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N-Widths and ε-Dimensions for High-Dimensional Approximations

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Abstract

In this paper, we study linear trigonometric hyperbolic cross approximations, Kolmogorov n-widths d n (W,H γ), and ε-dimensions n ε (W,H γ) of periodic d-variate function classes W with anisotropic smoothness, where d may be large. We are interested in finding the accurate dependence of d n (W,H γ) and n ε (W,H γ) as a function of two variables n, d and ε, d, respectively. Recall that n, the dimension of the approximating subspace, is the main parameter in the study of convergence rates with respect to n going to infinity. However, the parameter d may seriously affect this rate when d is large. We construct linear approximations of functions from W by trigonometric polynomials with frequencies from hyperbolic crosses and prove upper bounds for the error measured in isotropic Sobolev spaces H γ. Furthermore, in order to show the optimality of the proposed approximation, we prove upper and lower bounds of the corresponding n-widths d n (W,H γ) and ε-dimensions n ε (W,H γ). Some of the received results imply that the curse of dimensionality can be broken in some relevant situations.

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References

  1. K.I. Babenko, On the approximation of periodic functions of several variables by trigonometric polynomials, Dokl. Akad. Nauk SSSR 132, 247–250 (1960). English transl. in Soviet Math. Dokl. 1 (1960).

    Google Scholar 

  2. R. Bellmann, Dynamic Programming (Princeton University Press, Princeton, 1957).

    Google Scholar 

  3. H.-J. Bungartz, M. Griebel, A note on the complexity of solving Poisson’s equation for spaces of bounded mixed derivatives, J. Complex. 15, 167–199 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  4. H.-J. Bungartz, M. Griebel, Sparse grids, Acta Numer. 13, 147–269 (2004).

    Article  MathSciNet  Google Scholar 

  5. J. Céa, Approximation variationnelle des problémes aux limites, Ann. Inst. Fourier 14, 345–444 (1964).

    Article  MATH  Google Scholar 

  6. A. Chernov, Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading, Appl. Numer. Math. 62, 360–377 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Chernov, C. Schwab, Sparse p-version BEM for first kind boundary integral equations with random loading, Appl. Numer. Math. 59, 2698–2712 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  8. R.A. DeVore, S.V. Konyagin, V.N. Temlyakov, Hyperbolic wavelet approximation, Constr. Approx. 14, 1–26 (1998).

    Article  MathSciNet  Google Scholar 

  9. R. DeVore, G. Petrova, P. Wojtaszcyk, Approximating functions of few variables in high dimensions, Constr. Approx. 33, 125–143 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Dũng, Some approximative characteristics of the classes of smooth functions of several variables in the metric of L 2, Usp. Mat. Nauk 34, 189–190 (1979).

    Google Scholar 

  11. D. Dũng, Mean ε-dimension of the functional class B G,p , Mat. Zametki 28, 727–736 (1980).

    MathSciNet  Google Scholar 

  12. D. Dũng, Approximation of functions of several variables on a torus by trigonometric polynomials, Mat. Sb. (N.S.) 131(2), 251–271 (1986).

    Google Scholar 

  13. D. Dũng, On optimal recovery of multivariate periodic functions, in Harmonic Analysis, ed. by S. Igary (Springer, Berlin, 1991), pp. 96–105.

    Google Scholar 

  14. D. Dũng, Optimal recovery of functions of a certain mixed smoothness, Vietnam J. Math. 20(2), 18–32 (1992).

    MATH  Google Scholar 

  15. D. Dũng, B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness, J. Complex. 27, 541–567 (2011).

    Article  MATH  Google Scholar 

  16. D. Dũng, D. Dryanov, The minimal number of sample values for recovery of non-bandlimited functions, Atti Semin. Mat. Fis. Univ. Modena 39, 423–431 (1991).

    MATH  Google Scholar 

  17. J. Garcke, M. Griebel, M. Thess, Data mining with sparse grids, Computing 67, 225–253 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  18. T. Gerstner, Sparse grid quadrature methods for computational finance. Habilitation, Institute for Numerical Simulation, University of Bonn, 2007.

  19. T. Gerstner, M. Griebel, Sparse grids, in Encyclopedia of Quantitative Finance, ed. by R. Cont (Wiley, New York, 2010).

    Google Scholar 

  20. M. Griebel, J. Hamaekers, Tensor product multiscale many-particle spaces with finite-order weights for the electronic Schrödinger equation, Z. Phys. Chem. 224, 527–543 (2010).

    Article  Google Scholar 

  21. M. Griebel, S. Knapek, Optimized tensor-product approximation spaces, Constr. Approx. 16, 525–540 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Griebel, S. Knapek, Optimized general sparse grid approximation spaces for operator equations, Math. Comput. 78(268), 2223–2257 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  23. A.N. Kolmogorov, Über die beste Annäherung von Funktionen einer Funktionklasse, Ann. Math. 37, 107–111 (1936).

    Article  MathSciNet  MATH  Google Scholar 

  24. A.N. Kolmogorov, Mathematics and Mechanics. Selected Papers, vol. I (Nauka, Moscow, 1985) (in Russian).

    MATH  Google Scholar 

  25. P.D. Lax, A.N. Milgram, “Parabolic equations”. Contributions to the theory of partial differential equations, Ann. Math. Stud. 33, 167–190 (1954).

    MathSciNet  MATH  Google Scholar 

  26. E. Novak, H. Woźniakowski, Tractability of Multivariate Problems, Volume I: Linear Information. EMS Tracts in Mathematics, vol. 6 (Eur. Math. Soc. Publ. House, Zürich, 2008).

    Book  Google Scholar 

  27. E. Novak, H. Woźniakowski, Approximation of infinitely differentiable multivariate functions is intractable, J. Complex. 25, 398–404 (2009).

    Article  MATH  Google Scholar 

  28. E. Novak, H. Woźniakowski, Optimal order of convergence and (in)tractability of multivariate approximation of smooth functions, Constr. Approx. 30, 457–473 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  29. E. Novak, H. Woźniakowski, Tractability of Multivariate Problems, Volume II: Standard Information for Functionals. EMS Tracts in Mathematics, vol. 12 (Eur. Math. Soc. Publ. House, Zürich, 2010).

    Book  Google Scholar 

  30. A. Pinkus, N-Widths in Approximation Theory (Springer, New York, 1985).

    Book  MATH  Google Scholar 

  31. C. Schwab, R.-A. Todor, Sparse finite elements for elliptic problems with stochastic loading, Numer. Math. 95, 707–734 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  32. C. Schwab, R.-A. Todor, Sparse finite elements for stochastic elliptic problems higher order moments, Computing 71, 43–63 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  33. L. Schwartz, Théorie des Distributions (Hermann & Cie, Paris, 1966).

    MATH  Google Scholar 

  34. W. Sickel, T. Ullrich, The Smolyak algorithm, sampling on sparse grids and function spaces of dominating mixed smoothness, East J. Approx. 13, 387–425 (2007).

    MathSciNet  Google Scholar 

  35. W. Sickel, T. Ullrich, Tensor products of Sobolev–Besov spaces and applications to approximation from the hyperbolic cross, J. Approx. Theory 161, 748–786 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  36. W. Sickel, T. Ullrich, Spline interpolation on sparse grids, Appl. Anal. 90, 337–383 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  37. S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 148, 1042–1045 (1963).

    MathSciNet  MATH  Google Scholar 

  38. V.N. Temlyakov, Approximation of Periodic Functions. Computational Mathematics and Analysis Series (Nova Science, Commack, 1993).

    MATH  Google Scholar 

  39. V.M. Tikhomirov, Widths of sets in function spaces and the theory of best approximations, Usp. Mat. Nauk 15(3), 81–120 (1960). English translation in Russian Math. Survey, 15, 1960.

    MathSciNet  Google Scholar 

  40. V.M. Tikhomirov, Some problems in approximation theory. Moscow State University, 1985 (in Russian).

  41. R.-A. Todor, A new approach to energy-based sparse finite-element spaces, IMA J. Numer. Anal. 29(1), 72–85 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  42. T. Ullrich, Smolyak’s algorithm, sampling on sparse grids and Sobolev spaces of dominating mixed smoothness, East J. Approx. 14, 1–38 (2008).

    MathSciNet  MATH  Google Scholar 

  43. G. Wasilkowski, H. Woźniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complex. 11, 1–56 (1995).

    Article  MATH  Google Scholar 

  44. P. Wojtaszcyk, Complexity of approximation of functions of few variables in high dimensions, J. Complex. 27, 141–150 (2011).

    Article  Google Scholar 

  45. H. Yserentant, On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives, Numer. Math. 98, 731–759 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  46. H. Yserentant, Sparse grid spaces for the numerical solution of the electronic Schrödinger equation, Numer. Math. 101, 381–389 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  47. H. Yserentant, Regularity and Approximability of Electronic Wave Functions. Lecture Notes in Mathematics, vol. 2000 (Springer, Berlin, 2010).

    Book  MATH  Google Scholar 

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Acknowledgements

The work of the first named author was supported by Grant 102.01-2012.15 of the National Foundation for Development of Science and Technology (Vietnam). The both authors would like to thank the Hausdorff Research Institute for Mathematics (HIM) and the organizers of the HIM Trimester Program “Analysis and Numerics for High Dimensional Problems”, where this paper was initiated, for providing a fruitful research environment and additional financial support. Last but not least, the authors would like to thank the referees for a critical reading of the manuscript and for several valuable suggestions which helped to improve its presentation.

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Authors and Affiliations

  1. Information Technology Institute, Vietnam National University, Hanoi, 144 Xuan Thuy, Hanoi, Vietnam

    Dinh Dũng

  2. Hausdorff-Center for Mathematics and Institute for Numerical Simulation, 53115, Bonn, Germany

    Tino Ullrich

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  1. Dinh Dũng
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  2. Tino Ullrich
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Correspondence to Dinh Dũng.

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Communicated by Wolfgang Dahmen.

Dedicated to the memory of Professor S.M. Nikol’skij.

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Dũng, D., Ullrich, T. N-Widths and ε-Dimensions for High-Dimensional Approximations. Found Comput Math 13, 965–1003 (2013). https://doi.org/10.1007/s10208-013-9149-9

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  • DOI: https://doi.org/10.1007/s10208-013-9149-9

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