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Projekt
Sparse Fast Fourier Transforms
Antragsteller
Professor Dr. Stefan Kunis; Professor Dr. Daniel Potts
Fachliche Zuordnung
Mathematik
Förderung
Förderung von 2008 bis 2016
Projektkennung
Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 79725622
A straightforward discretisation of high dimensional problems often leads to a curse of dimensions and thus the use of sparsity has become a popular tool. Efficient algorithms like the fast Fourier transform (FFT) have to be customised to these thinner discretisations and we focus on three major topics regarding the Fourier analysis of high dimensional functions: We will develop stable and effective spatial discretisations of hyperbolic cross FFTs by the numerical optimisation of rank-1 lattice rules. If the sparsity pattern is unknown, we study greedy methods and Prony type methods for the sparserecovery of signals from samples. Besides a comparison of both approaches, we will study recent deterministic compressed sensing approaches, extend the stable Prony method to complex and closely neighbouring frequencies, and suggest new d-variate generalisations based on the combination of solutions to low dimensional subproblems. Finally, we will proceed in the development of a stable variant of the so-called butter y sparse FFT which is used if the support of the Fourier coefficients as well as the spatial sampling set are subsets of `smooth' curves and surfaces. In summary, we are interested in computational methods for the Fourier analysis of sparse high dimensional signals and continue our research based on results of the first period. All algorithms will be implemented within Matlab and C software packages and we will continue cooperations with application experts within but not restricted to nuclear magnetic resonance spectroscopy, photoacoustic imaging, and magnetic resonance imaging.
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