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SUMMARY:Metric Approximation of Set-Valued Functions - Elena Berdysheva  (
 Justus-Liebig-Universität Gießen)
DTSTART:20190312T150000Z
DTEND:20190312T163000Z
UID:TALK121501@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:We study approximation of set-valued functions (SVFs) | functi
 ons mapping a real interval to compact sets in Rd. In addition to the theo
 retical interest in this subject\, it is relevant to various applications 
 in &#12\;elds where SVFs are used\, such as economy\, optimization\, dynam
 ical systems\, control theory\, game theory\, di&#11\;erential inclusions\
 , geometric modeling. In particular\, SVFs are relevant to the problem of 
 the reconstruction of 3D objects from their parallel cross-sections. The i
 mages (values) of the related SVF are the cross-sections of the 3D object\
 , and the graph of this SVF is the 3D object. Adaptations of classical sam
 ple-based approximation operators\, in particular\, of positive operators 
 for approximation of SVFs with convex images were intensively studied by a
  number of authors. For example\, R.A Vitale studied an adaptation of the 
 classical Bernstein polynomial operator based on Minkowski linear combinat
 ion of sets which converges to the convex hull of the image. Thus\, the li
 mit SVF is always a function with<br>convex images\, even if the original 
 function is not. This e&#11\;ect is called convexi&#12\;cation and is obse
 rved in various adaptations based on Minkowski linear combinations. Clearl
 y such adaptations work for set-valued functions with convex images\, but 
 are useless for the approximation of SFVs with non-convex images. Also the
  standard construction of an integral of set-valued functions | the Aumann
  integral | possesses the property of convexi&#12\;cation. Dyn\, Farkhi an
 d Mokhov developed in a series of work a new approach that is free of conv
 exi&#12\;cation | the so-called metric linear combinations and the metric 
 integral.<br>Adaptations of classical approximation operators to continuou
 s SFVs were studied by Dyn\, Farkhi and Mokhov. Here\, we develop methods 
 for approximation of SFVs that are not necessarily contin- uous. As the &#
 12\;rst step\, we consider SVFs of bounded variation in the Hausdor&#11\; 
 metric.<br>In particular\, we adapt to SVFs local operators such as the sy
 mmetric Schoenberg spline operator\, the Bernstein polynomial operator and
  the Steklov function. Error bounds\, obtained in the averaged Hausdor&#11
 \; metric\, provide rates of approximation similar to those for real-value
 d functions of bounded variation.<br>Joint work with Nira Dyn\, Elza Farkh
 i and Alona Mokhov (Tel Aviv University\, Israel).<br>
LOCATION:Seminar Room 2\, Newton Institute
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