A rigorous optimization framework for high-resolution parameter estimation on the continuum

Many areas of science depend on the ability to recover accurate information about a parametric model from incomplete or noisy measurements. A common approach is to formulate an optimization problem and use computational methods to search for the best explanation of the observed data. However, existing optimization methods often produce unreliable results because they can potentially find a spurious minimum and/or the global minimum is inaccurate due to noise. This project develops new mathematical foundations for designing optimization methods that are both accurate and reliable. The resulting framework will help researchers better understand when these methods can be trusted and will provide practical algorithms for a broad class of scientific problems. The project will also broaden participation in mathematical research by supporting graduate students through research experiences and seminars, while making all software freely available to encourage reproducible research and future scientific advances. This project develops a general optimization framework for continuous parameter estimation by constructing objective functions with provably favorable geometric properties. The research seeks to establish conditions under which informative minima lie close to the true parameters, possess large basins of attraction, and can be located efficiently using local optimization methods, while spurious minima remain shallow and can be avoided through a rejection strategy. A central goal is to show that these properties are governed by an underlying correlation kernel, providing a unified theoretical framework for a broad class of parameter estimation problems. The framework will be analyzed and validated through three representative applications: wave-based imaging, sparse decompositions, and reconstruction from noisy random rotations. For each application, the project will establish theoretical recovery guarantees and develop computational algorithms that exploit the underlying problem structure to achieve high-resolution parameter estimation. 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.