Searcharxiv⌕ Search

arXiv subjects

Josef Dick

Publications and source records attributed to Josef Dick.

At least 73 records · Page 4Linked to original sources
arXiv2014

Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality

In this paper we prove a correspondence principle between multivariate functions of bounded variation in the sense of Hardy and Krause and signed measures of finite total variation, which allows us to obtain a simple proof of a generalized Koksma--Hlawka inequality for non-uniform measures. Applications of this inequality to importance sampling in Quasi-Monte Carlo integration and tractability theory are given. Furthermore, we discuss the problem of transforming a low-discrepancy sequence with respect to the uniform measure into a sequence with low discrepancy with respect to a general measure $μ$, and show the limitations of a method suggested by Chelson.

math.CA↗
arXiv2014

Proof Techniques in Quasi-Monte Carlo Theory

In this survey paper we discuss some tools and methods which are of use in quasi-Monte Carlo (QMC) theory. We group them in chapters on Numerical Analysis, Harmonic Analysis, Algebra and Number Theory, and Probability Theory. We do not provide a comprehensive survey of all tools, but focus on a few of them, including reproducing and covariance kernels, Littlewood-Paley theory, Riesz products, Minkowski's fundamental theorem, exponential sums, diophantine approximation, Hoeffding's inequality and empirical processes, as well as other tools. We illustrate the use of these methods in QMC using examples.

math.NA↗
arXiv2014

Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling

In this paper we introduce a reproducing kernel Hilbert space defined on $\mathbb{R}^{d+1}$ as the tensor product of a reproducing kernel defined on the unit sphere $\mathbb{S}^{d}$ in $\mathbb{R}^{d+1}$ and a reproducing kernel defined on $[0,\infty)$. We extend Stolarsky's invariance principle to this case and prove upper and lower bounds for numerical integration in the corresponding reproducing kernel Hilbert space. The idea of separating the direction from the distance from the origin can also be applied to the construction of quadrature methods. An extension of the area-preserving Lambert transform is used to generate points on $\mathbb{S}^{d-1}$ via lifting Sobol' points in $[0,1)^{d}$ to the sphere. The $d$-th component of each Sobol' point, suitably transformed, provides the distance information so that the resulting point set is normally distributed in $\mathbb{R}^{d}$. Numerical tests provide evidence of the usefulness of constructing Quasi-Monte Carlo type methods for integration in such spaces. We also test this method on examples from financial applications (option pricing problems) and compare the results with traditional methods for numerical integration in $\mathbb{R}^{d}$.

math.NA↗
arXiv2014

Discrepancy Estimates for Acceptance-Rejection Samplers Using Stratified Inputs

In this paper we propose an acceptance-rejection sampler using stratified inputs as diver sequence. We estimate the discrepancy of the points generated by this algorithm. First we show an upper bound on the star discrepancy of order $N^{-1/2-1/(2s)}$. Further we prove an upper bound on the $q$-th moment of the $L_q$-discrepancy $(\mathbb{E}[N^{q}L^{q}_{q,N}])^{1/q}$ for $2\le q\le \infty$, which is of order $N^{(1-1/s)(1-1/q)}$. We also present an improved convergence rate for a deterministic acceptance-rejection algorithm using $(t,m,s)-$nets as driver sequence.

stat.CO↗
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

The inverse of the star-discrepancy problem and the generation of pseudo-random numbers

The inverse of the star-discrepancy problem asks for point sets $P_{N,s}$ of size $N$ in the $s$-dimensional unit cube $[0,1]^s$ whose star-discrepancy $D^\ast(P_{N,s})$ satisfies $$D^\ast(P_{N,s}) \le C \sqrt{s/N},$$ where $C> 0$ is a constant independent of $N$ and $s$. The first existence results in this direction were shown by Heinrich, Novak, Wasilkowski, and Woźniakowski in 2001, and a number of improvements have been shown since then. Until now only proofs that such point sets exist are known. Since such point sets would be useful in applications, the big open problem is to find explicit constructions of suitable point sets $P_{N,s}$. We review the current state of the art on this problem and point out some connections to pseudo-random number generators.

math.NA↗
arXiv2014

A Discrepancy Bound for a Deterministic Acceptance-Rejection Sampler

We consider an acceptance-rejection sampler based on a deterministic driver sequence. The deterministic sequence is chosen such that the discrepancy between the empirical target distribution and the target distribution is small. We use quasi-Monte Carlo (QMC) point sets for this purpose. The empirical evidence shows convergence rates beyond the crude Monte Carlo rate of $N^{-1/2}$. We prove that the discrepancy of samples generated by the QMC acceptance-rejection sampler is bounded from above by $N^{-1/s}$. A lower bound shows that for any given driver sequence, there always exists a target density such that the star discrepancy is at most $N^{-2/(s+1)}$. For a general density, whose domain is the real state space $\mathbb{R}^{s-1}$, the inverse Rosenblatt transformation can be used to convert samples from the $(s-1)-$dimensional cube to $\mathbb{R}^{s-1}$. We show that this transformation is measure preserving. This way, under certain conditions, we obtain the same convergence rate for a general target density defined in $\mathbb{R}^{s-1}$. Moreover, we also consider a deterministic reduced acceptance-rejection algorithm recently introduced by Barekat and Caflisch [F. Barekat and R.Caflisch. Simulation with Fluctuation and Singular Rates. ArXiv:1310.4555[math.NA], 2013.]

math.ST↗
arXiv2014

A reduced fast component-by-component construction of lattice points for integration in weighted spaces with fast decreasing weights

Lattice rules and polynomial lattice rules are quadrature rules for approximating integrals over the $s$-dimensional unit cube. Since no explicit constructions of such quadrature methods are known for dimensions $s > 2$, one usually has to resort to computer search algorithms. The fast component-by-component approach is a useful algorithm for finding suitable quadrature rules. We present a modification of the fast component-by-component algorithm which yields savings of the construction cost for (polynomial) lattice rules in weighted function spaces. The idea is to reduce the size of the search space for coordinates which are associated with small weights and are therefore of less importance to the overall error compared to coordinates associated with large weights. We analyze tractability conditions of the resulting QMC rules. Numerical results demonstrate the effectiveness of our method.

math.NA↗
arXiv2014

The weighted star discrepancy of Korobov's $p$-sets

We analyze the weighted star discrepancy of so-called $p$-sets which go back to definitions due to Korobov in the 1950s and Hua and Wang in the 1970s. Since then, these sets have largely been ignored since a number of other constructions have been discovered which achieve a better convergence rate. However, it has recently been discovered that the $p$-sets perform well in terms of the dependence on the dimension. We prove bounds on the weighted star discrepancy of the $p$-sets which hold for any choice of weights. For product weights we give conditions under which the discrepancy bounds are independent of the dimension $s$. This implies strong polynomial tractability for the weighted star discrepancy. We also show that a very weak condition on the product weights suffices to achieve polynomial tractability.

math.NT↗
arXiv2014

Numerical integration of Hölder continuous, absolutely convergent Fourier-, Fourier cosine-, and Walsh series

We introduce quasi-Monte Carlo rules for the numerical integration of functions $f$ defined on $[0,1]^s$, $s \ge 1$, which satisfy the following properties: the Fourier-, Fourier cosine- or Walsh coefficients of $f$ are absolutely summable and $f$ satisfies a Hölder condition of order $α$, for some $0 < α\le 1$. We show a convergent rate of the integration error of order $\max((s-1) N^{-1/2}, s^{α/2} N^{-α} )$. The construction of the quadrature points is explicit and is based on Weil sums.

math.NA↗
arXiv2013

Applications of geometric discrepancy in numerical analysis and statistics

In this paper we discuss various connections between geometric discrepancy measures, such as discrepancy with respect to convex sets (and convex sets with smooth boundary in particular), and applications to numerical analysis and statistics, like point distributions on the sphere, the acceptance-rejection algorithm and certain Markov chain Monte Carlo algorithms.

math.NA↗
arXiv2013

Discrepancy bounds for infinite-dimensional order two digital sequences over $\mathbb{F}_2$

In this paper we provide explicit constructions of digital sequences over the finite field of order 2 in the infinite dimensional unit cube whose first $N$ points projected onto the first $s$ coordinates have $\mathcal{L}_q$ discrepancy bounded by $r^{3/2-1/q} \sqrt{m_1^{s-1} + m_2^{s-1} + \cdots + m_r^{s-1}} N^{-1}$ for all $N = 2^{m_1} + 2^{m_2} + \cdots + 2^{m_r} \ge 2$ and $2 \le q < \infty$. In particular we have for $N = 2^m$ that the $\mathcal{L}_q$ discrepancy is of order $m^{(s-1)/2} 2^{-m}$ for all $2 \le q < \infty$.

math.NT↗
arXiv2013

Low-discrepancy point sets for non-uniform measures

In the present paper we prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure $μ$ on the $d$-dimensional unit cube. We improve a theorem of Beck, by showing that for any $d \geq 1$, $N \geq 1,$ and any non-negative, normalized Borel measure $μ$ on $[0,1]^d$ there exists a point set $x_1, \dots, x_N \in [0,1]^d$ whose star-discrepancy with respect to $μ$ is of order $$ D_N^*(x_1, \dots, x_N; μ) \ll \frac{(\log N)^{(3d+1)/2}}{N}. $$ For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear discrepancy of hypergraphs. Furthermore, the theory of large deviation bounds for empirical processes indexed by sets is discussed, and we prove a numerically explicit upper bound for the inverse of the discrepancy for Vapnik--Červonenkis classes. Finally, using a recent version of the Koksma--Hlawka inequality due to Brandolini, Colzani, Gigante and Travaglini, we show that our results imply the existence of cubature rules yielding fast convergence rates for the numerical integration of functions having discontinuities of a certain form.

math.NT↗
arXiv2013

Explicit constructions of point sets and sequences with low discrepancy

In this article we survey recent results on the explicit construction of finite point sets and infinite sequences with optimal order of $\mathcal{L}_q$ discrepancy. In 1954 Roth proved a lower bound for the $\mathcal{L}_2$ discrepancy of finite point sets in the unit cube of arbitrary dimension. Later various authors extended Roth's result to lower bounds also for the $\mathcal{L}_q$ discrepancy and for infinite sequences. While it was known already from the early 1980s on that Roth's lower bound is best possible in the order of magnitude, it was a longstanding open question to find explicit constructions of point sets and sequences with optimal order of $\mathcal{L}_2$ discrepancy. This problem was solved by Chen and Skriganov in 2002 for finite point sets and recently by the authors of this article for infinite sequences. These constructions can also be extended to give optimal order of the $\mathcal{L}_q$ discrepancy of finite point sets for $q \in (1,\infty)$. The main aim of this article is to give an overview of these constructions and related results.

math.NT↗
arXiv2013

Higher order Sobol' indices

Sobol' indices measure the dependence of a high dimensional function on groups of variables defined on the unit cube $[0,1]^d$. They are based on the ANOVA decomposition of functions, which is an $L^2$ decomposition. In this paper we discuss generalizations of Sobol' indices which yield $L^p$ measures of the dependence of $f$ on subsets of variables. Our interest is in values $p>2$ because then variable importance becomes more about reaching the extremes of $f$. We introduce two methods. One based on higher order moments of the ANOVA terms and another based on higher order norms of a spectral decomposition of $f$, including Fourier and Haar variants. Both of our generalizations have representations as integrals over $[0,1]^{kd}$ for $k\ge 1$, allowing direct Monte Carlo or quasi-Monte Carlo estimation. We find that they are sensitive to different aspects of $f$, and thus quantify different notions of variable importance.

math.NA↗
arXiv2013

Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition

We study the numerical integration problem for functions with infinitely many variables. The function spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms related to the ANOVA decomposition of the integrands. The weights model the relative importance of different groups of variables. We investigate randomized quadrature algorithms and measure their quality by estimating the randomized worst-case integration error. In particular, we provide lower error bounds for a very general class of randomized algorithms that includes non-linear and adaptive algorithms. Furthermore, we propose new randomized changing dimension algorithms and present favorable upper error bounds. For product weights and finite-intersection weights our lower and upper error bounds match and show that our changing dimension algorithms are optimal in the sense that they achieve convergence rates arbitrarily close to the best possible convergence rate. As more specific examples, we discuss unanchored Sobolev spaces of different degrees of smoothness and randomized changing dimension algorithms that use as building blocks scrambled polynomial lattice rules. Our analysis extends the analysis given in [J. Baldeaux, M. Gnewuch. Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition. arXiv:1209.0882v1 [math.NA], Preprint 2012]. In contrast to the previous article we now investigate a different cost model for algorithms. With respect to that cost model, randomized multilevel algorithms cannot, in general, achieve optimal convergence rates, but, as we prove for important classes of weights, changing dimension algorithms actually can.

math.NA↗
arXiv2013

Optimal $\mathcal{L}_2$ discrepancy bounds for higher order digital sequences over the finite field $\mathbb{F}_2$

We show that the $\mathcal{L}_2$ discrepancy of the explicitly constructed infinite sequences of points $(\boldsymbol{x}_0,\boldsymbol{x}_1, \boldsymbol{x}_2,...)$ in $[0,1)^s$ over $\mathbb{F}_2$ introduced in [J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal., {\bf 46}, 1519--1553, 2008] satisfy $$\mathcal{L}_{2,N}(\{\boldsymbol{x}_0,\boldsymbol{x}_1,..., \boldsymbol{x}_{N-1}\}) \le C_s N^{-1} (\log N)^{s/2} \quad {for all} N \ge 2,$$ and $$\mathcal{L}_{2,2^m}(\{\boldsymbol{x}_0,\boldsymbol{x}_1,..., \boldsymbol{x}_{2^m-1}\}) \le C_s 2^{-m} m^{(s-1)/2} \quad {for all} m \ge 1,$$ where $C_s > 0$ is a constant independent of $N$ and $m$. These results are best possible by lower bounds in [P.D. Proinov, On the $L^2$ discrepancy of some infinite sequences. Serdica, {\bf 11}, 3--12, 1985] and [K. F. Roth, On irregularities of distribution. Mathematika, {\bf 1}, 73--79, 1954]. Further, for every $N \ge 2$ we explicitly construct finite point sets $\{\boldsymbol{y}_0,..., \boldsymbol{y}_{N-1}\}$ in $[0,1)^s$ such that $$\mathcal{L}_{2,N}(\{\boldsymbol{y}_0,\boldsymbol{y}_1,..., \boldsymbol{y}_{N-1}\}) \le C_s N^{-1} (\log N)^{(s-1)/2}.$$ Another solution for finite point sets by a different construction was previously shown in [W. W. L. Chen and M. M. Skriganov, Explicit constructions in the classical mean squares problem in irregularity of point distribution. J. Reine Angew. Math., {\bf 545}, 67--95, 2002].

math.NT↗