Mathematics Colloquia and Seminars

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Fourier integral operators are oscillatory integrals related to linear wave propagation. They are a central object in most imaging problems involving waves, like seismic or ultrasound imaging; and also in some tomography problems, like electron microscopy. In most cases, the quality of imaging hinges on the practicality of applying these oscillatory integrals on a large computational scale. In this talk, I will present a fast algorithm for the computation of large classes of Fourier integral operators. The algorithm is based on factorization and separation of the oscillatory kernel over adequate angular neighborhoods in the frequency variable, and inspired by modern constructions in applied harmonic analysis. We prove that the complexity of computing and applying a Fourier integral operator on an N-by-N grid, is O(N^{2.5} log N) in time, and as low as O(sqrt N) in storage. The constants in front of these estimates are small and depend weakly on the desired accuracy. We illustrate the properties and potential of the algorithm with several numerical examples.