Integrated Multiscale Modeling of Molecular Computing Devices Page: 2 of 8
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Specific tasks of the original proposal
* Formulation of appropriate open and periodic boundary conditions within the multiwavelet approach.
* Comparison of various representations of functions and operators in the multiwavelet basis as a way of
finding more effective ways of computing.
* Exploration of alternative types of multiresolution bases.
* Development of efficient representations for the Green's function at positive energies.
* Exploration of alternative implementations of the Green's function approach within the multiresolution
Additional task of the non-cost extension
* Development of algorithms based on separated representations for boundary value problems.
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 below.
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
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 ap-
plying 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
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 func-
tions by exponential sums and/or their rational representations.
1. Gregory Beylkin, PI, partially supported.
2. Fernando P6rez, Research Associate, partially supported.
3. Lucas Monzon, Research Associate, partially supported.
4. Christopher Kurcz, partially supported as graduate student and then as a postdoc.
5. Terry Haut, partially supported as a postdoc.
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Beylkin, Gregory. Integrated Multiscale Modeling of Molecular Computing Devices, report, March 23, 2012; United States. (digital.library.unt.edu/ark:/67531/metadc840610/m1/2/: accessed December 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.