Abstract:
This presentation will cover generalized accuracy bounds for solutions to Inverse Problems (IPs) and describes the application of these bounds to single-molecule fluorescence microscopy and to the super-resolution of multi-spectral satellite data.
IPs, defined as the reconstruction of an unknown quantity from noisy measurements, are pervasive throughout the applied sciences. Representative examples include medical imaging, radar inverse scattering, and astronomy. The underlying mathematical formulation is typically an ill-posed, potentially nonlinear reconstruction problem, commonly referred to as an ill-posed inverse problem. A broad range of methodologies has been developed to construct approximate inverse maps, including optimization-based approaches such as compressed sensing, Bayesian inference techniques, and data-driven methods such as deep learning.
The presentation begins with introductory examples drawn from Earth observation and medical imaging and then examines the fundamental accuracy–stability tradeoff that governs the computation of solutions to inverse problems. For all stable approximate inverse maps that compute solutions to ill-posed inverse problems, nonzero lower bounds on achievable accuracy arise as a direct consequence of this tradeoff, as established in Gottschling et al., SIAM Review, 67(1), 2025, and Colbrook et al., Proceedings of the National Academy of Sciences, 119(12), 2022. However, the accuracy–stability tradeoff produces bounds that are not directly computable in practice. The diversity of existing reconstruction methodologies underscores the need for a unifying theoretical framework to enable the principled selection of approximate inverse maps that approach the theoretical optimum. At present, computable accuracy bounds applicable to general inverse problems and independent of the specific reconstruction method are not available in the literature.
This seminar introduces computable and sharp accuracy bounds for the reconstruction error associated with solution methods for inverse problems. These bounds are method-independent and depend solely on the dataset of signals, the forward model, and the noise model.
To facilitate practical implementation in scientific applications, the presentation includes an algorithmic framework and accompanying software. The framework is validated using two inverse problems drawn from distinct application domains: fluorescence localization microscopy and the super-resolution of multispectral satellite data. The computation of these accuracy bounds enables informed optimization of datasets and forward models prior to investing resources in the design and implementation of approximate inverse maps.
Speaker’s Bio:
Dr. Nina M. Gottschling is a Wigner Fellow at the Oak Ridge National Laboratory. Previously, she served as a team lead and postdoctoral researcher at the German Aerospace Center near Munich, Germany. She earned her Ph.D. in Applied Mathematics from the University of Cambridge, United Kingdom. Her doctoral thesis, titled “On existence, stability, accuracy and learning of approximate decoders for ill-posed inverse problems,” examined theoretical foundations and learning-based approaches to reconstruction methods for ill-posed inverse problems. Dr. Gottschling holds a M.Sc. in Theoretical and Mathematical Physics from Ludwig Maximilian University of Munich. She also earned a B.Sc. in Physics, and a B.A. in Philosophy from the Ludwig Maximilian University of Munich. As an early-career scientist, she was selected to participate in the 73rd Lindau Nobel Laureate Meeting in Physics in 2024.
