arXiv subjects

Takashi Goda

Publications and source records attributed to Takashi Goda.

72 records · Page 4Linked to original sources

arXiv2017

Quasi-Monte Carlo integration for twice differentiable functions over a triangle

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.

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arXiv2016

Construction of interlaced polynomial lattice rules for infinitely differentiable functions

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.

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arXiv2016

Optimal order quadrature error bounds for infinite-dimensional higher order digital sequences

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 $s$.

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arXiv2016

Computing the variance of a conditional expectation via non-nested Monte Carlo

Computing the variance of a conditional expectation has often been of importance in uncertainty quantification. Sun et al. has introduced an unbiased nested Monte Carlo estimator, which they call $1\frac{1}{2}$-level simulation since the optimal inner-level sample size is bounded as the computational budget increases. In this letter we construct unbiased non-nested Monte Carlo estimators based on the so-called pick-freeze scheme due to Sobol'. An extension of our approach to compute higher order moments of a conditional expectation is also discussed.

stat.CO↗

arXiv2016

Unbiased Monte Carlo estimation for the expected value of partial perfect information

The expected value of partial perfect information (EVPPI) denotes the value of eliminating uncertainty on a subset of unknown parameters involved in a decision model. The EVPPI can be regarded as a decision-theoretic sensitivity index, and has been widely used for identifying relatively important unknown parameters. It follows from Jensen's inequality, however, that the standard nested Monte Carlo computation of the EVPPI results in biased estimates. In this paper we introduce two unbiased Monte Carlo estimators for the EVPPI based on multilevel Monte Carlo method, introduced by Heinrich (1998) and Giles (2008), and its extension by Rhee and Glynn (2012, 2015). Our unbiased estimators are simple and straightforward to implement, and thus are of highly practical use. Numerical experiments show that even the convergence behaviors of our unbiased estimators are superior to that of the standard nested Monte Carlo estimator.

stat.CO↗

arXiv2016

An explicit construction of optimal order quasi-Monte Carlo rules for smooth integrands

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.

math.NA↗

arXiv2016

Quasi-Monte Carlo integration using digital nets with antithetics

Antithetic sampling, which goes back to the classical work by Hammersley and Morton (1956), is one of the well-known variance reduction techniques for Monte Carlo integration. In this paper we investigate its application to digital nets over $\mathbb{Z}_b$ for quasi-Monte Carlo (QMC) integration, a deterministic counterpart of Monte Carlo, of functions defined over the $s$-dimensional unit cube. By looking at antithetic sampling as a geometric technique in a compact totally disconnected abelian group, we first generalize the notion of antithetic sampling from base $2$ to an arbitrary base $b\ge 2$. Then we analyze the QMC integration error of digital nets over $\mathbb{Z}_b$ with $b$-adic antithetics. Moreover, for a prime $b$, we prove the existence of good higher order polynomial lattice point sets with $b$-adic antithetics for QMC integration of smooth functions in weighted Sobolev spaces. Numerical experiments based on Sobol' point sets up to $s=100$ show that the rate of convergence can be improved for smooth integrands by using antithetic sampling technique, which is quite encouraging beyond the reach of our theoretical result and motivates future work to address.

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arXiv2016

Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness

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.

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arXiv2015

The $b$-adic symmetrization of digital nets for quasi-Monte Carlo integration

The notion of symmetrization, also known as Davenport's reflection principle, is well known in the area of the discrepancy theory and quasi-Monte Carlo (QMC) integration. In this paper we consider applying a symmetrization technique to a certain class of QMC point sets called digital nets over $\mathbb{Z}_{b}$. Although symmetrization has been recognized as a geometric technique in the multi-dimensional unit cube, we give another look at symmetrization as a geometric technique in a compact totally disconnected abelian group with dyadic arithmetic operations. Based on this observation we generalize the notion of symmetrization from base 2 to an arbitrary base $b\in \mathbb{N}$, $b\ge 2$. Subsequently, we study the QMC integration error of symmetrized digital nets over $\mathbb{Z}_{b}$ in a reproducing kernel Hilbert space. The result can be applied to component-by-component construction or Korobov construction for finding good symmetrized (higher order) polynomial lattice rules which achieve high order convergence of the integration error for smooth integrands at the expense of an exponential growth of the number of points with the dimension. Moreover, we consider two-dimensional symmetrized Hammersley point sets in prime base $b$, and prove that the minimum Dick weight is large enough to achieve the best possible order of $L_p$ discrepancy for all $1\le p< \infty$.

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arXiv2015

Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration

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.

math.NA↗

arXiv2015

Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces

Quadrature rules using higher order digital nets and sequences are known to exploit the smoothness of a function for numerical integration and to achieve an improved rate of convergence as compared to classical digital nets and sequences for smooth functions. A construction principle of higher order digital nets and sequences based on a digit interlacing function was introduced in [J. Dick, SIAM J. Numer. Anal., 45 (2007) pp.~2141--2176], which interlaces classical digital nets or sequences whose number of components is a multiple of the dimension. In this paper, we study the use of polynomial lattice point sets for interlaced components. We call quadrature rules using such point sets {\em interlaced polynomial lattice rules}. We consider weighted Walsh spaces containing smooth functions and derive two upper bounds on the worst-case error for interlaced polynomial lattice rules, both of which can be employed as a quality criterion for the construction of interlaced polynomial lattice rules. We investigate the component-by-component construction and the Korobov construction as a means of explicit constructions of good interlaced polynomial lattice rules that achieve the optimal rate of the worst-case error. Through this approach we are able to obtain a good dependence of the worst-case error bounds on the dimension under certain conditions on the weights, while significantly reducing the construction cost as compared to higher order polynomial lattice rules.

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arXiv2014

The Mean Square Quasi-Monte Carlo Error for Digitally Shifted Digital Nets

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.

math.NA↗

arXiv2014

The $b$-adic tent transformation for quasi-Monte Carlo integration using digital nets

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.

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arXiv2014

On the $L_p$ discrepancy of two-dimensional folded Hammersley point sets

We give an explicit construction of two-dimensional point sets whose $L_p$ discrepancy is of best possible order for all $1\le p\le \infty$. It is provided by folding Hammersley point sets in base $b$ by means of the $b$-adic baker's transformation which has been introduced by Hickernell (2002) for $b=2$ and Goda, Suzuki and Yoshiki (2013) for arbitrary $b\in \mathbb{N}$, $b\ge 2$. We prove that both the minimum Niederreiter-Rosenbloom-Tsfasman weight and the minimum Dick weight of folded Hammersley point sets are large enough to achieve the best possible order of $L_p$ discrepancy for all $1\le p\le \infty$.

math.NA↗

arXiv2014

Constructing good higher order polynomial lattice rules with modulus of reduced degree

In this paper we investigate multivariate integration in weighted unanchored Sobolev spaces of smoothness of arbitrarily high order. As quadrature points we employ higher order polynomial lattice point sets over $\mathbb{F}_{2}$ which are randomly digitally shifted and then folded using the tent transformation. We first prove the existence of good higher order polynomial lattice rules which achieve the optimal rate of the mean square worst-case error, while reducing the required degree of modulus by half as compared to higher order polynomial lattice rules whose quadrature points are randomly digitally shifted but not folded using the tent transformation. Thus we are able to restrict the search space of generating vectors significantly. We then study the component-by-component construction as an explicit means of obtaining good higher order polynomial lattice rules. In a way analogous to [J. Baldeaux, J. Dick, G. Leobacher, D. Nuyens, F. Pillichshammer, Numer. Algorithms, 59 (2012) 403--431], we show how to calculate the quality criterion efficiently and how to obtain the fast component-by-component construction using the fast Fourier transform. Our result generalizes the previous result shown by [L.L. Cristea, J. Dick, G. Leobacher, F. Pillichshammer, Numer. Math., 105 (2007) 413--455], in which the degree of smoothness is fixed at 2 and classical polynomial lattice rules are considered.

math.NA↗

arXiv2014

Fast construction of higher order digital nets for numerical integration in weighted Sobolev spaces

Higher order digital nets are special classes of point sets for quasi-Monte Carlo rules which achieve the optimal convergence rate for numerical integration of smooth functions. An explicit construction of higher order digital nets was proposed by Dick, which is based on digitally interlacing in a certain way the components of classical digital nets whose number of components is a multiple $ds$ of the dimension $s$. In this paper we give a fast computer search algorithm to find good classical digital nets suitable for interlaced components by using polynomial lattice point sets. We consider certain weighted Sobolev spaces of smoothness of arbitrarily high order, and derive an upper bound on the mean square worst-case error for digitally shifted higher order digital nets. Employing this upper bound as a quality criterion, we prove that the component-by-component construction can be used efficiently to find good polynomial lattice point sets suitable for interlaced components. Through this approach we are able to get some tractability results under certain conditions on the weights. Fast construction using the fast Fourier transform requires the construction cost of $O(dsN \log N)$ operations using $O(N)$ memory, where $N$ is the number of points and $s$ is the dimension. This implies a significant reduction in the construction cost as compared to higher order polynomial lattice point sets. Numerical experiments confirm that the performance of our constructed point sets often outperforms those of higher order digital nets with Sobol' sequences and Niederreiter-Xing sequences used for interlaced components, indicating the usefulness of our algorithm.

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arXiv2014

Construction of interlaced scrambled polynomial lattice rules of arbitrary high order

Higher order scrambled digital nets are randomized quasi-Monte Carlo rules which have recently been introduced in [J. Dick, Ann. Statist., 39 (2011), 1372--1398] and shown to achieve the optimal rate of convergence of the root mean square error for numerical integration of smooth functions defined on the $s$-dimensional unit cube. The key ingredient there is a digit interlacing function applied to the components of a randomly scrambled digital net whose number of components is $ds$, where the integer $d$ is the so-called interlacing factor. In this paper, we replace the randomly scrambled digital nets by randomly scrambled polynomial lattice point sets, which allows us to obtain a better dependence on the dimension while still achieving the optimal rate of convergence. Our results apply to Owen's full scrambling scheme as well as the simplifications studied by Hickernell, Matoušek and Owen. We consider weighted function spaces with general weights, whose elements have square integrable partial mixed derivatives of order up to $α\ge 1$, and derive an upper bound on the variance of the estimator for higher order scrambled polynomial lattice rules. Employing our obtained bound as a quality criterion, we prove that the component-by-component construction can be used to obtain explicit constructions of good polynomial lattice point sets. By first constructing classical polynomial lattice point sets in base $b$ and dimension $ds$, to which we then apply the interlacing scheme of order $d$, we obtain a construction cost of the algorithm of order $\mathcal{O}(dsmb^m)$ operations using $\mathcal{O}(b^m)$ memory in case of product weights, where $b^m$ is the number of points in the polynomial lattice point set.

math.NA↗

arXiv2014

Construction of scrambled polynomial lattice rules over $\mathbb{F}_2$ with small mean square weighted $\mathcal{L}_2$ discrepancy

The $\mathcal{L}_2$ discrepancy is one of several well-known quantitative measures for the equidistribution properties of point sets in the high-dimensional unit cube. The concept of weights was introduced by Sloan and Woźniakowski to take into account the relative importance of the discrepancy of lower dimensional projections. As known under the name of quasi-Monte Carlo methods, point sets with small weighted $\mathcal{L}_2$ discrepancy are useful in numerical integration. This study investigates the component-by-component construction of polynomial lattice rules over the finite field $\mathbb{F}_2$ whose scrambled point sets have small mean square weighted $\mathcal{L}_2$ discrepancy. An upper bound on this discrepancy is proved, which converges at almost the best possible rate of $N^{-2+δ}$ for all $δ>0$, where $N$ denotes the number of points. Numerical experiments confirm that the performance of our constructed polynomial lattice point sets is comparable or even superior to that of Sobol' sequences.

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