arXiv subjects

Michael Gnewuch

Publications and source records attributed to Michael Gnewuch.

28 records · Page 2Linked to original sources

arXiv2016

Equivalence of Weighted Anchored and ANOVA Spaces of Functions with Mixed Smoothness of Order one in $L_p$

We consider $γ$-weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in a weighted $L_p$ norm with $1 \leq p \leq \infty$. The domain of the functions is $D^d$, where $D \subseteq \mathbb{R}$ is a bounded or unbounded interval. We provide conditions on the weights $γ$ that guarantee that anchored and ANOVA spaces are equal (as sets of functions) and have equivalent norms with equivalence constants uniformly or polynomially bounded in $d$. Moreover, we discuss applications of these results to integration and approximation of functions on $D^d$.

math.NA↗

arXiv2016

Embeddings of Weighted Hilbert Spaces and Applications to Multivariate and Infinite-Dimensional Integration

We study embeddings and norm estimates for tensor products of weighted reproducing kernel Hilbert spaces. These results lead to a transfer principle that is directly applicable to tractability studies of multivariate problems as integration and approximation, and to their infinite-dimensional counterparts. In an application we consider weighted tensor product Sobolev spaces of mixed smoothness of any integer order, equipped with the classical, the anchored, or the ANOVA norm. Here we derive new results for multivariate and infinite-dimensional integration.

math.NA↗

arXiv2014

Calculation of Discrepancy Measures and Applications

In this book chapter we survey known approaches and algorithms to compute discrepancy measures of point sets. After providing an introduction which puts the calculation of discrepancy measures in a more general context, we focus on the geometric discrepancy measures for which computation algorithms have been designed. In particular, we explain methods to determine $L_2$-discrepancies and approaches to tackle the inherently difficult problem to calculate the star discrepancy of given sample sets. We also discuss in more detail three applications of algorithms to approximate discrepancies.

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↗

arXiv2012

Infinite-Dimensional Integration in Weighted Hilbert Spaces: Anchored Decompositions, Optimal Deterministic Algorithms, and Higher Order Convergence

We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel algorithms and changing dimension algorithms. More precisely, the spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms induced by an underlying anchored function space decomposition. Here the weights model the relative importance of different groups of variables. The error criterion used is the deterministic worst case error. We study two cost models for function evaluation which depend on the number of active variables of the chosen sample points, and two classes of weights, namely product and order-dependent (POD) weights and the newly introduced weights with finite active dimension. We show for these classes of weights that multilevel algorithms achieve the optimal rate of convergence in the first cost model while changing dimension algorithms achieve the optimal convergence rate in the second model. As an illustrative example, we discuss the anchored Sobolev space with smoothness parameter $α$ and provide new optimal quasi-Monte Carlo multilevel algorithms and quasi-Monte Carlo changing dimension algorithms based on higher-order polynomial lattice rules.

math.NA↗

arXiv2012

Lower Error Bounds for Randomized Multilevel and Changing Dimension Algorithms

We provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on weighted Hilbert spaces with an underlying anchored decomposition and arbitrary weights. We focus on randomized algorithms and the randomized worst case error. We study two cost models for function evaluation which depend on the number of active variables of the chosen sample points. Multilevel algorithms behave very well with respect to the first cost model, while changing dimension algorithms and also dimension-wise quadrature methods, which are based on a similar idea, can take advantage of the more generous second cost model. We prove the first non-trivial lower error bounds for randomized algorithms in these cost models and demonstrate their quality in the case of product weights. In particular, we show that the randomized changing dimension algorithms provided in [L. Plaskota, G. W. Wasilkowski, J. Complexity 27 (2011), 505--518] achieve convergence rates arbitrarily close to the optimal convergence rate.

math.NA↗

arXiv2012

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

In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of ANOVA-type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first non-trivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or non-linear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. Müller-Gronbach, B. Niu, K. Ritter, J. Complexity 26 (2010), 229-254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi-Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.

math.NA↗

arXiv2011

A Randomized Algorithm Based on Threshold Accepting to Approximate the Star Discrepancy

We present a new algorithm for estimating the star discrepancy of arbitrary point sets. Similar to the algorithm for discrepancy approximation of Winker and Fang [SIAM J. Numer. Anal. 34 (1997), 2028--2042] it is based on the optimization algorithm threshold accepting. Our improvements include, amongst others, a non-uniform sampling strategy which is more suited for higher-dimensional inputs, and rounding steps which transform axis-parallel boxes, on which the discrepancy is to be tested, into \emph{critical test boxes}. These critical test boxes provably yield higher discrepancy values, and contain the box that exhibits the maximum value of the local discrepancy. We provide comprehensive experiments to test the new algorithm. Our randomized algorithm computes the exact discrepancy frequently in all cases where this can be checked (i.e., where the exact discrepancy of the point set can be computed in feasible time). Most importantly, in higher dimension the new method behaves clearly better than all previously known methods.

cs.DS↗

arXiv2008

Construction of Minimal Bracketing Covers for Rectangles

We construct explicit $δ$-bracketing covers with minimal cardinality for the set system of (anchored) rectangles in the two dimensional unit cube. More precisely, the cardinality of these $δ$-bracketing covers are bounded from above by $δ^{-2} + o(δ^{-2})$. A lower bound for the cardinality of arbitrary $δ$-bracketing covers for $d$-dimensional anchored boxes from [M. Gnewuch, Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, J. Complexity 24 (2008) 154-172] implies the lower bound $δ^{-2}+O(δ^{-1})$ in dimension $d=2$, showing that our constructed covers are (essentially) optimal. We study also other $δ$-bracketing covers for the set system of rectangles, deduce the coefficient of the most significant term $δ^{-2}$ in the asymptotic expansion of their cardinality, and compute their cardinality for explicit values of $δ$.

math.CO↗

arXiv2006

Discrepancy of Symmetric Products of Hypergraphs

For a hypergraph ${\mathcal H} = (V,{\mathcal E})$, its $d$--fold symmetric product is $Δ^d {\mathcal H} = (V^d,\{E^d |E \in {\mathcal E}\})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound ${disc}(Δ^d {\mathcal H},2) \le {disc}({\mathcal H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of {C}artesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and ${disc}(Δ^d {\mathcal H},c) = Ω_d({disc}({\mathcal H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product ${\mathcal H}^d$, which satisfies ${disc}({\mathcal H}^d,c) = O_{c,d}({disc}({\mathcal H},c)^d)$.

math.CO↗