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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Reduced order models for spectral domain inversion
: embedding into the continuous problem and genera
tion of internal data.* - Shari Moskow (Drexel Uni
versity)
DTSTART;TZID=Europe/London:20191212T160000
DTEND;TZID=Europe/London:20191212T163000
UID:TALK135625AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/135625
DESCRIPTION:We generate data-driven reduced order models (ROMs
) for inversion of the one and two dimensional Sch
r\\"odinger equation in the spectral domain given
boundary data at a few frequencies. The ROM is the
Galerkin projection of the Schr\\"odinger operato
r onto the space spanned by solutions at these sam
ple frequencies\, and corresponds to a rational in
terpolant of the Neumann to Dirichlet map. The RO
M matrix is in general full\, and not good for ext
racting the potential. However\, using an orthogon
al change of basis via Lanczos iteration\, we can
transform the ROM to a block triadiagonal form fro
m which it is easier to extract the unknown coeffi
cient. In one dimension\, the tridiagonal matrix c
orresponds to a three-point staggered finite-diffe
rence system for the Schr\\"odinger operator discr
etized on a so-called spectrally matched grid whi
ch is almost independent of the medium. In higher
dimensions\, the orthogonalized basis functions pl
ay the role of the grid steps. The orthogonalized
basis functions are localized and also depend only
very weakly on the medium\, and thus by embedding
into the continuous problem\, the reduced order m
odel yields highly accurate internal solutions. Th
at is to say\, we can obtain\, just from boundary
data\, very good approximations of the solution of
the Schr\\"odinger equation in the whole domain f
or a spectral interval that includes the sample fr
equencies. We present inversion experiments based
on the internal solutions in one and two dimension
s.

* joint with L. Borcea\, V. Druskin\,
A. Mamonov\, M. Zaslavsky
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
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