On the approximation of smooth functions using generalized digital nets

In this paper, we study an approximation algorithm which firstly approximates certain Walsh coefficients of the function under consideration and consequently uses a Walsh polynomial to approximate the function. A similar approach has previously been used for approximating periodic functions, using lattice rules (and Fourier polynomials), and for approximating functions in Walsh Korobov spaces, using digital nets. Here, the key ingredient is the use of generalized digital nets (which have recently been shown to achieve higher order convergence rates for the integration of smooth functions). This allows us to approximate functions with square integrable mixed partial derivatives of order @a>1 in each variable. The approximation error is studied in the worst case setting in the L"2 norm. We also discuss tractability of our proposed approximation algorithm, investigate its computational complexity, and present numerical examples.

[1]  R. Caflisch Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[2]  Josef Dick,et al.  Equidistribution Properties of Generalized Nets and Sequences , 2009 .

[3]  H. Woxniakowski Information-Based Complexity , 1988 .

[4]  H. Faure Discrépance de suites associées à un système de numération (en dimension s) , 1982 .

[5]  Henryk Wozniakowski,et al.  When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? , 1998, J. Complex..

[6]  H. E. Chrestenson A class of generalized Walsh functions , 1955 .

[7]  I. Sobol On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[8]  Harald Niederreiter,et al.  Monte Carlo and quasi-Monte Carlo methods 2004 , 2006 .

[9]  Xiaoqun Wang,et al.  Strong tractability of multivariate integration using quasi-Monte Carlo algorithms , 2003, Math. Comput..

[10]  Josef Dick,et al.  Explicit Constructions of Quasi-Monte Carlo Rules for the Numerical Integration of High-Dimensional Periodic Functions , 2007, SIAM J. Numer. Anal..

[11]  Henryk Wozniakowski,et al.  Lattice rule algorithms for multivariate approximation in the average case setting , 2008, J. Complex..

[12]  Peter Kritzer,et al.  Duality theory and propagation rules for generalized digital nets , 2010, Math. Comput..

[13]  Henryk Wozniakowski,et al.  Information-based complexity , 1987, Nature.

[14]  Harald Niederreiter,et al.  Monte Carlo and Quasi-Monte Carlo Methods 2006 , 2007 .

[15]  F. J. Hickernell,et al.  Spline Methods Using Low Discrepancy Designs , 2008 .

[16]  I. Sloan,et al.  Lattice Rules for Multivariate Approximation in the Worst Case Setting , 2006 .

[17]  Josef Dick,et al.  QMC Rules of Arbitrary High Order: Reproducing Kernel Hilbert Space Approach , 2009 .

[18]  Henryk Wozniakowski,et al.  Multivariate L∞ approximation in the worst case setting over reproducing kernel Hilbert spaces , 2008, J. Approx. Theory.

[19]  I. Sloan Lattice Methods for Multiple Integration , 1994 .

[20]  E. Novak,et al.  Tractability of Multivariate Problems , 2008 .

[21]  F. J. Hickernell,et al.  Trigonometric spectral collocation methods on lattices , 2003 .

[22]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[23]  H. Niederreiter Point sets and sequences with small discrepancy , 1987 .

[24]  J. Walsh A Closed Set of Normal Orthogonal Functions , 1923 .

[25]  Henryk Wozniakowski,et al.  On the power of standard information for multivariate approximation in the worst case setting , 2009, J. Approx. Theory.

[26]  Gerhard Larcher,et al.  On the numerical integration of Walsh series by number-theoretic methods , 1994 .

[27]  J. Dick,et al.  Approximation of Functions Using Digital Nets , 2008 .

[28]  Harald Niederreiter,et al.  Nets, (t, s)-Sequences, and Codes , 2008 .

[29]  Josef Dick,et al.  Walsh Spaces Containing Smooth Functions and Quasi-Monte Carlo Rules of Arbitrary High Order , 2008, SIAM J. Numer. Anal..

[30]  Henryk Wozniakowski,et al.  On the Power of Standard Information for Weighted Approximation , 2001, Found. Comput. Math..

[31]  Gottlieb Pirsic,et al.  A Software Implementation of Niederreiter-Xing Sequences , 2002 .

[32]  H. Niederreiter,et al.  Rational Points on Curves Over Finite Fields: Theory and Applications , 2001 .

[33]  Fred J. Hickernell,et al.  Monte Carlo and Quasi-Monte Carlo Methods 2000 , 2002 .

[34]  Fred J. Hickernell,et al.  Error Analysis of Splines for Periodic Problems Using Lattice Designs , 2006 .

[35]  Henryk Wozniakowski,et al.  Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers , 2004, Found. Comput. Math..

[36]  Fred J. Hickernell,et al.  A multivariate fast discrete Walsh transform with an application to function interpolation , 2008, Math. Comput..

[37]  Harald Niederreiter,et al.  Constructions of (t, m, s)-nets and (t, s)-sequences , 2005, Finite Fields Their Appl..

[38]  Henryk Wozniakowski,et al.  Weighted Tensor Product Algorithms for Linear Multivariate Problems , 1999, J. Complex..

[39]  Josef Dick,et al.  Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules , 2007, J. Complex..

[40]  Josef Dick,et al.  Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces , 2005, J. Complex..

[41]  J. Dick THE DECAY OF THE WALSH COEFFICIENTS OF SMOOTH FUNCTIONS , 2009, Bulletin of the Australian Mathematical Society.

[42]  Fred J. Hickernell,et al.  The existence of good extensible rank-1 lattices , 2003, J. Complex..