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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Metric Approximation of Set-Valued Functions - Ele
na Berdysheva (Justus-Liebig-Universität Gießen)
DTSTART;TZID=Europe/London:20190312T150000
DTEND;TZID=Europe/London:20190312T163000
UID:TALK121501AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/121501
DESCRIPTION:We study approximation of set-valued functions (SV
Fs) | functions mapping a real interval to compact
sets in Rd. In addition to the theoretical intere
st in this subject\, it is relevant to various app
lications in \;elds where SVFs are used\, such
as economy\, optimization\, dynamical systems\, c
ontrol theory\, game theory\, di\;erential inc
lusions\, geometric modeling. In particular\, SVFs
are relevant to the problem of the reconstruction
of 3D objects from their parallel cross-sections.
The images (values) of the related SVF are the cr
oss-sections of the 3D object\, and the graph of t
his SVF is the 3D object. Adaptations of classical
sample-based approximation operators\, in particu
lar\, of positive operators for approximation of S
VFs with convex images were intensively studied by
a number of authors. For example\, R.A Vitale stu
died an adaptation of the classical Bernstein poly
nomial operator based on Minkowski linear combinat
ion of sets which converges to the convex hull of
the image. Thus\, the limit SVF is always a functi
on with

convex images\, even if the original fu
nction is not. This e\;ect is called convexi&#
12\;cation and is observed in various adaptations
based on Minkowski linear combinations. Clearly su
ch adaptations work for set-valued functions with
convex images\, but are useless for the approximat
ion of SFVs with non-convex images. Also the stand
ard construction of an integral of set-valued func
tions | the Aumann integral | possesses the proper
ty of convexi\;cation. Dyn\, Farkhi and Mokhov
developed in a series of work a new approach that
is free of convexi\;cation | the so-called me
tric linear combinations and the metric integral.<
br>Adaptations of classical approximation operator
s to continuous SFVs were studied by Dyn\, Farkhi
and Mokhov. Here\, we develop methods for approxim
ation of SFVs that are not necessarily contin- uou
s. As the \;rst step\, we consider SVFs of bou
nded variation in the Hausdor\; metric.

In
particular\, we adapt to SVFs local operators such
as the symmetric Schoenberg spline operator\, the
Bernstein polynomial operator and the Steklov fun
ction. Error bounds\, obtained in the averaged Hau
sdor\; metric\, provide rates of approximation
similar to those for real-valued functions of bou
nded variation.

Joint work with Nira Dyn\, Elza
Farkhi and Alona Mokhov (Tel Aviv University\, Is
rael).

LOCATION:Seminar Room 2\, Newton Institute
CONTACT:info@newton.ac.uk
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