Theory, Implementation, and Applications of Sublinear-Time Fourier Transform Algorithms

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The problem of computing a Fourier representation of data is afundamental one, that arises in many areas of mathematics, science,and technology. Recently, a class of algorithms has been proposed toapproximate the Fourier transform, with provable quality guarantees,in dramatically less time than required by traditional exactalgorithms. The investigator and his colleagues evaluate competingvariations of the algorithm, both proposed and new, and choose thebest from among them for general purpose use as well as for specificapplication to Magnetic Resonence Imaging and solution of partialdifferential equations, including advection-diffusion equations.The Fourier Transform is a fundamental technique with manyapplications. The investigator and his team investigate recentapproximate methods that are dramatically more efficient thantraditional methods. The team directly addresses applications tobiomedical Magnetic Resonance Imaging and the modeling of liquid andgas behavior, which has further applications to studying theatmosphere and bodies of water in the environment, as well asapplications to manufacturing. The investigator and his team alsomake available fundamental tools that will be useful to others in abroad range of scientific, educational, and engineering settingsincluding modeling the interactions of tiny particles, reliabletelecommunications, and imaging generally. The project combinescutting-edge techniques from several disciplines and trains a graduatestudent in these techniques and their integration.