CAREER: Fast and Accurate Algorithms for Uncertainty Quantification in Large-Scale Inverse Problems

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The need to visualize regions that are impossible to see with the naked eye is pervasive in everyday life. For example, in medicine, accurate visualization of tissue is needed to diagnose and treat tumors. A key step in imaging technologies requires one to solve an inverse problem in order to transform measured data into detailed image reconstructions of the quantities of interest. However, image reconstruction is inherently uncertain, in part, due to noisy measurements from sensors. Ignoring the uncertainty in the imaging process can lead to undesirable outcomes, such as misjudging the location and spread of a suspected tumor. Uncertainty Quantification (UQ) in imaging is in its infancy and hence, the potential for impact in research contributions is high. UQ for imaging is computationally challenging since thousands of inversions are needed beyond the initial inversion to generate accurate statistics of the uncertainty. Current approaches for UQ are inadequate because they either fail to deliver solutions in a reasonable computational time or they lack the applicability across a broad range of imaging technologies.The project is on the development of fast algorithms for UQ in large-scale inverse problems that are applicable to a broad range of imaging technologies. These algorithms are expected to bring down the computational cost by at least an order of magnitude while maintaining the accuracy of the solutions. More specifically, the project will (1) Advance image reconstruction and UQ techniques for incorporating prior information based on fractional partial differential equation (PDE) and Bayesian level set approaches; and (2) Develop new algorithms and analysis for data-driven dimensionality techniques for UQ in Bayesian inverse problems, using randomized and Krylov subspace methods. The algorithms developed here will be rigorously analyzed and validated on several model problems and applications, including diffuse optical and photoacoustic tomography (in biomedicine) and hydraulic tomography and satellite data fusion (in geoscience). The algorithms developed here are also applicable to other imaging-based inverse problems in biomedicine, geophysics, materials science, etc. Outside of imaging applications, these mathematical advances will be of interest to scientists working in many areas of computational science, for example, fractional partial differential equations (PDEs), model reduction, tensor decompositions, and principal component analysis. Lastly, the PI's education and outreach activities will make UQ and imaging technologies more modular, accessible, and easier to understand for pre-service and early career K-12 educators, undergraduate students, and graduate students. Specifically, the educational program of this project will: (1) Strengthen STEM education through teacher training workshops and practical research experiences for pre-service and early career K-12 teachers, which will result in reproducible teaching modules for use in K-12 education; and (2) Enhance undergraduate and graduate curriculum at North Carolina State University by creating accessible seminar talks and new course content.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.