QMC rules are equal weight quadrature rules for approximating integrals over $[0,1]^s$ . One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on $[0,1]^s$ with square integrable partial mixed derivatives of order $α$ . Using Parseval's identity, this smoothness can be defined for all real numbers $α> 1/2$ . This condition is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined. This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with $α$ arbitrarily close to $1/2$ . To obtain such estimates we introduce a $\log$ -scale for functions with smoothness close to $1/2$ , which we call $\log$ -Korobov spaces. We show that lattice rules can be used to obtain an integration error of order $\mathcal{O}(N^{-1/2} (\log N)^{-μ(1-λ)/2})$ for any $1/μ<λ\le 1$ , where $μ>1$ is a power in the $\log$ -scale. We also consider tractability of numerical integration for weighted Korobov spaces with product weights $(γ_j)_{j \in \mathbb{N}}$ . It is known that if $\sum_{j=1}^\infty γ_j^τ< \infty$ for some $1/(2α) < τ\le 1$ one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where $τ$ is close to $1/(2 α)$ , namely we show dimension independent error bounds under the condition that $\sum_{j=1}^\infty γ_j \max\{1, \log γ_j^{-1}\}^{μ(1-λ)} < \infty$ for some $1/μ< λ\le 1$ . The essential tool in our analysis is a $\log$ -scale Jensen's inequality. The results described above also apply to integration in $\log$ -cosine spaces using tent-transformed lattice rules.QMC rules are equal weight quadrature rules for approximating integrals over $[0,1]^s$. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on $[0,1]^s$ with square integrable partial mixed derivatives of order $α$. Using Parseval's identity, this smoothness can be defined for all real numbers $α> 1/2$. This condition is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined. This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with $α$ arbitrarily close to $1/2$. To obtain such estimates we introduce a $\log$-scale for functions with smoothness close to $1/2$, which we call $\log$-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order $\mathcal{O}(N^{-1/2} (\log N)^{-μ(1-λ)/2})$ for any $1/μ<λ\le 1$, where $μ>1$ is a power in the $\log$-scale. We also consider tractability of numerical integration for weighted Korobov spaces with product weights $(γ_j)_{j \in \mathbb{N}}$. It is known that if $\sum_{j=1}^\infty γ_j^τ< \infty$ for some $1/(2α) < τ\le 1$ one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where $τ$ is close to $1/(2 α)$, namely we show dimension independent error bounds under the condition that $\sum_{j=1}^\infty γ_j \max\{1, \log γ_j^{-1}\}^{μ(1-λ)} < \infty$ for some $1/μ< λ\le 1$. The essential tool in our analysis is a $\log$-scale Jensen's inequality. The results described above also apply to integration in $\log$-cosine spaces using tent-transformed lattice rules.

arXiv:1411.2715Nov 11, 20140 citations

Approximation of analytic functions in Korobov spaces

Josef Dick, Peter Kritzer, Friedrich Pillichshammer +1 more

Journal of ComplexityMay 13, 201355 citations

Integration and approximation in cosine space

Ronald F. A. Cools, Frances Y. Kuo, Dirk Nuyens +1 more

We analyse the worst case error (w.c.e) for approximation of d-variate non-periodic functions belonging to the weighted half-period cosine space. We use the so called "tent-transformation" on rank-1 lattice rules for the same. We briefly present some known results about cubature in these spaces and then present the results for approximation. We will start with the analysis of the approximation operator App:H_d -> L_2 where H_d is the half period cosine space. It can be shown that the w.c.e in these spaces can be related to the w.c.e in the Korobov space.We analyse the worst case error (w.c.e) for approximation of d-variate non-periodic functions belonging to the weighted half-period cosine space. We use the so called "tent-transformation" on rank-1 lattice rules for the same. We briefly present some known results about cubature in these spaces and then present the results for approximation. We will start with the analysis of the approximation operator App:H_d -> L_2 where H_d is the half period cosine space. It can be shown that the w.c.e in these spaces can be related to the w.c.e in the Korobov space.

Jul 7, 20150 citations

Integration and approximation in arbitrary dimensions

Fred J. Hickernell, H. Woźniakowski

Advances in Computational MathematicsJan 1, 2000101 citations

Numerical integration in

\log

-Korobov and

\log

-cosine spaces

QMC rules are equal weight quadrature rules for approximating integrals over

[0,1]^s

. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on

[0,1]^s

with square integrable partial mixed derivatives of order

α

. Using Parseval's identity, this smoothness can be defined for all real numbers

α> 1/2

. This condition is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined. This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with

α

arbitrarily close to

1/2

. To obtain such estimates we introduce a

\log

-scale for functions with smoothness close to

1/2

, which we call

\log

-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order

\mathcal{O}(N^{-1/2} (\log N)^{-μ(1-λ)/2})

for any

1/μ<λ\le 1

, where

μ>1

is a power in the

\log

-scale. We also consider tractability of numerical integration for weighted Korobov spaces with product weights

(γ_j)_{j \in \mathbb{N}}

. It is known that if

\sum_{j=1}^\infty γ_j^τ< \infty

for some

1/(2α) < τ\le 1

one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where

τ

is close to

1/(2 α)

, namely we show dimension independent error bounds under the condition that

\sum_{j=1}^\infty γ_j \max\{1, \log γ_j^{-1}\}^{μ(1-λ)} < \infty

for some

1/μ< λ\le 1

. The essential tool in our analysis is a

\log

-scale Jensen's inequality. The results described above also apply to integration in

\log

-cosine spaces using tent-transformed lattice rules.