Advances in Computational Mathematics, ISSN 1019-7168, 2/2016, Volume 42, Issue 1, pp. 55 - 84

We study multivariate integration of functions that are invariant under permutations (of subsets) of their arguments. We find an upper bound for the nth...

Visualization | 68Q25 | Computational Mathematics and Numerical Analysis | 68W40 | Numerical integration | Mathematical and Computational Biology | Mathematics | 65Y20 | Computational Science and Engineering | Quasi-Monte Carlo methods | Quadrature | Cubature | 65D32 | Mathematical Modeling and Industrial Mathematics | Rank-1 lattice rules | QUASI-MONTE CARLO | MATHEMATICS, APPLIED | Monte Carlo method | Lattice theory | Multivariate analysis | Analysis | Mathematics - Numerical Analysis

Visualization | 68Q25 | Computational Mathematics and Numerical Analysis | 68W40 | Numerical integration | Mathematical and Computational Biology | Mathematics | 65Y20 | Computational Science and Engineering | Quasi-Monte Carlo methods | Quadrature | Cubature | 65D32 | Mathematical Modeling and Industrial Mathematics | Rank-1 lattice rules | QUASI-MONTE CARLO | MATHEMATICS, APPLIED | Monte Carlo method | Lattice theory | Multivariate analysis | Analysis | 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

Annali di Matematica Pura ed Applicata (1923 -), ISSN 0373-3114, 2/2018, Volume 197, Issue 1, pp. 109 - 126

We prove upper bounds on the order of convergence of lattice-based algorithms for numerical integration in function spaces of dominating mixed smoothness on...

65D30 | 11K31 | Rank-1 lattice points | Numerical integration | Kronecker lattice | 65D32 | Mathematics, general | Mathematics | Quasi-Monte Carlo | Zaremba index | Besov space | QUADRATURE-FORMULAS | MATHEMATICS, APPLIED | CUBATURE FORMULAS | NUMBERS | APPROXIMATION | OPTIMAL COEFFICIENTS | MATHEMATICS | MULTIPLE INTEGRATION | DISCREPANCY | ERROR | RULES | BY-COMPONENT CONSTRUCTION | Monte Carlo method | Analysis | Algorithms

65D30 | 11K31 | Rank-1 lattice points | Numerical integration | Kronecker lattice | 65D32 | Mathematics, general | Mathematics | Quasi-Monte Carlo | Zaremba index | Besov space | QUADRATURE-FORMULAS | MATHEMATICS, APPLIED | CUBATURE FORMULAS | NUMBERS | APPROXIMATION | OPTIMAL COEFFICIENTS | MATHEMATICS | MULTIPLE INTEGRATION | DISCREPANCY | ERROR | RULES | BY-COMPONENT CONSTRUCTION | Monte Carlo method | Analysis | Algorithms

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 05/2016, Volume 496, pp. 438 - 451

Characterisations are provided of the Hermite normal form of certain integer circulant and skew-circulant matrices. These matrices are associated with rank-1...

Hermite normal form | Rank-1 circulant lattice rules | Rank-1circulant lattice rules | MATHEMATICS | MATHEMATICS, APPLIED | MATRICES | Online searching | Rankings | Internet/Web search services | Database searching

Hermite normal form | Rank-1 circulant lattice rules | Rank-1circulant lattice rules | MATHEMATICS | MATHEMATICS, APPLIED | MATRICES | Online searching | Rankings | Internet/Web search services | Database searching

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

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

Applied and Computational Harmonic Analysis, ISSN 1063-5203, 11/2016, Volume 41, Issue 3, pp. 713 - 748

In this paper, we suggest approximate algorithms for the reconstruction of sparse high-dimensional trigonometric polynomials, where the support in frequency...

Trigonometric polynomials | Lattice rule | Sparse fast Fourier transform | Rank-1 lattice | FFT | Approximation of multivariate functions | MATHEMATICS, APPLIED | PURSUIT | GRIDS | PARAMETER-ESTIMATION | FOURIER-TRANSFORM | STABILITY | Cytokinins | Algorithms

Trigonometric polynomials | Lattice rule | Sparse fast Fourier transform | Rank-1 lattice | FFT | Approximation of multivariate functions | MATHEMATICS, APPLIED | PURSUIT | GRIDS | PARAMETER-ESTIMATION | FOURIER-TRANSFORM | STABILITY | Cytokinins | Algorithms

Journal Article

Applied and Computational Harmonic Analysis, ISSN 1063-5203, 11/2019, Volume 47, Issue 3, pp. 702 - 729

The paper discusses the construction of high dimensional spatial discretizations for arbitrary multivariate trigonometric polynomials, where the frequency...

Sparse multivariate trigonometric polynomials | Lattice rule | Multiple rank-1 lattice | Fast Fourier transform | MATHEMATICS, APPLIED | PERIODIC-FUNCTIONS | APPROXIMATION | RANK-1 LATTICE

Sparse multivariate trigonometric polynomials | Lattice rule | Multiple rank-1 lattice | Fast Fourier transform | MATHEMATICS, APPLIED | PERIODIC-FUNCTIONS | APPROXIMATION | RANK-1 LATTICE

Journal Article

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

10.
Full Text
Multiple Rank-1 Lattices as Sampling Schemes for Multivariate Trigonometric Polynomials

Journal of Fourier Analysis and Applications, ISSN 1069-5869, 2/2018, Volume 24, Issue 1, pp. 17 - 44

We present a new sampling method that allows for the unique reconstruction of (sparse) multivariate trigonometric polynomials. The crucial idea is to use...

Abstract Harmonic Analysis | Mathematical Methods in Physics | Sparse multivariate trigonometric polynomials | Lattice rule | 65T50 | Fourier Analysis | Signal,Image and Speech Processing | Fast Fourier transform | Approximations and Expansions | Mathematics | Multiple rank-1 lattice | Partial Differential Equations | INTERPOLATION | QUASI-MONTE CARLO | MATHEMATICS, APPLIED | APPROXIMATION | FOURIER-TRANSFORM | SPARSE GRIDS | ALGORITHMS | Algorithms

Abstract Harmonic Analysis | Mathematical Methods in Physics | Sparse multivariate trigonometric polynomials | Lattice rule | 65T50 | Fourier Analysis | Signal,Image and Speech Processing | Fast Fourier transform | Approximations and Expansions | Mathematics | Multiple rank-1 lattice | Partial Differential Equations | INTERPOLATION | QUASI-MONTE CARLO | MATHEMATICS, APPLIED | APPROXIMATION | FOURIER-TRANSFORM | SPARSE GRIDS | ALGORITHMS | Algorithms

Journal Article

Journal of Approximation Theory, ISSN 0021-9045, 10/2019, Volume 246, pp. 1 - 27

In this work, we consider the approximate reconstruction of high-dimensional periodic functions based on sampling values. As sampling schemes, we utilize...

Trigonometric polynomials | Lattice rule | Generalized hyperbolic cross | Fast Fourier transform | Approximation of multivariate periodic functions | Multiple rank-1 lattice | Generalized mixed smoothness | MATHEMATICS | ALGORITHMS | Algorithms

Trigonometric polynomials | Lattice rule | Generalized hyperbolic cross | Fast Fourier transform | Approximation of multivariate periodic functions | Multiple rank-1 lattice | Generalized mixed smoothness | MATHEMATICS | ALGORITHMS | Algorithms

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

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

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2015, Volume 37, Issue 3, pp. A1436 - A1450

Quasi-Monte Carlo (QMC) rules 1/N Sigma(N-1)(n=0) f(y(n)A) can be used to approximate integrals of the form integral([0,1]s) f(yA) dy, where A is a matrix and...

Lattice rule | Korobov p-set | PDEs with random input | Fast Fourier transform | High-dimensional integration | Polynomial lattice rule | Quasi-Monte Carlo | MATHEMATICS, APPLIED | polynomial lattice rule | INTEGRATION | RANK-1 LATTICE RULES | QUASI-MONTE-CARLO | quasi-Monte Carlo | high-dimensional integration | fast Fourier transform | lattice rule | BY-COMPONENT CONSTRUCTION

Lattice rule | Korobov p-set | PDEs with random input | Fast Fourier transform | High-dimensional integration | Polynomial lattice rule | Quasi-Monte Carlo | MATHEMATICS, APPLIED | polynomial lattice rule | INTEGRATION | RANK-1 LATTICE RULES | QUASI-MONTE-CARLO | quasi-Monte Carlo | high-dimensional integration | fast Fourier transform | lattice rule | BY-COMPONENT CONSTRUCTION

Journal Article

Journal of Complexity, ISSN 0885-064X, 06/2015, Volume 31, Issue 3, pp. 424 - 456

In this paper, we present error estimates for the approximation of multivariate periodic functions in periodic Hilbert spaces of isotropic and dominating mixed...

Hyperbolic cross | Trigonometric polynomials | Lattice rule | Rank-1 lattice | Approximation of multivariate functions | Fast Fourier transform | INTERPOLATION | MATHEMATICS | MATHEMATICS, APPLIED | FOURIER-TRANSFORM | SPACES | SPARSE GRIDS

Hyperbolic cross | Trigonometric polynomials | Lattice rule | Rank-1 lattice | Approximation of multivariate functions | Fast Fourier transform | INTERPOLATION | MATHEMATICS | MATHEMATICS, APPLIED | FOURIER-TRANSFORM | SPACES | SPARSE GRIDS

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

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 1/2012, Volume 50, Issue 6, pp. 3351 - 3374

In this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the...

Error rates | Elliptic differential equations | Approximation | Error analysis | Function spaces | Logical givens | Mathematical lattices | Error bounds | Mathematical independent variables | Coefficients | Karhunen-Loève expansion | Elliptic partial differential equations with random coefficients | Infinite dimensional integration | Finite element methods | Quasi-Monte Carlo methods | MATHEMATICS, APPLIED | quasi-Monte Carlo methods | RANK-1 LATTICE RULES | APPROXIMATIONS | ALGORITHMS | TRACTABILITY | MULTIVARIATE INTEGRATION | SOBOLEV SPACES | elliptic partial differential equations with random coefficients | finite element methods | Karhunen-Loeve expansion | CONVERGENCE | infinite dimensional integration | DIMENSIONAL INTEGRATION | BROWNIAN BRIDGE | BY-COMPONENT CONSTRUCTION | Finite element method | Monte Carlo methods | Partial differential equations | Mathematical analysis | Mathematical models | Convergence

Error rates | Elliptic differential equations | Approximation | Error analysis | Function spaces | Logical givens | Mathematical lattices | Error bounds | Mathematical independent variables | Coefficients | Karhunen-Loève expansion | Elliptic partial differential equations with random coefficients | Infinite dimensional integration | Finite element methods | Quasi-Monte Carlo methods | MATHEMATICS, APPLIED | quasi-Monte Carlo methods | RANK-1 LATTICE RULES | APPROXIMATIONS | ALGORITHMS | TRACTABILITY | MULTIVARIATE INTEGRATION | SOBOLEV SPACES | elliptic partial differential equations with random coefficients | finite element methods | Karhunen-Loeve expansion | CONVERGENCE | infinite dimensional integration | DIMENSIONAL INTEGRATION | BROWNIAN BRIDGE | BY-COMPONENT CONSTRUCTION | Finite element method | Monte Carlo methods | Partial differential equations | Mathematical analysis | Mathematical models | Convergence

Journal Article

Numerical Algorithms, ISSN 1017-1398, 2/2012, Volume 59, Issue 2, pp. 161 - 183

Quasi-Monte Carlo integration rules, which are equal-weight sample averages of function values, have been popular for approximating multivariate integrals due...

Digital nets and sequences | Algebra | Algorithms | Numerical integration | Computer Science | Numeric Computing | Mathematics, general | Theory of Computation | Lattice rules and sequences | Quasi-Monte Carlo | Higher order convergence | MATHEMATICS, APPLIED | RANK-1 LATTICE RULES | BY-COMPONENT CONSTRUCTION

Digital nets and sequences | Algebra | Algorithms | Numerical integration | Computer Science | Numeric Computing | Mathematics, general | Theory of Computation | Lattice rules and sequences | Quasi-Monte Carlo | Higher order convergence | MATHEMATICS, APPLIED | RANK-1 LATTICE RULES | 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

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

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