Description: Significant advances were made on all objectives of the research program. We have developed fast multiresolution methods for performing electronic structure calculations with emphasis on constructing efficient representations of functions and operators. We extended our approach to problems of scattering in solids, i.e. constructing fast algorithms for computing above the Fermi energy level. Part of the work was done in collaboration with Robert Harrison and George Fann at ORNL. Specific results (in part supported by this grant) are listed here and are described in greater detail. (1) We have implemented a fast algorithm to apply the Green's function for the free space (oscillatory) Helmholtz kernel. The algorithm maintains its speed and accuracy when the kernel is applied to functions with singularities. (2) We have developed a fast algorithm for applying periodic and quasi-periodic, oscillatory Green's functions and those with boundary conditions on simple domains. Importantly, the algorithm maintains its speed and accuracy when applied to functions with singularities. (3) We have developed a fast algorithm for obtaining and applying multiresolution representations of periodic and quasi-periodic Green's functions and Green's functions with boundary conditions on simple domains. (4) We have implemented modifications to improve the speed of adaptive multiresolution algorithms for applying operators which are represented via a Gaussian expansion. (5) We have constructed new nearly optimal quadratures for the sphere that are invariant under the icosahedral rotation group. (6) We obtained new results on approximation of functions by exponential sums and/or rational functions, one of the key methods that allows us to construct separated representations for Green's functions. (7) We developed a new fast and accurate reduction algorithm for obtaining optimal approximation of functions by exponential sums and/or their rational representations.
Date: March 23, 2012
Creator: Beylkin, Gregory
Item Type: Refine your search to only Report
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