Structured compressed sensing algorithms: design, analysis and applications

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Many problems in science and engineering require the reconstruction of an object - an image, signal or high-dimensional function, for example - from a collection of measurements. Due to time, cost or other constraints, one is often severely limited by the amount of data that can be collected, which significantly affects ones ability to recover the unknown object accurately. This research program involves the development, analysis and application of new algorithms for this problem, based on the theory and techniques of compressed sensing (CS). Examples of relevant applications include medical imaging, microscopy, uncertainty quantification in physical systems, machine learning and the numerical solution of PDEs. Its overarching objective is to introduce new, computationally-efficient numerical optimization techniques for such applications that possess both better accuracy and lower acquisition time and cost. ****Specific objectives***1) To design and implement a new generation of CS-based algorithms for imaging that incorporate additional structure in both the sampling and recovery process. By leveraging such structure, this work is expected to bring substantial improvements over current state-of-the-art algorithms, yielding tangible benefits in key imaging technologies such as MRI, X-ray CT and electron and fluorescence microscopy.***2) To develop and study new CS-based methods for high-dimensional approximation that exploit structured smoothness-sparsity priors and randomized sampling techniques to enhance accuracy. As data collection becomes easier and more widespread, there is a pressing need in science and engineering to understand increasingly complex phenomena by approximating high-dimensional functions. The outcomes of this work will be improved techniques for this problem, bringing benefits to important practical tasks such as uncertainty quantification in physical (e.g. biological, mechanical or fluid) systems.***3) To investigate the limits of stability and accuracy for sampling-based algorithms in scientific computing, and to design new computational methods based on conformal mappings that attain such limits. Sampling-based algorithms have a variety of uses in scientific computing, including surface reconstruction, numerical methods for PDEs and numerical software. This work will enhance knowledge through a better understanding of the theoretical limits achievable by any algorithm for this problem, and its benefit will be an improved set of methods based on such limits.***Overall, the focus of this research program is the development of new algorithms for challenging data-oriented problems. It aims to bring benefits in a range of applications in key areas of national need in science, engineering and medicine. This research will also contribute to the pressing need for skills in these areas, both in academia and industry, through the training of HQP.**