Collaborative Research: Mathematical Methods for Optimal Polynomial Recovery of High-Dimensional Systems from Sparse and Noisy Data

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Current problems in approximation that are driven by applications in science and engineering, are typically formulated in very high dimensions. This project involves the study of different problems related to high-dimensional approximation, that arise in a large number of applications including neutron, tomographic and magnetic resonance image reconstruction, uncertainty quantification, optimal control and parameter identification, as well as in important areas of energy and material science. The approaches used in this work will result in substantially improved and mathematically well-founded methodologies for computer simulations of solutions to real-world problems. The project will be centered around the interdisciplinary training of graduate students in computational data science and engineering. The results obtained will be disseminated through journal articles, conference talks, and a collaborative website. In this effort we propose to develop novel mathematical techniques for approximation of high-dimensional systems from a limited amount of sparse and noisy data. The results of this effort will enable scientists to understand what are the number realizations of a nonlinear manifold that required to recover the entire high-dimensional solution map, with optimal approximation guarantees and minimal computational cost. Our rigorous mathematical approach includes: Novel weighted convex optimization and iterative thresholding techniques for optimal polynomial recovery, established via an improved estimate of the restricted isometry property; and Advanced multi-index methods that alleviate complexity and accelerate convergence of solutions by constructing model hierarchies with the use of reduced-basis techniques.