CDS&E-MSS: Recovery of High-Dimensional Structured Functions
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Many scientific problems of crucial importance for the United States involve a very large number of parameters. This high dimensionality has long been a challenge for the numerical treatment of physical, chemical, and biological processes, and now it also represents an obstacle in data science, especially for the task of extracting useful information from only a limited amount of observations. Propitiously, the high-dimensional objects prevailing in real-life problems often possess some underlying structure simplifying their manipulation. The purpose of this project is to exploit this structure in order to develop a consistent theoretical framework and to conceive novel computational methods for the exact recovery (or good approximation) of the high-dimensional objects sketchily acquired.More specifically, the objects considered in this project are functions of very many variables. Handling them via traditional numerical methods is doomed by the so-called curse of dimensionality. But modern ideas such as sparsity and variable reduction make it possible to bypass the curse and thus are revolutionizing the approach to high-dimensional problems. Building upon the fundamentals from the theory of compressive sensing, the project will consider sparse recovery and simultaneously structured recovery as part of the more general recovery of high-dimensional functions depending on few reduced variables. The research strategy starts by investigating the theoretical limitations of any recovery method within a model, then continues by refining the model through confrontation with real-life problems, and finishes by implementing the algorithms proven to perform optimally. Since the project features interactions with several applied fields (in particular, Engineering and Bioinformatics), the novel numerical methods are to be tested in these areas, which will in turn provide fresh mathematical insight. A final part of the project is devoted to the integration of emerging concepts into the culture of the next scientific generation and in particular to the training of mathematicians in computational and data-related aspects.