Skip to main content
SpringerLink
Log in

Kolmogorov and Linear Widths on Generalized Besov Classes in the Monte Carlo Setting

  • Conference paper
Theoretical and Mathematical Foundations of Computer Science (ICTMF 2011)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 164))

  • 1941 Accesses

Abstract

In this paper, we studied the Kolmogorov and the linear widths on the generalized Besov classes \(B^\Omega_{p,\theta}\) in the norm of L q in the Monte Carlo setting. Applying the discretization technique and some properties of pseudo-s-scale, we determined the exact asymptotic orders of the Kolmogorov and the linear widths for certain values of the parameters p, q, θ.

Project supported by the NSF of China(Grant No.10926056 and No.10971251).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bakhvalov, N.S.: On the approximate computation of multiple integrals. Vestnik Moskov Univ.Ser. Mat. Mekh. Astr. Fiz. Khim 4, 3–18 (1959)

    MathSciNet  Google Scholar 

  2. Traub, J.F., Wasilkowski, G.W., Wo′zniakowski, H.: Information-based Complexity. Academic Press, New York (1988)

    Google Scholar 

  3. Fang, G.S., Ye, P.X.: Integration Error for Multivariate Functions From Anisotropic Classes. J. Complexity 19, 610–627 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fang, G.S., Ye, P.X.: Complexity of deterministic and randomized methods for multivariate integration problem for the class \(H^{\wedge}_{p}(I^{d})\). IMA Journal of Numerical Analysis 25, 473–485 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Heinrich, S.: Lower Bounds for the Complexity of Monte Carlo Function Approximation. J. Complexity 8, 277–300 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Heinrich, S.: Random approximation in numerical analysis. In: Bierstedt, K.D., Bonet, J., Horvath, J., et al. (eds.) Functional Analysis: Proceedings of the Essen Conference. Lect. Notes in Pure and Appl. Math., vol. 150, pp. 123–171. Chapman and Hall/CRC, Boca Raton (1994)

    Google Scholar 

  7. Heinrich, S.: Monte Carlo approximation of weakly singular integral operators. J. Complexity 22, 192–219 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Math′e, P.: Random approximation of Sobolev embedding. J. Complexity 7, 261–281 (1991)

    Article  MathSciNet  Google Scholar 

  9. Math′e, P.: Approximation Theory of Stochastic Numerical Methods. Habilitationsschrift, Fachbereich Mathematik, Freie Universität Berlin (1994)

    Google Scholar 

  10. Novak, E.: Deterministic and stochastic error bounds in numerical analysis. Lecture Notes in Mathematics, vol. 1349. Springer, Berlin (1988)

    MATH  Google Scholar 

  11. Fang, G.S., Duan, L.Q.: The complexity of function approximation on Sobolev spaces with bounded mixed derivative by linear Monte Carlo methods. J. Complexity 24, 398–409 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fang, G.S., Duan, L.Q.: The information-based complexity of approximation problem by adaptive-Monte Carlo methods. Science in China A: Mathematics 51, 1679–1689 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nikolskii, S.M.: Approximation of Functions of Several Variables and Imbeddings Theorems. Springer, Berlin (1975)

    Book  Google Scholar 

  14. Romanyuk, A.S.: On estimate of the Kolmogorov widths of the classes \(B^{r}_{p,\theta}\) in the space Lq. Ukr. Math. J. 53, 1189–1196 (2001)

    Article  MathSciNet  Google Scholar 

  15. Romanyuk, A.S.: Linear widths of the Besov classes of periodic functions of many variables.II. Ukr. Math. J. 53, 965–977 (2001)

    Article  MathSciNet  Google Scholar 

  16. Romanyuk, A.S.: Approximation of classes \(B^{r}_{p,\theta}\) by linear methods and best approximations. Ukr. Math. J. 54, 825–838 (2002)

    Article  MathSciNet  Google Scholar 

  17. Pustovoitov, N.N.: Representation and approximation of multivariate periodic functions with a given mixed modulus of smoothness. Analysis Math. 20, 35–48 (1994)

    Article  MathSciNet  Google Scholar 

  18. Sun, Y.S., Wang, H.P.: Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness. In: Proc. Steklov Inst. Math., vol. 219, pp. 350–371 (1997)

    Google Scholar 

  19. Amanov, T.I.: Representation and imbedding theorems for the function spaces \(S^{r}_{p,\theta}B(R^{n}), S^{*r}_{p,\theta}B(0{\leq}x_{j}{\leq}2{\pi}, j = 1,., n)\). Trudy Mat. Inst. Akad. Nauk SSSR 77, 5–34 (1965)

    MathSciNet  Google Scholar 

  20. Stasyuk, S.A.: Best approximations and Kolmogorov and trigonometric widths of the classes \(B^{\Omega}_{p,\theta} (T^{d})\) of periodic functions of many variables. Ukr. Math. J. 56, 1849–1863 (2004)

    Article  MathSciNet  Google Scholar 

  21. Fedunyk, O.V.: Linear widths of the classes \(\rm B^{\Omega}_{p,\theta} (T^{d})\) of periodic functions of many variables in the space Lq. Ukr. Math. J. 58, 103–117 (2006)

    Article  MATH  Google Scholar 

  22. Duan, L.Q.: The best m-term approximations on generalized Besov classes \(MB^{\Omega}_{q,\theta}\) with regard to orthogonal dictionaries. J. Approx. Theory 162, 1964–1981 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pinkus, A.: N-widths in Approximation Theory. Springer, New York (1985)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Hangzhou Dianzi University, Hangzhou, 310018, China

    Liqin Duan

  2. School of Mathematics, Nankai University, Tianjin, 300071, China

    Peixin Ye

Authors
  1. Liqin Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peixin Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Southwestern University of Finance and Economics, No. 55 Guanghuacun Street, 610074, Chengdu, China

    Qihai Zhou

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Duan, L., Ye, P. (2011). Kolmogorov and Linear Widths on Generalized Besov Classes in the Monte Carlo Setting. In: Zhou, Q. (eds) Theoretical and Mathematical Foundations of Computer Science. ICTMF 2011. Communications in Computer and Information Science, vol 164. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24999-0_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-24999-0_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-24998-3

  • Online ISBN: 978-3-642-24999-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation