RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Guidelines for authors Publishing Ethics Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 CMFD: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (318 kB)
References: PDF file   HTML file

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
\Bibitem{Osk07} \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. \mathnet{http://mi.mathnet.ru/cmfd111} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2342543} \zmath{https://zbmath.org/?q=an:1195.41020} \transl \jour Journal of Mathematical Sciences \yr 2008 \vol 155 \issue 1 \pages 129--152 \crossref{https://doi.org/10.1007/s10958-008-9212-2} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-55749109006} 

• http://mi.mathnet.ru/eng/cmfd111
• http://mi.mathnet.ru/eng/cmfd/v25/p126

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
•  Number of views: This page: 308 Full text: 91 References: 36