On the $$L_2$$-discrepancy of Latin hypercubes

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Type: Article
Publication Date: 2025-04-25
Citations: 0
DOI: https://doi.org/10.1007/s00605-025-02065-2

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  • Monatshefte für Mathematik
The inner distance of a Latin square was defined by myself and six others during an REU in the Summer of 2020 at Moravian College. Since then, I have been … The inner distance of a Latin square was defined by myself and six others during an REU in the Summer of 2020 at Moravian College. Since then, I have been curious about its possible connections to other combinatorial mathematics. The inner distance of a matrix is the minimum value of the distance between entries in adjacent cells, where our distance metric is distance modulo $n$. Intuitively, one expects that most Latin squares have inner distance 1, for example there probably exists a pair of adjacent cells with consecutive integers. And very few should have \textit{maximum} inner distance; the maximum inner distance was found by construction for all $n\geq 3$ to be exactly $\left \lfloor\frac{n-1}{2}\right \rfloor$. In this paper we also establish existence for all smaller inner distances. Much of our introductory work is showcased in \cite{inner distance}, with a primary focus on determining the maximum inner distance for Latin squares and Sudoku Latin squares. In this paper, we focus on enumerating maximum inner distance squares, and calculated the exact number for any $n$.
We show that the sets of $d$-dimensional Latin hypercubes over a non-empty set $X$, with $d$ running over the positive integers, determine an operad which is isomorphic to a sub-operad … We show that the sets of $d$-dimensional Latin hypercubes over a non-empty set $X$, with $d$ running over the positive integers, determine an operad which is isomorphic to a sub-operad of the endomorphism operad of $X$. We generalise this to categories with finite products, and then further to internal versions for certain Cartesian closed monoidal categories with pullbacks.
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the 'Save PDF' … An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the 'Save PDF' action button.
We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order~11, (2) answer some questions of Alter by … We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order~11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order $n$ is divisible by $f!$ where $f$ is a particular integer close to $\frac12n$, (3) provide a formula for the number of Latin squares in terms of permanents of $(+1,-1)$-matrices, (4) find the extremal values for the number of 1-factorisations of $k$-regular bipartite graphs on $2n$ vertices whenever $1\leq k\leq n\leq11$, (5) show that the proportion of Latin squares with a non-trivial symmetry group tends quickly to zero as the order increases.
We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order~11, (2) answer some questions of Alter by … We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order~11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order $n$ is divisible by $f!$ where $f$ is a particular integer close to $\frac12n$, (3) provide a formula for the number of Latin squares in terms of permanents of $(+1,-1)$-matrices, (4) find the extremal values for the number of 1-factorisations of $k$-regular bipartite graphs on $2n$ vertices whenever $1\leq k\leq n\leq11$, (5) show that the proportion of Latin squares with a non-trivial symmetry group tends quickly to zero as the order increases.
This paper contains a collection of results on Latin hypercube sampling. The first result is a Berry-Esseen-type bound for the multivariate central limit theorem of the sample mean $\hat{\mu}_n$ based … This paper contains a collection of results on Latin hypercube sampling. The first result is a Berry-Esseen-type bound for the multivariate central limit theorem of the sample mean $\hat{\mu}_n$ based on a Latin hypercube sample. The second establishes sufficient conditions on the convergence rate in the strong law for $\hat{\mu}_n$. Finally motivated by the concept of empirical likelihood, a way of constructing nonparametric confidence regions based on Latin hypercube samples is proposed for vector means.
Abstract We establish that the logarithm of the number of latin d ‐cubes of order n is and the logarithm of the number of sets of t ( is fixed) … Abstract We establish that the logarithm of the number of latin d ‐cubes of order n is and the logarithm of the number of sets of t ( is fixed) orthogonal latin squares of order n is . Similar estimations are obtained for systems of mutually strongly orthogonal latin d ‐cubes. As a consequence, we construct a set of Steiner quadruple systems of order n such that the logarithm of its cardinality is as and .
This chapter discusses a general design approach to planning computer experiments, which seeks design points that fill a bounded design region as uniformly as possible. Such designs are broadly referred … This chapter discusses a general design approach to planning computer experiments, which seeks design points that fill a bounded design region as uniformly as possible. Such designs are broadly referred to as space-filling designs.
This chapter discusses a general design approach to planning computer experiments, which seeks design points that fill a bounded design region as uniformly as possible. Such designs are broadly referred … This chapter discusses a general design approach to planning computer experiments, which seeks design points that fill a bounded design region as uniformly as possible. Such designs are broadly referred to as space-filling designs.
This chapter discusses a general design approach to planning computer experiments, which seeks design points that fill a bounded design region as uniformly as possible. Such designs are broadly referred … This chapter discusses a general design approach to planning computer experiments, which seeks design points that fill a bounded design region as uniformly as possible. Such designs are broadly referred to as space-filling designs.
A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of … A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial rectangles with minimum discrepancy (0 or 1 for each line depending on the parity). Specifically, we get explicit formula for the number of matrices with minimum L^infinity-discrepancy up to 4 rows, and establish the order of magnitude of the number of such matrices with m rows and n columns while m is fixed and n approaches infinity. By considering the number of column-good matrices with a fixed row-sum vector, we have developed a theory of decreasing criterion on based row-sum vectors with majorization relation, which turns out to be a helpful tool in the proof of our main theorems.
For a graph $G$, its \emph{cubicity} $cub(G)$ is the minimum dimension $k$ such that $G$ is representable as the intersection graph of (axis--parallel) cubes in $k$--dimensional space. Chandran, Mannino and … For a graph $G$, its \emph{cubicity} $cub(G)$ is the minimum dimension $k$ such that $G$ is representable as the intersection graph of (axis--parallel) cubes in $k$--dimensional space. Chandran, Mannino and Oriolo showed that for a $d$--dimensional hypercube $H_d$, $\frac{d-1}{\log d} \le cub(H_d) \le 2d$. In this paper, we show that $cub(H_d) = Θ(\frac{d}{\log d})$.The parameter \emph{boxicity} generalizes cubicity: the boxicity $box(G)$ of a graph $G$ is defined as the minimum dimension $k$ such that $G$ is representable as the intersection graph of axis parallel boxes in $k$ dimensional space. Since $box(G) \le cub(G)$ for any graph $G$, our result implies that $box(H_d) = O(\frac{d}{\log d})$. The problem of determining a non-trivial lower bound for $box(H_d)$ is left open.
Doyle (circa 1980) found a formula for the number of $$k \times n$$ Latin rectangles $$L_{k,n}$$ . This formula remained dormant until it was recently used for counting $$k \times … Doyle (circa 1980) found a formula for the number of $$k \times n$$ Latin rectangles $$L_{k,n}$$ . This formula remained dormant until it was recently used for counting $$k \times n$$ Latin rectangles, where $$k \in \{4,5,6\}$$ . We give a formal proof of Doyle’s formula for arbitrary k. We also improve a previous implementation of this formula, which we use to find $$L_{k,n}$$ when $$k=4$$ and $$n \le 150$$ , when $$k=5$$ and $$n \le 40$$ and when $$k=6$$ and $$n \le 15$$ . Motivated by computational data for $$3 \le k \le 6$$ , some research problems and conjectures about the divisors of $$L_{k,n}$$ are presented.
A Latin square of order $n$ is an $n\times n$ array which contains $n$ distinct symbols exactly once in each row and column. We define the adjacent distance between two … A Latin square of order $n$ is an $n\times n$ array which contains $n$ distinct symbols exactly once in each row and column. We define the adjacent distance between two adjacent cells (containing integers) to be their difference modulo $n$, and inner distance of a Latin square to be the minimum of adjacent distances in the Latin square. By first establishing upper bounds and then constructing squares with said inner distance, we found the maximum inner distance of an $n \times n$ Latin square to be $\left\lfloor\frac{n-1}{2}\right\rfloor$. We then studied special kinds of Latin squares such as pandiagonals (also known as Knut-Vik designs), as well as Sudoku Latin squares. This research was conducted at the REU at Moravian College on Research Challenges of Computational and Experimental Mathematics, with support from the National Science Foundation.
A Latin square of order $n$ is an $n\times n$ array which contains $n$ distinct symbols exactly once in each row and column. We define the adjacent distance between two … A Latin square of order $n$ is an $n\times n$ array which contains $n$ distinct symbols exactly once in each row and column. We define the adjacent distance between two adjacent cells (containing integers) to be their difference modulo $n$, and inner distance of a Latin square to be the minimum of adjacent distances in the Latin square. By first establishing upper bounds and then constructing squares with said inner distance, we found the maximum inner distance of an $n \times n$ Latin square to be $\left\lfloor\frac{n-1}{2}\right\rfloor$. We then studied special kinds of Latin squares such as pandiagonals (also known as Knut-Vik designs), as well as Sudoku Latin squares. This research was conducted at the REU at Moravian College on Research Challenges of Computational and Experimental Mathematics, with support from the National Science Foundation.
Large deviations theory is a well-studied area which has shown to have numerous applications. The typical results, however, assume that the underlying random variables are either i.i.d. or exhibit some … Large deviations theory is a well-studied area which has shown to have numerous applications. The typical results, however, assume that the underlying random variables are either i.i.d. or exhibit some form of Markovian dependence. Our interest in this paper is to study the validity of large deviations results in the context of estimators built with Latin hypercube sampling, a well-known sampling technique for variance reduction. We show that a large deviation principle holds for Latin hypercube sampling for functions in one dimension and for separable multidimensional functions. Moreover, the upper bound of the probability of a large deviation in these cases is no higher under Latin hypercube sampling than it is under Monte Carlo sampling. We extend the latter property to functions that preserve negative dependence (such as functions that are monotone in each argument). Numerical experiments illustrate the theoretical results presented in the paper.
Large deviations theory is a well-studied area which has shown to have numerous applications. The typical results, however, assume that the underlying random variables are either i.i.d. or exhibit some … Large deviations theory is a well-studied area which has shown to have numerous applications. The typical results, however, assume that the underlying random variables are either i.i.d. or exhibit some form of Markovian dependence. Our interest in this paper is to study the validity of large deviations results in the context of estimators built with Latin Hypercube sampling, a well-known sampling technique for variance reduction. We show that a large deviation principle holds for Latin Hypercube sampling for functions in one dimension and for separable multi-dimensional functions. Moreover, the upper bound of the probability of a large deviation in these cases is no higher under Latin Hypercube sampling than it is under Monte Carlo sampling. We extend the latter property to functions that preserve negative dependence (such as functions that are monotone in each argument). Numerical experiments illustrate the theoretical results presented in the paper.
Latin Hypercube Sampling is a specific Monte Carlo estimator for numerical integration of functions on ${\mathbb R}^{d}$ with respect to some product probability distribution function. Previous analysis established that Latin … Latin Hypercube Sampling is a specific Monte Carlo estimator for numerical integration of functions on ${\mathbb R}^{d}$ with respect to some product probability distribution function. Previous analysis established that Latin Hypercube Sampling is superior to independent sampling, at least asymptotically; especially, if the function to be integrated allows a good additive fit. We propose an explicit approach to Latin Hypercube Sampling, based on orthogonal projections in an appropriate Hilbert space, related to the ANOVA decomposition, which allows a rigorous error analysis. Moreover, we indicate why convergence cannot be uniformly superior to independent sampling on the class of square integrable functions. We establish a general condition under which uniformity can be achieved, thereby indicating the rôle of certain Sobolev spaces.
The $p$-adic diaphony as introduced by Hellekalek is a quantitative measure for the irregularity of distribution of a sequence in the unit cube. In this paper we show how this … The $p$-adic diaphony as introduced by Hellekalek is a quantitative measure for the irregularity of distribution of a sequence in the unit cube. In this paper we show how this notion of diaphony can be interpreted as worst-case integration error in a certain reproducing kernel Hilbert space. Our main result is an upper bound on the $p$-adic diaphony of the Halton sequence.
MathematikaVolume 3, Issue 2 p. 131-135 Research Article Note on irregularities of distribution H. Davenport, H. Davenport University College, LondonSearch for more papers by this author H. Davenport, H. Davenport … MathematikaVolume 3, Issue 2 p. 131-135 Research Article Note on irregularities of distribution H. Davenport, H. Davenport University College, LondonSearch for more papers by this author H. Davenport, H. Davenport University College, LondonSearch for more papers by this author First published: 01 December 1956 https://doi.org/10.1112/S0025579300001807Citations: 63AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume3, Issue2December 1956Pages 131-135 RelatedInformation
The goal of this paper is to study point distributions in the multi-dimensional unit cube which possess the structure of finite abelian groups with respect to certain p-ary arithmetic operations. … The goal of this paper is to study point distributions in the multi-dimensional unit cube which possess the structure of finite abelian groups with respect to certain p-ary arithmetic operations. Such distributions can be thought of as finite subgroups in a compact totally disconnected group of the Cantor type. We apply the methods of Lq harmonic analysis to estimate very precisely the Lq-discrepancies for such distributions. Following this approach, we explicitly construct point distributions with the minimal order of the Lq-discrepancy for each q, 1 < q < ∞.
This paper contains a collection of results on Latin hypercube sampling. The first result is a Berry-Esseen-type bound for the multivariate central limit theorem of the sample mean $\hat{\mu}_n$ based … This paper contains a collection of results on Latin hypercube sampling. The first result is a Berry-Esseen-type bound for the multivariate central limit theorem of the sample mean $\hat{\mu}_n$ based on a Latin hypercube sample. The second establishes sufficient conditions on the convergence rate in the strong law for $\hat{\mu}_n$. Finally motivated by the concept of empirical likelihood, a way of constructing nonparametric confidence regions based on Latin hypercube samples is proposed for vector means.
Let hR denote an L∞-normalized Haar function adapted to a dyadic rectangle R⊂[0,1]3. We show that there is a positive η<1/2 so that for all integers n and coefficients α(R), … Let hR denote an L∞-normalized Haar function adapted to a dyadic rectangle R⊂[0,1]3. We show that there is a positive η<1/2 so that for all integers n and coefficients α(R), we have 2-n∑|R|=2-n|α(R)|≲n1-η‖∑|R|=2-nα(R)hR‖∞. This is an improvement over the trivial estimate by an amount of n-η, while the small ball conjecture says that the inequality should hold with η=1/2. There is a corresponding lower bound on the L∞-norm of the discrepancy function of an arbitrary distribution of a finite number of points in the unit cube in three dimensions. The prior result, in dimension three, is that of József Beck [1, Theorem 1.2], in which the improvement over the trivial estimate was logarithmic in n. We find several simplifications and extensions of Beck's argument to prove the result above
Abstract For a point set in the multidimensional unit torus we introduce an L k -measure of uniformity of distribution, which for k =2 reduces to diaphony (and thus in … Abstract For a point set in the multidimensional unit torus we introduce an L k -measure of uniformity of distribution, which for k =2 reduces to diaphony (and thus in this case essentially coincides with Weyl L 2 -discrepancy). For k ∈ [1, 2] we establish a sharp asymptotic for this new measure as the number of points of the set tends to infinity. Upper and lower-bound estimates are given also for k &gt;2.
We count all latin cubes of order $n\le6$ and latin hypercubes of order $n\le5$ and dimension $d\le5$. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For … We count all latin cubes of order $n\le6$ and latin hypercubes of order $n\le5$ and dimension $d\le5$. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For the same values of n and d we classify all d-ary quasigroups of order n into isomorphism classes and also count them according to the number of identity elements they possess (meaning we have counted the d-ary loops). We also give an exact formula for the number of (isomorphism classes of) d-ary quasigroups of order 3 for every d. Then we give a number of constructions for d-ary quasigroups with a specific number of identity elements. In the process, we prove that no 3-ary loop of order n can have exactly $n-1$ identity elements (but no such result holds in dimensions other than 3). Finally, we give some new examples of latin cuboids which cannot be extended to latin cubes.
Quasi-random (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error … Quasi-random (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the s-dimensional unit cube is measured by its discrepancy, which is of size $(\log N)^s N^{ - 1} $ for large N, as opposed to discrepancy of size $(\log \log N)^{1/2} N^{ - 1/2} $ for a random sequence (i.e., for almost any randomly chosen sequence). Several types of discrepancies, one of which is new, are defined and analyzed. A critical discussion of the theoretical bounds on these discrepancies is presented. Computations of discrepancies are presented for a wide choice of dimension s, number of points N, and different quasi-random sequences. In particular for moderate or large s, there is an intermediate regime in which the discrepancy of a quasi-random sequence is almost exactly the same as that of a randomly chosen sequence. A simplified proof is given for Woźniakowski's result relating discrepancy and average integration error, and this result is generalized to other measures on function space.
We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness $\alpha \in \mathbb{N}$, $\alpha \ge 2$, defined over the $s$-dimensional unit cube. … We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness $\alpha \in \mathbb{N}$, $\alpha \ge 2$, defined over the $s$-dimensional unit cube. We prove that randomly digitally shifted order $\beta$ digital nets can achieve the convergence of the root mean square worst-case error of order $N^{-\alpha}(\log N)^{(s-1)/2}$ when $\beta \ge 2\alpha$. 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 $s\alpha /2$. Our result implies the existence of a digitally shifted order $\beta$ digital net achieving the convergence of the worst-case error of order $N^{-\alpha}(\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.
A new sampling algorithm is presented based on repeated additions of Latin Hypercube Sampling (LHS) where the new samples are drawn at predetermined bin fractions. The method is motivated by … A new sampling algorithm is presented based on repeated additions of Latin Hypercube Sampling (LHS) where the new samples are drawn at predetermined bin fractions. The method is motivated by a desire to have incremental growth in the sample size and maintain uniform marginal distributions. The method can be coupled with a correlation control algorithm as well as a discrepancy reduction algorithm to ensure a uniform distribution of points in the sample space. The overall size of the sample space increases by a user specified base size (typically 25 to 100 points). At the completion of each base sample, the algorithm can estimate the convergence or error in the simulation to assess when sufficient samples have been executed to achieve a specified tolerance. As presented here, the replicated Latin Hypercube Sampling (rLHS) scheme is commonly coupled with correlation control, and discrepancy control. The preferred correlation control algorithm is by Iman & Conover that can be used to minimize spurious correlation between inputs or to induce a desired correlation. The discrepancy control algorithm is based on a simple accept/reject scheme and is only used in this work to demonstrate the effect of reducing the discrepancy for a number of test cases. The preferred discrepancy measure is based on the wrap-around L2. This discrepancy measure allows the algorithm to clean-up (i.e., reduce the discrepancy) the most important faces of the hypercube based on the importance of each input variable. The algorithm is applied to several application problems where it is observed to yield up to four orders of magnitude lower error than simple random sampling (SRS) when estimating the mean of the response. The improvement in the percentile estimates are less dramatic, yet is often one order of magnitude lower error than for SRS.
We give an exact formula for the L 2 discrepancy of a class of generalized two-dimensional Hammersley point sets in base b, namely generalized Zaremba point sets. These point sets … We give an exact formula for the L 2 discrepancy of a class of generalized two-dimensional Hammersley point sets in base b, namely generalized Zaremba point sets. These point sets are digitally shifted Hammersley point sets with an arbitrary number of different digital shifts in base b. The Zaremba point set introduced by White in 1975 is the special case where the b shifts are taken repeatedly in sequential order, hence needing at least b b points to obtain the optimal order of L 2 discrepancy. On the contrary, our study shows that only one non-zero shift is enough for the same purpose, whatever the base b is.
The theory of numbers plays a central role in several areas of great importance to computer science, such as cryptography, pseudo-random number generation, complexity theory, algebraic problems, coding theory, and … The theory of numbers plays a central role in several areas of great importance to computer science, such as cryptography, pseudo-random number generation, complexity theory, algebraic problems, coding theory, and combinatorics, to name just a few. We have already seen that relatively simple properties of prime numbers allow us to devise k-wise independent variables (Chapter 3), and number-theoretic ideas are at the heart of the algebraic techniques in randomization discussed in Chapter 7.
Abstract We study the periodic $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> -discrepancy of point sets in the d -dimensional torus. This discrepancy is intimately connected with the root-mean-square $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> -discrepancy of … Abstract We study the periodic $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> -discrepancy of point sets in the d -dimensional torus. This discrepancy is intimately connected with the root-mean-square $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> -discrepancy of shifted point sets, with the notion of diaphony, and with the worst-case error of cubature formulas for the integration of periodic functions in Sobolev spaces of mixed smoothness. In discrepancy theory, many results are based on averaging arguments. In order to make such results relevant for applications, one requires explicit constructions of point sets with “average” discrepancy. In our main result, we study Korobov’s p -sets and show that this point sets have periodic $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> -discrepancy of average order. This result is related to an open question of Novak and Woźniakowski.
We study the extreme and the periodic&nbsp;$L_2$ discrepancy of plane point sets. The extreme discrepancy is based on arbitrary rectangles as test sets whereas the periodic discrepancy uses "periodic intervals", … We study the extreme and the periodic&nbsp;$L_2$ discrepancy of plane point sets. The extreme discrepancy is based on arbitrary rectangles as test sets whereas the periodic discrepancy uses "periodic intervals", which can be seen as intervals on the t
We study the extreme and the periodic $L_p$ discrepancy of point sets in the $d$-dimensional unit cube. The extreme discrepancy uses arbitrary subintervals of the unit cube as test sets, … We study the extreme and the periodic $L_p$ discrepancy of point sets in the $d$-dimensional unit cube. The extreme discrepancy uses arbitrary subintervals of the unit cube as test sets, whereas the periodic discrepancy is based on periodic intervals modu
Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasi–Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic … Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasi–Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic concepts and then advances to current methods used in research. As deterministic versions of the Monte Carlo method, quasi–Monte Carlo rules have increased in popularity, with many fruitful applications in mathematical practice. These rules require nodes with good uniform distribution properties, and digital nets and sequences in the sense of Niederreiter are known to be excellent candidates. Besides the classical theory, the book contains chapters on reproducing kernel Hilbert spaces and weighted integration, duality theory for digital nets, polynomial lattice rules, the newest constructions by Niederreiter and Xing and many more. The authors present an accessible introduction to the subject based mainly on material taught in undergraduate courses with numerous examples, exercises and illustrations.