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Piecewise approximation and neural networks

Martina Révayová, Csaba Török (2007)

Kybernetika

The paper deals with the recently proposed autotracking piecewise cubic approximation (APCA) based on the discrete projective transformation, and neural networks (NN). The suggested new approach facilitates the analysis of data with complex dependence and relatively small errors. We introduce a new representation of polynomials that can provide different local approximation models. We demonstrate how APCA can be applied to especially noisy data thanks to NN and local estimations. On the other hand,...

Pointwise inequalities and approximation in fractional Sobolev spaces

David Swanson (2002)

Studia Mathematica

We prove that a function belonging to a fractional Sobolev space L α , p ( ℝ ⁿ ) may be approximated in capacity and norm by smooth functions belonging to C m , λ ( ℝ ⁿ ) , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

Pointwise inequalities for Sobolev functions and some applications

Bogdan Bojarski, Piotr Hajłasz (1993)

Studia Mathematica

We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer’s theorem which states that Sobolev functions can be approximated by C m functions both in norm and capacity.

Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

Houman Owhadi, Lei Zhang, Leonid Berlyand (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (L∞) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution H) minimizing the L2 norm of the source...

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