Advanced Studies in Pure Mathematics

High-order processing of singular data

Yosef Yomdin and Gal Zahavi

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Abstract

This paper provides a survey of some recent results (mostly in [68, 71, 72, 73, 74]) concerning high-order representation and processing of singular data. We present these results from a certain general point of view which we call a “Model-Net” approach: this is a method of representation and processing of various types of mathematical data, based on the explicit recovery of the hierarchy of data singularities. As an example we use a description of singularities and normal forms of level surfaces of “product functions” recently obtained in [68, 34] and on this base describe in detail the structure of the Model-net representation of such surfaces.

Then we discuss a “Taylor-net” representation of smooth functions consisting of a net of Taylor polynomials of a prescribed degree $k$ (or $k$-jets) of this function stored at a certain grid in its domain. Following [72, 74] we present results on the stability of Hermite fitting, which is the main tool in acquisition of Taylor-net data.

Next we present (following [71, 73, 74]) a method for numerical solving PDE's based on Taylor-net representation of the unknown function. We extend this method also to the case of the Burgers equation near a formed shock-wave.

Finally, we shortly discuss (following [28, 56]) the problem of a non-linear acquisition of Model-nets from measurements, as well as some additional implementations of the Model-net approach.

Article information

Source
Noncommutativity and Singularities: Proceedings of French–Japanese symposia held at IHÉS in 2006, J.-P. Bourguignon, M. Kotani, Y. Maeda and N. Tose, eds. (Tokyo: Mathematical Society of Japan, 2009), 173-207

Dates
Received: 24 October 2007
Revised: 19 June 2008
First available in Project Euclid: 28 November 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1543447908

Digital Object Identifier
doi:10.2969/aspm/05510173

Mathematical Reviews number (MathSciNet)
MR2463497

Zentralblatt MATH identifier
1180.68280

Citation

Yomdin, Yosef; Zahavi, Gal. High-order processing of singular data. Noncommutativity and Singularities: Proceedings of French–Japanese symposia held at IHÉS in 2006, 173--207, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05510173. https://projecteuclid.org/euclid.aspm/1543447908


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