Foundations of Computational Mathematics, ISSN 1615-3375, 10/2015, Volume 15, Issue 5, pp. 1245 - 1278

Higher order scrambled digital nets are randomized quasi-Monte Carlo rules which have recently been introduced by Dick (Ann Stat 39:1372–1398, 2011) and shown...

Interlaced scrambled polynomial lattice rules | Economics general | Weighted function spaces | Tractability of multivariate integration | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Secondary 65D30, 65D32 | Numerical Analysis | Randomized quasi-Monte Carlo | Primary 65C05 | Applications of Mathematics | Math Applications in Computer Science | Computer Science, general | Component-by-component construction | MATHEMATICS, APPLIED | SMOOTH FUNCTIONS | SEQUENCES | SPACES | DIGITAL NETS | ALGORITHMS | TRACTABILITY | MULTIVARIATE INTEGRATION | MATHEMATICS | NUMERICAL-INTEGRATION | LOW-DISCREPANCY | COMPUTER SCIENCE, THEORY & METHODS | BY-COMPONENT CONSTRUCTION | Approximation theory | Polynomials | Lattice theory | Analysis | Mathematics - Numerical Analysis

Interlaced scrambled polynomial lattice rules | Economics general | Weighted function spaces | Tractability of multivariate integration | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Secondary 65D30, 65D32 | Numerical Analysis | Randomized quasi-Monte Carlo | Primary 65C05 | Applications of Mathematics | Math Applications in Computer Science | Computer Science, general | Component-by-component construction | MATHEMATICS, APPLIED | SMOOTH FUNCTIONS | SEQUENCES | SPACES | DIGITAL NETS | ALGORITHMS | TRACTABILITY | MULTIVARIATE INTEGRATION | MATHEMATICS | NUMERICAL-INTEGRATION | LOW-DISCREPANCY | COMPUTER SCIENCE, THEORY & METHODS | BY-COMPONENT CONSTRUCTION | Approximation theory | Polynomials | Lattice theory | Analysis | Mathematics - Numerical Analysis

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 10/2008, Volume 77, Issue 264, pp. 2345 - 2373

It has been shown by Hickernell and Niederreiter that there exist generating vectors for integration lattices which yield small integration errors for n = p,...

Integers | Numerical quadratures | Algorithms | Integrands | Mathematical lattices | Hilbert spaces | Markovs inequality | Mathematical vectors | Polynomials | Sobolev spaces | Extensible lattice rule | Numerical integration | Component-by-component construction | Quasi-Monte Carlo | MATHEMATICS, APPLIED | numerical integration | component-by-component construction | SEQUENCES | WEIGHTED KOROBOV | ALGORITHMS | MULTIVARIATE INTEGRATION | SOBOLEV SPACES | QUADRATURE | CONVERGENCE | DISCREPANCY | RULES | extensible lattice rule | BY-COMPONENT CONSTRUCTION

Integers | Numerical quadratures | Algorithms | Integrands | Mathematical lattices | Hilbert spaces | Markovs inequality | Mathematical vectors | Polynomials | Sobolev spaces | Extensible lattice rule | Numerical integration | Component-by-component construction | Quasi-Monte Carlo | MATHEMATICS, APPLIED | numerical integration | component-by-component construction | SEQUENCES | WEIGHTED KOROBOV | ALGORITHMS | MULTIVARIATE INTEGRATION | SOBOLEV SPACES | QUADRATURE | CONVERGENCE | DISCREPANCY | RULES | extensible lattice rule | BY-COMPONENT CONSTRUCTION

Journal Article

Journal of Complexity, ISSN 0885-064X, 02/2016, Volume 32, Issue 1, pp. 74 - 80

The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several...

Lattice point sets | Component-by-component algorithm | Polynomial lattice point sets | MATHEMATICS, APPLIED | RANK-1 LATTICE RULES | SPACES | EQUATIONS | WEIGHTED KOROBOV | ALGORITHMS | MATHEMATICS | QUASI-MONTE CARLO | INTEGRATION | CONVERGENCE | POINTS | Algorithms

Lattice point sets | Component-by-component algorithm | Polynomial lattice point sets | MATHEMATICS, APPLIED | RANK-1 LATTICE RULES | SPACES | EQUATIONS | WEIGHTED KOROBOV | ALGORITHMS | MATHEMATICS | QUASI-MONTE CARLO | INTEGRATION | CONVERGENCE | POINTS | Algorithms

Journal Article

Mathematics of Computation, ISSN 0025-5718, 4/2006, Volume 75, Issue 254, pp. 903 - 920

We reformulate the original component-by-component algorithm for rank-1 lattices in a matrix-vector notation so as to highlight its structural properties. For...

Prime numbers | Error rates | Function spaces | Mathematical lattices | Mathematical vectors | Matrices | Mathematical functions | Sobolev spaces | Factorization | Construction costs | Numerical integration | Component-by-component construction | Quasi-Monte Carlo | Rank-1 lattice rules | Fast algorithms | MATHEMATICS, APPLIED | numerical integration | SOBOLEV SPACES | rank-1 lattice rules | component-by-component construction | fast algorithms | quasi-Monte Carlo

Prime numbers | Error rates | Function spaces | Mathematical lattices | Mathematical vectors | Matrices | Mathematical functions | Sobolev spaces | Factorization | Construction costs | Numerical integration | Component-by-component construction | Quasi-Monte Carlo | Rank-1 lattice rules | Fast algorithms | MATHEMATICS, APPLIED | numerical integration | SOBOLEV SPACES | rank-1 lattice rules | component-by-component construction | fast algorithms | quasi-Monte Carlo

Journal Article

Constructive Approximation, ISSN 0176-4276, 4/2017, Volume 45, Issue 2, pp. 311 - 344

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens et al. (Adv...

68Q25 | 68W40 | Numerical integration | Mathematics | 65Y20 | Quasi-Monte Carlo methods | Quadrature | 65D30 | Cubature | Numerical Analysis | Analysis | 65D32 | 65C05 | Component-by-component construction | Rank-1 lattice rules | APPROXIMATION | MATHEMATICS | WEIGHTED SOBOLEV SPACES | ACHIEVE | COMPLEXITY | GOOD LATTICE RULES | BY-COMPONENT CONSTRUCTION | Computer science | Monte Carlo method | Algorithms | Resveratrol

68Q25 | 68W40 | Numerical integration | Mathematics | 65Y20 | Quasi-Monte Carlo methods | Quadrature | 65D30 | Cubature | Numerical Analysis | Analysis | 65D32 | 65C05 | Component-by-component construction | Rank-1 lattice rules | APPROXIMATION | MATHEMATICS | WEIGHTED SOBOLEV SPACES | ACHIEVE | COMPLEXITY | GOOD LATTICE RULES | BY-COMPONENT CONSTRUCTION | Computer science | Monte Carlo method | Algorithms | Resveratrol

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 03/2015, Volume 276, pp. 1 - 15

Lattice rules and polynomial lattice rules are quadrature rules for approximating integrals over the -dimensional unit cube. Since no explicit constructions of...

Weighted reproducing kernel Hilbert spaces | Numerical integration | Quasi-Monte Carlo methods | Component-by-component algorithm | Component-by-component | Hilbert spaces | Weighted reproducing kernel | Carlo methods | algorithm | Quasi-Monte | HILBERT-SPACES | MATHEMATICS, APPLIED | KOROBOV | ALGORITHMS | MULTIVARIATE INTEGRATION | QUASI-MONTE CARLO | SOBOLEV SPACES | CONVERGENCE | RULES | Analysis | Algorithms

Weighted reproducing kernel Hilbert spaces | Numerical integration | Quasi-Monte Carlo methods | Component-by-component algorithm | Component-by-component | Hilbert spaces | Weighted reproducing kernel | Carlo methods | algorithm | Quasi-Monte | HILBERT-SPACES | MATHEMATICS, APPLIED | KOROBOV | ALGORITHMS | MULTIVARIATE INTEGRATION | QUASI-MONTE CARLO | SOBOLEV SPACES | CONVERGENCE | RULES | Analysis | Algorithms

Journal Article

Journal of Complexity, ISSN 0885-064X, 02/2017, Volume 38, pp. 22 - 30

The standard method for constructing generating vectors for good lattice point sets is the component-by-component construction. Numerical experiments have...

Lattice point sets | Component-by-component algorithm | MATHEMATICS, APPLIED | INTEGRATION | SPACES | WEIGHTS | COMPUTER SCIENCE, THEORY & METHODS | RULES

Lattice point sets | Component-by-component algorithm | MATHEMATICS, APPLIED | INTEGRATION | SPACES | WEIGHTS | COMPUTER SCIENCE, THEORY & METHODS | RULES

Journal Article

Journal of Complexity, ISSN 0885-064X, 2006, Volume 22, Issue 1, pp. 4 - 28

The component-by-component construction algorithm constructs the generating vector for a rank-1 lattice one component at a time by minimizing the worst-case...

Rank-1 lattices | Numerical integration | Quasi-Monte Carlo | Analysis of algorithms | Quadrature and cubature formulas | Fast component-by-component construction | rank-1 lattices | MATHEMATICS, APPLIED | numerical integration | SOBOLEV SPACES | analysis of algorithms | quadrature and cubature formulas | quasi-Monte Carlo | CONVERGENCE | COMPUTER SCIENCE, THEORY & METHODS | fast component-by-component construction | Algorithms

Rank-1 lattices | Numerical integration | Quasi-Monte Carlo | Analysis of algorithms | Quadrature and cubature formulas | Fast component-by-component construction | rank-1 lattices | MATHEMATICS, APPLIED | numerical integration | SOBOLEV SPACES | analysis of algorithms | quadrature and cubature formulas | quasi-Monte Carlo | CONVERGENCE | COMPUTER SCIENCE, THEORY & METHODS | fast component-by-component construction | Algorithms

Journal Article

SIAM JOURNAL ON NUMERICAL ANALYSIS, ISSN 0036-1429, 2019, Volume 57, Issue 1, pp. 44 - 69

We study multivariate numerical integration of smooth functions in weighted Sobolev spaces with dominating mixed smoothness alpha >= 2 defined over the...

MATHEMATICS, APPLIED | polynomial lattice rule | component-by-component construction | MONTE CARLO INTEGRATION | quasi-Monte Carlo | WALSH COEFFICIENTS | Richardson extrapolation | ALGORITHMS | EFFICIENT | BY-COMPONENT CONSTRUCTION | Mathematics - Numerical Analysis

MATHEMATICS, APPLIED | polynomial lattice rule | component-by-component construction | MONTE CARLO INTEGRATION | quasi-Monte Carlo | WALSH COEFFICIENTS | Richardson extrapolation | ALGORITHMS | EFFICIENT | BY-COMPONENT CONSTRUCTION | Mathematics - Numerical Analysis

Journal Article

Analele Stiintifice ale Universitatii Al I Cuza din Iasi - Matematica, ISSN 1221-8421, 2011, Volume 57, Issue 1, pp. 235 - 247

This paper examines construction schemes of lattice rules based on the weighted star discrepancy with general weights. Two popular schemes are considered,...

Korobov lattice rules | Component-by-component construction | Weighted star discrepancy | INTEGRALS | MATHEMATICS | component-by-component construction | SPACES | ALGORITHMS | weighted star discrepancy

Korobov lattice rules | Component-by-component construction | Weighted star discrepancy | INTEGRALS | MATHEMATICS | component-by-component construction | SPACES | ALGORITHMS | weighted star discrepancy

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 2005, Volume 43, Issue 1, pp. 76 - 95

We introduce a new construction method for digital nets which yield point sets in the s-dimensional unit cube with low star discrepancy. The digital nets are...

Digital net | Component-by-component algorithm | Weighted star discrepancy | EXISTENCE | digital net | MATHEMATICS, APPLIED | SOBOLEV SPACES | INTEGRATION | component-by-component algorithm | LATTICE RULES | weighted star discrepancy

Digital net | Component-by-component algorithm | Weighted star discrepancy | EXISTENCE | digital net | MATHEMATICS, APPLIED | SOBOLEV SPACES | INTEGRATION | component-by-component algorithm | LATTICE RULES | weighted star discrepancy

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 05/2019, Volume 351, pp. 77 - 100

In this paper, we study an efficient algorithm for constructing node sets of high-quality quasi-Monte Carlo integration rules for weighted Korobov, Walsh, and...

Polynomial lattice points | Numerical integration | Lattice points | Quasi-Monte Carlo methods | Component-by-component construction | Weighted function spaces | MATHEMATICS, APPLIED | NUMBER | SPACES | EQUATIONS | WEIGHTED KOROBOV | MULTIVARIATE INTEGRATION | CONVERGENCE | DISCREPANCY | RULES | BY-COMPONENT CONSTRUCTION

Polynomial lattice points | Numerical integration | Lattice points | Quasi-Monte Carlo methods | Component-by-component construction | Weighted function spaces | MATHEMATICS, APPLIED | NUMBER | SPACES | EQUATIONS | WEIGHTED KOROBOV | MULTIVARIATE INTEGRATION | CONVERGENCE | DISCREPANCY | RULES | BY-COMPONENT CONSTRUCTION

Journal Article

Journal of Complexity, ISSN 0885-064X, 2011, Volume 27, Issue 5, pp. 449 - 465

We study the problem of constructing shifted rank-1 lattice rules for the approximation of high-dimensional integrals with a low weighted star discrepancy, for...

Star discrepancy | Tractability | Numerical integration | Quasi-Monte Carlo | Lattice rules | Component-by-component construction | MATHEMATICS | MATHEMATICS, APPLIED | QUADRATURE | ERROR-BOUNDS | MONTE CARLO ALGORITHMS | BY-COMPONENT CONSTRUCTION

Star discrepancy | Tractability | Numerical integration | Quasi-Monte Carlo | Lattice rules | Component-by-component construction | MATHEMATICS | MATHEMATICS, APPLIED | QUADRATURE | ERROR-BOUNDS | MONTE CARLO ALGORITHMS | BY-COMPONENT CONSTRUCTION

Journal Article

Journal of Complexity, ISSN 0885-064X, 10/2016, Volume 36, pp. 166 - 181

We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use...

Cosine series | Hyperbolic crosses | Quasi-Monte Carlo methods | Function approximation | Component-by-component construction | Rank-[formula omitted] lattice rules | Rank-1 lattice rules | MATHEMATICS, APPLIED | TRIGONOMETRIC POLYNOMIALS | ALGORITHMS | MATHEMATICS | ACHIEVE | CONVERGENCE | BY-COMPONENT CONSTRUCTION | Algorithms | Mathematics - Numerical Analysis

Cosine series | Hyperbolic crosses | Quasi-Monte Carlo methods | Function approximation | Component-by-component construction | Rank-[formula omitted] lattice rules | Rank-1 lattice rules | MATHEMATICS, APPLIED | TRIGONOMETRIC POLYNOMIALS | ALGORITHMS | MATHEMATICS | ACHIEVE | CONVERGENCE | BY-COMPONENT CONSTRUCTION | Algorithms | Mathematics - Numerical Analysis

Journal Article

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2006, Volume 28, Issue 6, pp. 2162 - 2188

Lattice rules are a family of equal-weight cubature formulae for approximating high-dimensional integrals. By now it is well established that good generating...

Embedded lattice rules | Numerical integration | Quasi-Monte Carlo methods | Extensible lattice sequences | Fast component-by-component construction | Rank-1 lattice rules | MATHEMATICS, APPLIED | numerical integration | NUMBER | rank-1 lattice rules | embedded lattice rules | extensible lattice sequences | quasi-Monte Carlo methods | WEIGHTED KOROBOV | fast component-by-component construction | SOBOLEV SPACES | PRIME | CONVERGENCE | BY-COMPONENT CONSTRUCTION

Embedded lattice rules | Numerical integration | Quasi-Monte Carlo methods | Extensible lattice sequences | Fast component-by-component construction | Rank-1 lattice rules | MATHEMATICS, APPLIED | numerical integration | NUMBER | rank-1 lattice rules | embedded lattice rules | extensible lattice sequences | quasi-Monte Carlo methods | WEIGHTED KOROBOV | fast component-by-component construction | SOBOLEV SPACES | PRIME | CONVERGENCE | BY-COMPONENT CONSTRUCTION

Journal Article

Mathematics and Computers in Simulation, ISSN 0378-4754, 01/2018, Volume 143, pp. 202 - 214

The component-by-component (CBC) algorithm is a method for constructing good generating vectors for lattice rules for the efficient computation of...

Quasi-Monte Carlo methods | Lattice rules | Component-by-component algorithm | COMPUTER SCIENCE, SOFTWARE ENGINEERING | QUASI-MONTE CARLO | MATHEMATICS, APPLIED | SOBOLEV SPACES | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | INTEGRATION | RANK-1 LATTICE RULES | CONVERGENCE | BY-COMPONENT CONSTRUCTION | Algorithms | Monte Carlo method

Quasi-Monte Carlo methods | Lattice rules | Component-by-component algorithm | COMPUTER SCIENCE, SOFTWARE ENGINEERING | QUASI-MONTE CARLO | MATHEMATICS, APPLIED | SOBOLEV SPACES | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | INTEGRATION | RANK-1 LATTICE RULES | CONVERGENCE | BY-COMPONENT CONSTRUCTION | Algorithms | Monte Carlo method

Journal Article

Springer Proceedings in Mathematics and Statistics, ISSN 2194-1009, 2018, Volume 241, pp. 197 - 215

Conference Proceeding

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 2009, Volume 232, Issue 2, pp. 240 - 251

We approximate weighted integrals over Euclidean space by using shifted rank-1 lattice rules with good bounds on the “generalised weighted star discrepancy”....

Component-by-component construction | Generalised weighted star discrepancy | Rank-1 lattice rules | MATHEMATICS, APPLIED | UNBOUNDED INTEGRANDS | CONSTRUCTION | ALGORITHMS | MULTIVARIATE INTEGRATION | STRONG TRACTABILITY

Component-by-component construction | Generalised weighted star discrepancy | Rank-1 lattice rules | MATHEMATICS, APPLIED | UNBOUNDED INTEGRANDS | CONSTRUCTION | ALGORITHMS | MULTIVARIATE INTEGRATION | STRONG TRACTABILITY

Journal Article

Springer Proceedings in Mathematics and Statistics, ISSN 2194-1009, 2018, Volume 241, pp. 377 - 394

Conference Proceeding

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 04/2007, Volume 76, Issue 258, pp. 989 - 1004

We study the problem of constructing rank-1 lattice rules which have good bounds on the ``weighted star discrepancy''. Here the non-negative weights are...

Integers | Total costs | Prime numbers | Numerical quadratures | Function spaces | Cardinality | Mathematical lattices | Mathematical vectors | Mathematical functions | Construction costs | Component-by-component construction | Weighted star discrepancy | Rank-1 lattice rules | EXISTENCE | MATHEMATICS, APPLIED | rank-1 lattice rules | component-by-component construction | SPACES | ALGORITHMS | TRACTABILITY | POINTS | weighted star discrepancy | MULTIVARIATE INTEGRATION

Integers | Total costs | Prime numbers | Numerical quadratures | Function spaces | Cardinality | Mathematical lattices | Mathematical vectors | Mathematical functions | Construction costs | Component-by-component construction | Weighted star discrepancy | Rank-1 lattice rules | EXISTENCE | MATHEMATICS, APPLIED | rank-1 lattice rules | component-by-component construction | SPACES | ALGORITHMS | TRACTABILITY | POINTS | weighted star discrepancy | MULTIVARIATE INTEGRATION

Journal Article

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