TRACTABILITY OF MULTIVARIATE INTEGRATION PROBLEM FOR PERIODIC CONTINUOUS FUNCTIONS

Fang Gensun1 (
¡¥¤)
Long Jingfan1,2 (
£¢ )
1.School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China 2.Department of Computer Science, Information Technology Institute, Beijing 100101, China E-mail: fanggs@
m∈Z\{0}
λm |f (m)|2
.
K1 (x, y ) = K1,γ (x, y ) = 1 + γ
m∈Z\{0}
λm exp{2π im(x − y )} =
λ (γ, m)
λ is a reproducing kernel in the Hilbert space H1 , where 1, if m = 0, ρλ (γ, m) = γ −1 λ−1 , if m = 0. m
1
= f
1,γ
= f, f
1/2 1
= f, f
1/2 1,γ
λ It is clear that if f ∈ H1 , then f has absolutely convergent Fourier series: thus, f is continuous, and
= |f (0)|2 + γ −1
∗ Received
No.4
Fang & Long: TRACTABILITY OF MULTIVARIATE INTEGRATION PROBLEM
791
of applications, and large d is used in financial mathematics, where it is quite common to integrate functions of 360 or more variables. For path integration, d = ∞, and an accurate approximation of the path integral requires approximations for d-dimensional integrals, where d can be arbitrarily large (see [15]). If d is large then often some variables are “more important” than others. To model such a situation, weighted tensor product norms, where different norms are taken for different variables, are considered in Refs.[8, 12, 13]. The authors concentrate on quasi-Monte Carlo formulas, prove upper and lower bounds with respect to discrepancies, and give a partial answer to why quasi-Monte Carlo algorithms can work well for arbitrarily large d. It is done by identifying classes of functions for which the effect of the dimension d is negligible. Motivated by Hickernell and Wo´ zniakowski [8] and Sloan and Wo´ zniakowski [13], we study the tractability of multivariate integration over the unit cube in weighted periodic continuous classes of functions with absolutely convergent Fourier coefficients for the worst case setting. λ λ Our function class is the unit ball on a Hilbert space Hd = Hd,γ , which is a tensor product of d λ λ different Hilbert spaces H1 = H1,γ of 1-periodic complex-valued L2 functions f defined on [0,1] λ with absolutely convergent Fourier series. The inner products of H1 have the following form f, g
(2)
∞
η (λ) :=
m=1
λm < ∞,
(3)
which characterizes the rate of decay of the Fourier coefficients,
1
f (m) =
0
exp(−2π imx)f (x)dx,
m ∈ Z,
and γ > 0 is a weight that may take different values in different components of the tensor λ product. The norm of H1 is given by 1/2 f
(4)
792
ACTA MATHEMATICA SCIENTIA
Vol.27 Ser.B
Indeed, by equation (1), we find that f, K1 (·, y )
1
=
m∈Z\{0}
ρλ (γ, m)f (m)K (·, y ) ρλ (γ, m)f (m) exp(2π imy ) = f (y ). ρλ (γ, m)
1
Introduction
Multivariate problem is of increasing importance in computational mathematics [2, 5, 6, 8, 12, 15, 16]. Especially, it is indeed quite common to approximate integrals of functions of d variables. All values of d seem to be of practical interests, and small d occurs in a number
October 19, 2005; revised March 10, 2006. Project supported by the National Natural Science Foundation of China (10671019), Research Fund for the Doctoral Program Higher Education(20050027007), and Beijing Educational Committee (2002Kj112) Corresponding author: Fang Gensun
Acta Mathematica Scientia 2007,27B(4):790–802
TRACTABILITY OF MULTIVARIATE INTEGRATION PROBLEM FOR PERIODIC CONTINUOUS FUNCTIONS∗
Abstract The authors study the tractability and strong tractability of a multivariate integration problem in the worst case setting for weighted 1-periodic continuous functions spaces of d coordinates with absolutely convergent Fourier series. The authors reduce the initial error by a factor ε for functions from the unit ball of the weighted periodic continuous functions spaces. Tractability is the minimal number of function samples required to solve the problem in polynomial in ε−1 and d, and the strong tractability is the presence of only a polynomial dependence in ε−1 . This problem has been recently studied for quasi-Monte Carlo quadrature rules, quadrature rules with non-negative coefficients, and rules for which all quadrature weights are arbitrary for weighted Korobov spaces of smooth periodic functions of d variables. The authors show that the tractability and strong tractability of a multivariate integration problem in worst case setting hold for the weighted periodic continuous functions spaces with absolutely convergent Fourier series under the same assumptions as in Ref.[14] on the weights of the Korobov space for quasi-Monte Carlo rules and rules for which all quadrature weights are non-negative. The arguments are not constructive. Key words Information-based complexity, tractability, Monte Carlo methods, multivariate integration 2000 MR Subject Classification 41A55, 41A63, 65D30, 41A25