Takehito Yoshiki

Josef Dick, Takashi Goda, Takehito Yoshiki

We study multivariate numerical integration of smooth functions in weighted Sobolev spaces with dominating mixed smoothness $α\geq 2$ defined over the $s$-dimensional unit cube. We propose a new quasi-Monte Carlo (QMC)-based quadrature rule, named \emph{extrapolated polynomial lattice rule}, which achieves the almost optimal rate of convergence. Extrapolated polynomial lattice rules are constructed in two steps: i) construction of classical polynomial lattice rules over $\mathbb{F}_b$ with $α$ consecutive sizes of nodes, $b^{m-α+1},\ldots,b^{m}$, and ii) recursive application of Richardson extrapolation to a chain of $α$ approximate values of the integral obtained by consecutive polynomial lattice rules. We prove the existence of good extrapolated polynomial lattice rules achieving the almost optimal order of convergence of the worst-case error in Sobolev spaces with general weights. Then, by restricting to product weights, we show that such good extrapolated polynomial lattice rules can be constructed by the fast component-by-component algorithm under a computable quality criterion. The required total construction cost is of order $(s+α)N\log N$, which improves the currently known result for interlaced polynomial lattice rule, that is of order $sαN\log N$. We also study the dependence of the worst-case error bound on the dimension. A big advantage of our method compared to interlaced polynomial lattice rules is that the fast QMC matrix vector method can be used in this setting, while still achieving the same rate of convergence. Such a method was previously not known. Numerical experiments for test integrands support our theoretical result.

Josef Dick, Takashi Goda, Kosuke Suzuki et al.

We study multivariate integration over the $s$-dimensional unit cube in a weighted space of infinitely differentiable functions. It is known from a recent result by Suzuki that there exists a good quasi-Monte Carlo (QMC) rule which achieves a super-polynomial convergence of the worst-case error in this function space, and moreover, that this convergence behavior is independent of the dimension under a certain condition on the weights. In this paper we provide a constructive approach to finding a good QMC rule achieving such a dimension-independent super-polynomial convergence of the worst-case error. Specifically, we prove that interlaced polynomial lattice rules, with an interlacing factor chosen properly depending on the number of points $N$ and the weights, can be constructed using a fast component-by-component algorithm in at most $O(sN(\log N)^2)$ arithmetic operations to achieve a dimension-independent super-polynomial convergence. The key idea for the proof of the worst-case error bound is to use a variant of Jensen's inequality with a purposely-designed concave function.

Matsumoto Makoto, Ryuichi Ohori, Takehito Yoshiki

In this paper, we consider Quasi-Monte Carlo (QMC) worst case error of weighted smooth function classes in $C^\infty[0,1]^s$ by a digital net over $\mathbb F_2$. We show that the ratio of the worst case error to the QMC integration error of an exponential function is bounded above and below by constants. This result provides us with a simple interpretation that a digital net with small QMC integration error for an exponential function also gives the small integration error for any function in this function space.

Takashi Goda, Kosuke Suzuki, Takehito Yoshiki

We investigate quasi-Monte Carlo (QMC) integration over the $s$-dimensional unit cube based on rank-1 lattice point sets in weighted non-periodic Sobolev spaces $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$ and their subspaces of high order smoothness $α>1$, where $\boldsymbolγ$ denotes a set of the weights. A recent paper by Dick, Nuyens and Pillichshammer has studied QMC integration in half-period cosine spaces with smoothness parameter $α>1/2$ consisting of non-periodic smooth functions, denoted by $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{cos}})$, and also in the sum of half-period cosine spaces and Korobov spaces with common parameter $α$, denoted by $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{kor}+\mathrm{cos}})$. Motivated by the results shown there, we first study embeddings and norm equivalences on those function spaces. In particular, for an integer $α$, we provide their corresponding norm-equivalent subspaces of $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$. This implies that $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{kor}+\mathrm{cos}})$ is strictly smaller than $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$ as sets for $α\geq 2$, which solves an open problem by Dick, Nuyens and Pillichshammer. Then we study the worst-case error of tent-transformed lattice rules in $\mathcal{H}(K_{2,\boldsymbolγ,s}^{\mathrm{sob}})$ and also the worst-case error of symmetrized lattice rules in an intermediate space between $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{kor}+\mathrm{cos}})$ and $\mathcal{H}(K_{α,\boldsymbolγ,s}^{\mathrm{sob}})$. We show that the almost optimal rate of convergence can be achieved for both cases, while a weak dependence of the worst-case error bound on the dimension can be obtained for the former case.

Takashi Goda, Kosuke Suzuki, Takehito Yoshiki

We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness $α\in \mathbb{N}$, $α\ge 2$, defined over the $s$-dimensional unit cube. We prove that randomly digitally shifted order $β$ digital nets can achieve the convergence of the root mean square worst-case error of order $N^{-α}(\log N)^{(s-1)/2}$ when $β\ge 2α$. The exponent of the logarithmic term, i.e., $(s-1)/2$, is improved compared to the known result by Baldeaux and Dick, in which the exponent is $sα/2$. Our result implies the existence of a digitally shifted order $β$ digital net achieving the convergence of the worst-case error of order $N^{-α}(\log N)^{(s-1)/2}$, which matches a lower bound on the convergence rate of the worst-case error for any cubature rule using $N$ function evaluations and thus is best possible.

Takashi Goda, Kosuke Suzuki, Takehito Yoshiki

Quasi-Monte Carlo (QMC) quadrature rules using higher order digital nets and sequences have been shown to achieve the almost optimal rate of convergence of the worst-case error in Sobolev spaces of arbitrary fixed smoothness $α\in \mathbb{N}$, $α\geq 2$. In a recent paper by the authors, it was proved that randomly-digitally-shifted order $2α$ digital nets in prime base $b$ achieve the best possible rate of convergence of the root mean square worst-case error of order $N^{-α}(\log N)^{(s-1)/2}$ for $N=b^m$, where $N$ and $s$ denote the number of points and the dimension, respectively, which implies the existence of an optimal order QMC rule. More recently, the authors provided an explicit construction of such an optimal order QMC rule by using Chen-Skriganov's digital nets in conjunction with Dick's digit interlacing composition. These results were for fixed number of points. In this paper we give a more general result on an explicit construction of optimal order QMC rules for arbitrary fixed smoothness $α\in \mathbb{N}$ including the endpoint case $α=1$. That is, we prove that the projection of any infinite-dimensional order $2α+1$ digital sequence in prime base $b$ onto the first $s$ coordinates achieves the best possible rate of convergence of the worst-case error of order $N^{-α}(\log N)^{(s-1)/2}$ for $N=b^m$. The explicit construction presented in this paper is not only easy to implement but also extensible in both $N$ and $s$.

Takashi Goda, Kosuke Suzuki, Takehito Yoshiki

In a recent paper by the authors, it is shown that there exists a quasi-Monte Carlo (QMC) rule which achieves the best possible rate of convergence for numerical integration in a reproducing kernel Hilbert space consisting of smooth functions. In this paper we provide an explicit construction of such an optimal order QMC rule. Our approach is to exploit both the decay and the sparsity of the Walsh coefficients of the reproducing kernel simultaneously. This can be done by applying digit interlacing composition due to Dick to digital nets with large minimum Hamming and Niederreiter-Rosenbloom-Tsfasman metrics due to Chen and Skriganov. To our best knowledge, our construction gives the first QMC rule which achieves the best possible convergence in this function space.

Josef Dick, Friedrich Pillichshammer, Kosuke Suzuki et al.

Frolov's cubature formula on the unit hypercube has been considered important since it attains an optimal rate of convergence for various function spaces. Its integration nodes are given by shrinking a suitable full rank $\mathbb{Z}$-lattice in $\mathbb{R}^d$ and taking all points inside the unit cube. The main drawback of these nodes is that they are hard to find computationally, especially in high dimensions.In such situations, quasi-Monte Carlo (QMC) rules based on digital nets have proven to be successful. However, there is still no construction known that leads to QMC rules which are optimal in the same generality as Frolov's. In this paper we investigate a polynomial analog of Frolov's cubature formula, which we expect to be important in this respect. This analog is defined in a field of Laurent series with coefficients in a finite field. A similar approach was previously studied in [M.~B.~Levin. Adelic constructions of low discrepancy sequences. Online Journal of Analytic Combinatorics. Issue 5, 2010.]. We show that our construction is a $(t,m,d)$-net, which also implies bounds on its star-discrepancy and the error of the corresponding cubature rule. Moreover, we show that our cubature rule is a QMC rule, whereas Frolov's is not, and provide an algorithm to determine its integration nodes explicitly. To this end we need to extend the notion of $(t,m,d)$-nets to fit the situation that the points can have infinite digit expansion and develop a duality theory. Additionally, we adapt the notion of admissible lattices to our setting and prove its significance.

Takashi Goda, Kosuke Suzuki, Takehito Yoshiki

We study quasi-Monte Carlo integration for twice differentiable functions defined over a triangle. We provide an explicit construction of infinite sequences of points including one by Basu and Owen (2015) as a special case, which achieves the integration error of order $N^{-1}(\log N)^3$ for any $N\geq 2$. Since a lower bound of order $N^{-1}$ on the integration error holds for any linear quadrature rule, the upper bound we obtain is best possible apart from the $\log N$ factor. The major ingredient in our proof of the upper bound is the dyadic Walsh analysis of twice differentiable functions over a triangle under a suitable recursive partitioning.

Josef Dick, Friedrich Pillichshammer, Kosuke Suzuki et al.

We prove upper bounds on the order of convergence of lattice based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov spaces of dominating mixed smoothness $\mathring{\mathbf{B}}^s_{p,θ}$, which also comprise the concept of Sobolev spaces of dominating mixed smoothness $\mathring{\mathbf{H}}^s_{p}$ as special cases. The considered algorithms are quasi-Monte Carlo rules with underlying nodes from $T_N(\mathbb{Z}^d) \cap [0,1)^d$, where $T_N$ is a real invertible generator matrix of size $d$. For such rules the worst-case error can be bounded in terms of the Zaremba index of the lattice $\mathbb{X}_N=T_N(\mathbb{Z}^d)$. We apply this result to Kronecker lattices and to rank-1 lattice point sets, which both lead to optimal error bounds up to $\log N$-factors for arbitrary smoothness $s$. The advantage of Kronecker lattices and classical lattice point sets is that the run-time of algorithms generating these point sets is very short.

Takashi Goda, Kosuke Suzuki, Takehito Yoshiki

In this paper we study quasi-Monte Carlo integration of smooth functions using digital nets. We fold digital nets over $\mathbb{Z}_{b}$ by means of the $b$-adic tent transformation, which has recently been introduced by the authors, and employ such \emph{folded digital nets} as quadrature points. We first analyze the worst-case error of quasi-Monte Carlo rules using folded digital nets in reproducing kernel Hilbert spaces. Here we need to permit digital nets with "infinite digit expansions," which are beyond the scope of the classical definition of digital nets. We overcome this issue by considering the infinite product of cyclic groups and the characters on it. We then give an explicit means of constructing good folded digital nets as follows: we use higher order polynomial lattice point sets for digital nets and show that the component-by-component construction can find good \emph{folded higher order polynomial lattice rules} that achieve the optimal convergence rate of the worst-case error in certain Sobolev spaces of smoothness of arbitrarily high order.

Takehito Yoshiki

In this paper we give a new Koksma-Hlawka type inequality for Quasi-Monte Carlo (QMC) integration. QMC integration of a function $f\colon[0,1)^s\rightarrow \mathbb{R}$ by a finite point set $\mathcal{P}\subset [0,1)^s$ is the approximation of the integral $I(f):=\int_{[0,1)^s}f(\mathbf{x})\,d\mathbf{x}$ by the average $I_{\mathcal{P}}(f):=\frac{1}{|\mathcal{P}|}\sum_{\mathbf{x} \in \mathcal{P}}f(\mathbf{x})$. We treat a certain class of point sets $\mathcal{P}$ called digital nets. A Koksma-Hlawka type inequality is an inequality bounding the integration error $\text{Err}(f;\mathcal{P}):=I(f)-I_{\mathcal{P}}(f)$ by a bound of the form $|\text{Err}(f;\mathcal{P})|\le C\cdot \|f\|\cdot D(\mathcal{P})$. We can obtain a Koksma-Hlawka type inequality by estimating bounds on $|\hat{f}(\mathbf{k})|$, where $\hat{f}(\mathbf{k})$ is a generalized Fourier coefficient with respect to the Walsh system. In this paper we prove bounds on Walsh coefficients $\hat{f}(\mathbf{k})$ by introducing an operator called `dyadic difference' $\partial_{i,n}$. By converting dyadic differences $\partial_{i,n}$ to derivatives $\frac{\partial }{\partial x_i}$, we get a new bound on $|\hat{f}(\mathbf{k})|$ for a function $f$ whose mixed partial derivatives up to order $α$ in each variable are continuous. This new bound is smaller than the known bound on $|\hat{f}(\mathbf{k})|$ under some condition. The new Koksma-Hlawka inequality is derived using this new bound on the Walsh coefficients.

Takehito Yoshiki

We give a lower bound on Walsh figure of merit (WAFOM), which is a parameter to estimate the integration error for quasi-Monte Carlo (QMC) integration by a point set called a digital net. This lower bound is optimal because the existence of point sets attaining the order was proved in [K. Suzuki, An explicit construction of point sets with large minimum Dick weight, Journal of Complexity 30, (2014), 347-354].

Takashi Goda, Ryuichi Ohori, Kosuke Suzuki et al.

In this paper, we study randomized quasi-Monte Carlo (QMC) integration using digitally shifted digital nets. We express the mean square QMC error of the $n$-th discrete approximation $f_n$ of a function $f\colon[0,1)^s\to \mathbb{R}$ for digitally shifted digital nets in terms of the Walsh coefficients of $f$. We then apply a bound on the Walsh coefficients for sufficiently smooth integrands to obtain a quality measure called Walsh figure of merit for root mean square error, which satisfies a Koksma-Hlawka type inequality on the root mean square error. Through two types of experiments, we confirm that our quality measure is of use for finding digital nets which show good convergence behaviors of the root mean square error for smooth integrands.

Takashi Goda, Kosuke Suzuki, Takehito Yoshiki

In this paper we investigate quasi-Monte Carlo (QMC) integration using digital nets over $\mathbb{Z}_b$ in reproducing kernel Hilbert spaces. The tent transformation, or the baker's transformation, was originally used for lattice rules by Hickernell (2002) to achieve higher order convergence of the integration error for smooth non-periodic integrands, and later, has been successfully applied to digital nets over $\mathbb{Z}_2$ by Cristea et al. (2007) and Goda (2014). The aim of this paper is to generalize the latter two results to digital nets over $\mathbb{Z}_b$ for an arbitrary prime $b$. For this purpose, we introduce the {\em $b$-adic tent transformation} for an arbitrary positive integer $b$ greater than 1, which is a generalization of the original (dyadic) tent transformation. Further, again for an arbitrary positive integer $b$ greater than 1, we analyze the mean square worst-case error of QMC rules using digital nets over $\mathbb{Z}_b$ which are randomly digitally shifted and then folded using the $b$-adic tent transformation in reproducing kernel Hilbert spaces. Using this result, for a prime $b$, we prove the existence of good higher order polynomial lattice rules over $\mathbb{Z}_b$ among the smaller number of candidates as compared to the result by Dick and Pillichshammer (2007), which achieve almost the optimal convergence rate of the mean square worst-case error in unanchored Sobolev spaces of smoothness of arbitrary high order.