitate more stable and efficient simulations.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.