Metric Approximation of Set-Valued Functions

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Description:	Berdysheva, E Tuesday 12th March 2019 - 15:00 to 16:30


Created:	2019-03-13 15:17
Collection:	Approximation, sampling and compression in data science
Publisher:	Isaac Newton Institute
Copyright:	Berdysheva, E
Language:	eng (English)
Distribution:	World (downloadable)
Explicit content:	No
Aspect Ratio:	16:9
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Abstract:	We study approximation of set-valued functions (SVFs) \| functions mapping a real interval to compact sets in Rd. In addition to the theoretical interest in this subject, it is relevant to various applications in elds where SVFs are used, such as economy, optimization, dynamical systems, control theory, game theory, dierential inclusions, 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 cross-sections of the 3D object, and the graph of this SVF is the 3D object. Adaptations of classical sample-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 combination of sets which converges to the convex hull of the image. Thus, the limit SVF is always a function with convex images, even if the original function is not. This eect is called convexication and is observed in various adaptations based on Minkowski linear combinations. Clearly 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 convexication. Dyn, Farkhi and Mokhov developed in a series of work a new approach that is free of convexication \| the so-called metric linear combinations and the metric integral. Adaptations of classical approximation operators to continuous SFVs were studied by Dyn, Farkhi and Mokhov. Here, we develop methods for approximation of SFVs that are not necessarily contin- uous. As the rst step, we consider SVFs of bounded 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 function. Error bounds, obtained in the averaged Hausdor metric, provide rates of approximation similar to those for real-valued functions of bounded variation. Joint work with Nira Dyn, Elza Farkhi and Alona Mokhov (Tel Aviv University, Israel).

Abstract:

We study approximation of set-valued functions (SVFs) | functions mapping a real interval to compact sets in Rd. In addition to the theoretical interest in this subject, it is relevant to various applications in elds where SVFs are used, such as economy, optimization, dynamical systems, control theory, game theory, dierential inclusions, 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 cross-sections of the 3D object, and the graph of this SVF is the 3D object. Adaptations of classical sample-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 combination of sets which converges to the convex hull of the image. Thus, the limit SVF is always a function with
convex images, even if the original function is not. This eect is called convexication and is observed in various adaptations based on Minkowski linear combinations. Clearly 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 convexication. Dyn, Farkhi and Mokhov developed in a series of work a new approach that is free of convexication | the so-called metric linear combinations and the metric integral.
Adaptations of classical approximation operators to continuous SFVs were studied by Dyn, Farkhi and Mokhov. Here, we develop methods for approximation of SFVs that are not necessarily contin- uous. As the rst step, we consider SVFs of bounded 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 function. Error bounds, obtained in the averaged Hausdor metric, provide rates of approximation similar to those for real-valued functions of bounded variation.
Joint work with Nira Dyn, Elza Farkhi and Alona Mokhov (Tel Aviv University, Israel).

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Metric Approximation of Set-Valued Functions