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High-order processing of singular data. (English) Zbl 1180.68280
Bourguignon, Jean-Pierre (ed.) et al., Noncommutativity and singularities. Proceedings of French-Japanese symposia held at IHÉS, Bures-sur-Yvette, France, November 20-23 and November 15–18, 2006. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-54-9/hbk). Advanced Studies in Pure Mathematics 55, 173-207 (2009).
Summary: This paper provides a survey of some recent results 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 [Y. Yomdin, Generic singularities of surfaces. Singapore: World Scientific. 357–375 (2007; Zbl 1125.58011) and D. Haviv and Y. Yomdin, Theor. Comput. Sci. 392, No. 1–3, 92–100 (2008; Zbl 1134.68061)] 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. We present results on the stability of Hermite fitting, which the main tool in acquisition of Taylor-Net data.
Next we present 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 the problem of a nonlinear acquisition of Model-Nets from measurements, as well as some additional implementations of the Model-Net approach.
For the entire collection see [Zbl 1173.14002].

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
58K05 Critical points of functions and mappings on manifolds