High-order processing of singular data

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.