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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:High Dimensional Approximation via Sparse Occupanc
y Trees - Peter Binev (University of South Caroli
na)
DTSTART;TZID=Europe/London:20190617T142000
DTEND;TZID=Europe/London:20190617T151000
UID:TALK126082AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/126082
DESCRIPTION:Adaptive domain decomposition is often used in fin
ite elements methods for solving partial different
ial equations in low space dimensions. The adaptiv
e decisions are usually described by a tree. Assum
ing that can find the (approximate) error for appr
oximating a function on each element of the partit
ion\, we have shown that a particular coarse-to-fi
ne method provides a near-best approximation. This
result can be extended to approximating point clo
uds any space dimension provided that we have rele
vant information about the errors and can organize
properly the data. Of course\, this is subject to
the curse of dimensionality and nothing can be do
ne in the general case. In case the intrinsic dime
nsionality of the data is much smaller than the sp
ace dimension\, one can define algorithms that def
y the curse. This is usually done by assuming that
the data domain is close to a low dimensional man
ifold and first approximating this manifold and th
en the function defined by it. A few years ago\, t
ogether with Philipp Lamby\, Wolfgang Dahmen\, and
Ron DeVore\, we proposed a direct method (without
specifically identifying any low dimensional set)
that we called "sparse occupancy trees". The meth
od defines a piecewise constant or linear approxim
ation on general simplicial partitions. This talk
considers an extension of this method to find a si
milar approximation on conforming simplicial parti
tions following an idea from a recent result toget
her with Francesca Fierro and Andreas Veeser about
near-best approximation on conforming triangulati
ons.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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