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Separated Representations and Their Applications
Applied Math| Speaker: | Gregory Beylkin, University of Colorado, Boulder |
| Location: | 1147 MSB |
| Start time: | Fri, Feb 10 2006, 4:10PM |
Description
We represent multidimensional operators in a separated form that can be viewed
as a numerical generalization of separation of variables. Since we separate
directions,
it does not preclude operators with high (operator) rank from having low
separation rank. We
introduce a concept of the condition number of a separated representation (to
measure the potential loss of significant digits) and make its control
a part of the algorithm for reduction of the separation rank.
Structurally, the separated representation is a tensor decomposition. However,
since its purpose and use are novel, it brings up a new set of examples and
questions regarding the mathematics of tensor
decompositions. Some of these questions lead to interesting problems in harmonic
analysis.
Since many physically significant operators have low separation rank, our basic
approach
has already found applications in quantum chemistry and fast multiresolution
algorithms.