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CMFD, 2007, Volume 25, Pages 126–148 (Mi cmfd111)  

This article is cited in 1 scientific paper (total in 1 paper)

Linear and Nonlinear Methods of Relief Approximation

K. I. Oskolkov

University of South Carolina

Abstract: In this article we compare the effectiveness of free (nonlinear) relief approximation, equidistant relief approximation, and polynomial approximation $\mathscr R^{\mathrm{fr}}_N[f]$, $\mathscr R^{\mathrm{eq}}_N[f]$, $\mathscr E_N[f]$ of an individual function $f(\mathbf{x})$ in the metric $\mathscr L^2(\mathbb B^2)$, where $\mathbb B^2$ is the unit ball $|\mathbf{x}|\le1$ in the plane $\mathbb R^2$. The notation we use is the following
\begin{gather*} \mathscr R^{\mathrm{fr}}_N[f] :=\inf_{R\in\mathscr W^{\mathrm{fr}}_N}\|f-R\|, \quad \mathscr R^{\mathrm{eq}}_N[f]:=\min_{R\in\mathscr W^{\mathrm{eq}}_N}\|f-R\|,
\mathscr E_N[f]:=\min_{P\in\mathscr{P}^2_{N-1}}\|f-P\|. \end{gather*}
Here $\mathscr W^{\mathrm{fr}}_N$ is the set of all $N$-term linear combinations of functions of the plane wave type
$$ R(\mathbf{x})=\sum_1^N W_j(\mathbf{x}\cdot\boldsymbol\theta_j) $$
with arbitrary profiles $W_j(x)$, $x\in\mathbb R^1$ and transmission directions $\{\boldsymbol\theta_j\}_1^N$; $\mathscr W^{\mathrm{eq}}_N$ is the subset of $\mathscr W^{\mathrm{fr}}_N$ associated with $N$ equidistant directions;
$$ \mathscr{P}^2_{N-1}:=\operatorname{Span}\{x_1^kx_2^l\}_{k+l<N} $$
denotes the subspace of algebraic polynomials of degree less or equal to $N-1$ in two real variables. Obviously, inequalities $\mathscr R^{\mathrm{fr}}_N[f] \le\mathscr R^{\mathrm{eq}}_N[f]\le\mathscr E_N[f]$ hold.
We state the following model problem. What are the functions which satisfy the relation $\mathscr R^{\mathrm{fr}}_N[f]=o(\mathscr R^{\mathrm{eq}}_N[f])$, i.e., where nonlinear approximation $\mathscr R^{\mathrm{fr}}$ is more effective than linear one? This effect have been proved for harmonic functions, namely, for any $\varepsilon>0$ there exists $c_\varepsilon>0$ such that if $\Delta f(\mathbf{x})=0$, $|\mathbf{x}|<1$, $f\in\mathscr L^2(\mathbb B^2)$, then
$$ \mathscr R^{\mathrm{fr}}_N[f] \le c_\varepsilon(\mathscr R^{\mathrm{eq}}_N[f]\exp(-N^\varepsilon)+\mathscr R^{\mathrm{eq}}_{N^{2-3\varepsilon}}[f]). $$
On the other hand, $\mathscr R^{\mathrm{fr}}_N[f]\ge\frac1c\mathscr R^{\mathrm{eq}}_{N^2}[f]$. Thus $\mathscr R^{\mathrm{fr}}_N[f]$ has an “almost squared effectiveness” of $\mathscr R^{\mathrm{eq}}_N[f]$ for $f=f_{\mathrm{harm}}$. However, this ultra-high order of approximation is obtained via a collaps of wave vectors.
On the other hand, the nonlinearity of $\mathscr R^{\mathrm{fr}}$ which corresponds to the freedom of choice of wave vectors, does not much improve the order of approximation, for instance, for all the radial functions. If $f(\mathbf{x})=f(|\mathbf{x}|)$, then $\mathscr E_{2N}[f]\ge\mathscr R^{\mathrm{eq}}_N[f]\ge\sqrt{\dfrac23}\mathscr E_{2N}(f)$ and $\mathscr R^{\mathrm{fr}}_N[f]\ge\sup\limits_{\varepsilon>0}\sqrt{\dfrac\varepsilon{3(1+\varepsilon)}}\mathscr R^{\mathrm{eq}}_{(1+\varepsilon)N}[f]$.
The technique we use is the Fourier–Chebyshev analysis (which is related to the inverse Radon transform on $\mathbb B^2$) and a duality between the relief approximation problem and the optimization of quadrature formulas in the sense of Kolmogorov–Nikolskii [1] for trigonometric polynomials classes.

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English version:
Journal of Mathematical Sciences, 2008, 155:1, 129–152

Bibliographic databases:

UDC: 517.5

Citation: K. I. Oskolkov, “Linear and Nonlinear Methods of Relief Approximation”, Theory of functions, CMFD, 25, PFUR, M., 2007, 126–148; Journal of Mathematical Sciences, 155:1 (2008), 129–152

Citation in format AMSBIB
\by K.~I.~Oskolkov
\paper Linear and Nonlinear Methods of Relief Approximation
\inbook Theory of functions
\serial CMFD
\yr 2007
\vol 25
\pages 126--148
\publ PFUR
\publaddr M.
\jour Journal of Mathematical Sciences
\yr 2008
\vol 155
\issue 1
\pages 129--152

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    This publication is cited in the following articles:
    1. S. V. Konyagin, A. A. Kuleshov, V. E. Maiorov, “Some problems in the theory of ridge functions”, Proc. Steklov Inst. Math., 301 (2018), 144–169  mathnet  crossref  crossref  isi  elib  elib
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