SIAM Journal on Numerical Analysis, ISSN 0036-1429, 2018, Volume 56, Issue 1, pp. 111 - 135

We analyze convergence rates of first-order quasi-Monte Carlo (QMC) integration with randomly shifted lattice rules and for higher-order, interlaced polynomial...

Uncertainty quantification | Elliptic partial differential equations with random input | Error estimates | Quasi–Monte carlo methods | High-dimensional quadrature | MATHEMATICS, APPLIED | uncertainty quantification | POLYNOMIAL-APPROXIMATION | quasi-Monte Carlo methods | RANK-1 LATTICE RULES | high-dimensional quadrature | ALGORITHMS | PETROV-GALERKIN DISCRETIZATION | HIGH-DIMENSIONAL INTEGRATION | OPERATOR-EQUATIONS | PARTIAL-DIFFERENTIAL-EQUATIONS | elliptic partial differential equations with random input | COEFFICIENTS | error estimates | BY-COMPONENT CONSTRUCTION

Uncertainty quantification | Elliptic partial differential equations with random input | Error estimates | Quasi–Monte carlo methods | High-dimensional quadrature | MATHEMATICS, APPLIED | uncertainty quantification | POLYNOMIAL-APPROXIMATION | quasi-Monte Carlo methods | RANK-1 LATTICE RULES | high-dimensional quadrature | ALGORITHMS | PETROV-GALERKIN DISCRETIZATION | HIGH-DIMENSIONAL INTEGRATION | OPERATOR-EQUATIONS | PARTIAL-DIFFERENTIAL-EQUATIONS | elliptic partial differential equations with random input | COEFFICIENTS | error estimates | BY-COMPONENT CONSTRUCTION

Journal Article

Journal of the Royal Statistical Society: Series B (Statistical Methodology), ISSN 1369-7412, 06/2015, Volume 77, Issue 3, pp. 509 - 579

Summary We derive and study sequential quasi Monte Carlo (SQMC), a class of algorithms obtained by introducing QMC point sets in particle filtering. SQMC is...

Particle filtering | Array‐randomized quasi Monte Carlo | Quasi Monte Carlo; Randomized quasi Monte Carlo; Sequential Monte Carlo | Low discrepancy | Array-randomized quasi Monte Carlo | Quasi Monte Carlo | SCRAMBLED NET QUADRATURE | STATISTICS & PROBABILITY | Randomized quasi Monte Carlo | SIMULATION | RANDOMIZATION | INTEGRATION | STATE-SPACE MODELS | AUXILIARY PARTICLE FILTERS | TIME-SERIES | Sequential Monte Carlo | DATA ASSIMILATION | BAYESIAN-INFERENCE | CENTRAL-LIMIT-THEOREM | Monte Carlo method | Algorithms | Studies | Statistical methods | Statistical analysis | Monte Carlo simulation

Particle filtering | Array‐randomized quasi Monte Carlo | Quasi Monte Carlo; Randomized quasi Monte Carlo; Sequential Monte Carlo | Low discrepancy | Array-randomized quasi Monte Carlo | Quasi Monte Carlo | SCRAMBLED NET QUADRATURE | STATISTICS & PROBABILITY | Randomized quasi Monte Carlo | SIMULATION | RANDOMIZATION | INTEGRATION | STATE-SPACE MODELS | AUXILIARY PARTICLE FILTERS | TIME-SERIES | Sequential Monte Carlo | DATA ASSIMILATION | BAYESIAN-INFERENCE | CENTRAL-LIMIT-THEOREM | Monte Carlo method | Algorithms | Studies | Statistical methods | Statistical analysis | Monte Carlo simulation

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 1/2007, Volume 45, Issue 5, pp. 2141 - 2176

In this paper, we give explicit constructions of point sets in the s-dimensional unit cube yielding quasi-Monte Carlo algorithms which achieve the optimal rate...

Integers | Numerical quadratures | Real numbers | Natural numbers | Mathematical lattices | Walsh function | Hilbert spaces | Mathematical vectors | Mathematical functions | Matrices | Digital net | Digital sequence | Lattice rule | Numerical integration | Quasi-Monte Carlo method | digital net | MATHEMATICS, APPLIED | numerical integration | WEIGHTED SOBOLEV SPACES | SEQUENCES | LATTICE RULES | DIGITAL NETS | DISCREPANCY | quasi-Monte Carlo method | digital sequence | lattice rule | POINT SETS | Mathematics - Numerical Analysis

Integers | Numerical quadratures | Real numbers | Natural numbers | Mathematical lattices | Walsh function | Hilbert spaces | Mathematical vectors | Mathematical functions | Matrices | Digital net | Digital sequence | Lattice rule | Numerical integration | Quasi-Monte Carlo method | digital net | MATHEMATICS, APPLIED | numerical integration | WEIGHTED SOBOLEV SPACES | SEQUENCES | LATTICE RULES | DIGITAL NETS | DISCREPANCY | quasi-Monte Carlo method | digital sequence | lattice rule | POINT SETS | Mathematics - Numerical Analysis

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 01/2019, Volume 77, Issue 1, pp. 144 - 172

We analyze combined Quasi-Monte Carlo quadrature and Finite Element approximations in Bayesian estimation of solutions to countably-parametric operator...

Infinite-dimensional quadrature | Bayesian inverse problems | Uncertainty quantification | SPOD weights | Parametric operator equations | Quasi-Monte Carlo | Lattice rules | CBC construction | Digital nets | MATHEMATICS, APPLIED | POLYNOMIAL-APPROXIMATION | ANALYTIC REGULARITY | WALSH COEFFICIENTS | PETROV-GALERKIN DISCRETIZATION | STOCHASTIC COLLOCATION METHOD | PARTIAL-DIFFERENTIAL-EQUATIONS | CONSTRUCTION | RULES | Finite element method | Domains | Monte Carlo method | Economic models | Parameter estimation | Dependence | Smoothness | Bayesian analysis | Regularity

Infinite-dimensional quadrature | Bayesian inverse problems | Uncertainty quantification | SPOD weights | Parametric operator equations | Quasi-Monte Carlo | Lattice rules | CBC construction | Digital nets | MATHEMATICS, APPLIED | POLYNOMIAL-APPROXIMATION | ANALYTIC REGULARITY | WALSH COEFFICIENTS | PETROV-GALERKIN DISCRETIZATION | STOCHASTIC COLLOCATION METHOD | PARTIAL-DIFFERENTIAL-EQUATIONS | CONSTRUCTION | RULES | Finite element method | Domains | Monte Carlo method | Economic models | Parameter estimation | Dependence | Smoothness | Bayesian analysis | Regularity

Journal Article

IMA JOURNAL OF NUMERICAL ANALYSIS, ISSN 0272-4979, 07/2019, Volume 39, Issue 3, pp. 1563 - 1593

Quasi-Monte Carlo (QMC) integration of output functionals of solutions of the diffusion problem with a log-normal random coefficient is considered. The random...

log-normal | SPARSE POLYNOMIAL-APPROXIMATION | MATHEMATICS, APPLIED | quasi-Monte Carlo methods | PARTIAL-DIFFERENTIAL-EQUATIONS | CONSERVATIVE TRANSPORT | partial differential equations with random coefficients | SIMULATION | FLOW

log-normal | SPARSE POLYNOMIAL-APPROXIMATION | MATHEMATICS, APPLIED | quasi-Monte Carlo methods | PARTIAL-DIFFERENTIAL-EQUATIONS | CONSERVATIVE TRANSPORT | partial differential equations with random coefficients | SIMULATION | FLOW

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 1/2008, Volume 46, Issue 3, pp. 1519 - 1553

We define a Walsh space which contains all functions whose partial mixed derivatives up to order 6 ≥ 1 exist and have finite variation. In particular, for a...

Integers | Series convergence | Prime numbers | Numerical integration | Real numbers | Walsh function | Mathematical functions | Matrices | Mathematics | Sobolev spaces | Digital nets and sequences | Quasi-Monte Carlo | Walsh functions | Walsh | MATHEMATICS, APPLIED | numerical integration | SERIES | functions | DIGITAL NETS | LATTICE RULES | ALGORITHMS | MULTIVARIATE INTEGRATION | SOBOLEV SPACES | quasi-Monte Carlo | digital nets and sequences | NUMERICAL-INTEGRATION | DISCREPANCY | WEIGHTS | CONSTRUCTIONS | Mathematics - Numerical Analysis

Integers | Series convergence | Prime numbers | Numerical integration | Real numbers | Walsh function | Mathematical functions | Matrices | Mathematics | Sobolev spaces | Digital nets and sequences | Quasi-Monte Carlo | Walsh functions | Walsh | MATHEMATICS, APPLIED | numerical integration | SERIES | functions | DIGITAL NETS | LATTICE RULES | ALGORITHMS | MULTIVARIATE INTEGRATION | SOBOLEV SPACES | quasi-Monte Carlo | digital nets and sequences | NUMERICAL-INTEGRATION | DISCREPANCY | WEIGHTS | CONSTRUCTIONS | Mathematics - Numerical Analysis

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 10/2016, Volume 304, pp. 26 - 42

Antithetic sampling, which goes back to the classical work by Hammersley and Morton (1956), is one of the well-known variance reduction techniques for Monte...

Higher order polynomial lattices | Quasi-Monte Carlo | Walsh functions | Antithetic sampling | Digital nets | Employee motivation | Analysis | Monte Carlo methods | Sobolev space | Mathematical analysis | Digital | Mathematical models | Polynomials | Sampling | Convergence | Mathematics - Numerical Analysis

Higher order polynomial lattices | Quasi-Monte Carlo | Walsh functions | Antithetic sampling | Digital nets | Employee motivation | Analysis | Monte Carlo methods | Sobolev space | Mathematical analysis | Digital | Mathematical models | Polynomials | Sampling | Convergence | Mathematics - Numerical Analysis

Journal Article

Journal of Approximation Theory, ISSN 0021-9045, 06/2015, Volume 194, pp. 62 - 86

In this paper we investigate quasi-Monte Carlo (QMC) integration using digital nets over Zb in reproducing kernel Hilbert spaces. The tent transformation...

Numerical integration | Higher order polynomial lattice point sets | Quasi-Monte Carlo | Tent transformation | Digital nets | MATHEMATICS | POLYNOMIAL LATTICE RULES | ARBITRARY HIGH-ORDER | CONSTRUCTIONS | ALGORITHMS | Mathematics - Numerical Analysis

Numerical integration | Higher order polynomial lattice point sets | Quasi-Monte Carlo | Tent transformation | Digital nets | MATHEMATICS | POLYNOMIAL LATTICE RULES | ARBITRARY HIGH-ORDER | CONSTRUCTIONS | ALGORITHMS | Mathematics - Numerical Analysis

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 11/2014, Volume 83, Issue 290, pp. 2821 - 2851

with an implied constant that depends on the \mathbb{H}^s( \mathbb{S}^d ). Here \sigma _d \mathbb{S}^d We provide methods for generation and numerical testing...

Integers | Spherical caps | Numerical quadratures | Mathematical sequences | Numerical integration | Hilbert spaces | Polynomials | Mathematical functions | Sobolev spaces | Degrees of polynomials | Quadrature | Sphere | Worst-case error | Discrepancy | Spherical design | QMC design | Quasi Monte Carlo | ARBITRARY DIMENSION | MATHEMATICS, APPLIED | numerical integration | quadrature | SPACES | DISTANCES | SUMS | CUBATURE ERROR | sphere | BOUNDS | SETS | NUMERICAL-INTEGRATION | S-2 | POINTS | worst-case error | spherical design | Mathematics - Numerical Analysis

Integers | Spherical caps | Numerical quadratures | Mathematical sequences | Numerical integration | Hilbert spaces | Polynomials | Mathematical functions | Sobolev spaces | Degrees of polynomials | Quadrature | Sphere | Worst-case error | Discrepancy | Spherical design | QMC design | Quasi Monte Carlo | ARBITRARY DIMENSION | MATHEMATICS, APPLIED | numerical integration | quadrature | SPACES | DISTANCES | SUMS | CUBATURE ERROR | sphere | BOUNDS | SETS | NUMERICAL-INTEGRATION | S-2 | POINTS | worst-case error | spherical design | Mathematics - Numerical Analysis

Journal Article

IEEE Transactions on Power Systems, ISSN 0885-8950, 08/2013, Volume 28, Issue 3, pp. 3335 - 3343

This paper presents a new quasi-Monte Carlo (QMC) based probabilistic small signal stability analysis (PSSSA) method to assess the dynamic effects of plug-in...

quasi-Monte Carlo | probabilistic small signal stability analysis | Sobol sequence | plug-in electric vehicle | Monte Carlo simulation | wind energy conversion system | DEMAND | DESIGN | GENERATION | ENGINEERING, ELECTRICAL & ELECTRONIC | Monte Carlo method | Usage | Stability | Analysis | Innovations | Eigenvalues | Distribution (Probability theory) | Electric power systems

quasi-Monte Carlo | probabilistic small signal stability analysis | Sobol sequence | plug-in electric vehicle | Monte Carlo simulation | wind energy conversion system | DEMAND | DESIGN | GENERATION | ENGINEERING, ELECTRICAL & ELECTRONIC | Monte Carlo method | Usage | Stability | Analysis | Innovations | Eigenvalues | Distribution (Probability theory) | Electric power systems

Journal Article

ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE, ISSN 0764-583X, 08/2019, Volume 53, Issue 5, pp. 1507 - 1552

We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian,...

MATHEMATICS, APPLIED | uncertainty quantification | APPROXIMATION | RANK-1 LATTICE RULES | SPACES | EQUATIONS | multilevel quasi-Monte Carlo | high-dimensional quadrature | GAUSSIAN FIELDS | Quasi-Monte Carlo methods | DISCRETIZATION | elliptic partial differential equations with lognormal input | MESH REFINEMENT | CONVERGENCE | error estimates | BY-COMPONENT CONSTRUCTION

MATHEMATICS, APPLIED | uncertainty quantification | APPROXIMATION | RANK-1 LATTICE RULES | SPACES | EQUATIONS | multilevel quasi-Monte Carlo | high-dimensional quadrature | GAUSSIAN FIELDS | Quasi-Monte Carlo methods | DISCRETIZATION | elliptic partial differential equations with lognormal input | MESH REFINEMENT | CONVERGENCE | error estimates | BY-COMPONENT CONSTRUCTION

Journal Article

Applied Numerical Mathematics, ISSN 0168-9274, 05/2018, Volume 127, pp. 110 - 124

In this paper we consider two sets of points for Quasi-Monte Carlo integration on two-dimensional manifolds. The first is the set of mapped low-discrepancy...

Low-discrepancy sequences | Greedy minimal Riesz s energy points | Measure preserving maps | Quasi-Monte Carlo method | Cubature on manifolds | MATHEMATICS, APPLIED | SPHERE | DISTANCES | SUMS

Low-discrepancy sequences | Greedy minimal Riesz s energy points | Measure preserving maps | Quasi-Monte Carlo method | Cubature on manifolds | MATHEMATICS, APPLIED | SPHERE | DISTANCES | SUMS

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 10/2017, Volume 454, Issue 1, pp. 361 - 384

We study quasi-Monte Carlo integration for twice differentiable functions defined over a triangle. We provide an explicit construction of infinite sequences of...

Digital nets and sequences | Quasi-Monte Carlo | Numerical integration on triangle | Dyadic Walsh analysis | MATHEMATICS | MATHEMATICS, APPLIED | Mathematics - Numerical Analysis

Digital nets and sequences | Quasi-Monte Carlo | Numerical integration on triangle | Dyadic Walsh analysis | MATHEMATICS | MATHEMATICS, APPLIED | Mathematics - Numerical Analysis

Journal Article

SIAM JOURNAL ON NUMERICAL ANALYSIS, ISSN 0036-1429, 2019, Volume 57, Issue 2, pp. 854 - 874

This paper studies the rate of convergence for conditional quasi-Monte Carlo (QMC), which is a counterpart of conditional Monte Carlo. We focus on...

MATHEMATICS, APPLIED | ANOVA decomposition | conditional quasi-Monte Carlo | INTEGRATION | singularities | discontinuities | smoothing

MATHEMATICS, APPLIED | ANOVA decomposition | conditional quasi-Monte Carlo | INTEGRATION | singularities | discontinuities | smoothing

Journal Article

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 02/2018, Volume 329, pp. 480 - 497

In this paper, an optimal multilevel randomized quasi-Monte-Carlo method to solve the stationary stochastic drift–diffusion-Poisson system is developed. We...

Randomized quasi-Monte-Carlo | Stochastic partial differential equation | Multilevel randomized quasi-Monte-Carlo | Multilevel Monte-Carlo | Optimal numerical method | Field-effect transistor | LATTICE RULES | WEIGHTED KOROBOV | GATE | MOSFETS | MULTIVARIATE INTEGRATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | PATH SIMULATION | MULTIPLE INTEGRATION | Monte Carlo method | Field-effect transistors | Analysis | Methods | Differential equations

Randomized quasi-Monte-Carlo | Stochastic partial differential equation | Multilevel randomized quasi-Monte-Carlo | Multilevel Monte-Carlo | Optimal numerical method | Field-effect transistor | LATTICE RULES | WEIGHTED KOROBOV | GATE | MOSFETS | MULTIVARIATE INTEGRATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | PATH SIMULATION | MULTIPLE INTEGRATION | Monte Carlo method | Field-effect transistors | Analysis | Methods | Differential equations

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

17.
Full Text
Higher order quasi-monte carlo integration for holomorphic, parametric operator equations

SIAM-ASA Journal on Uncertainty Quantification, ISSN 2166-2525, 2016, Volume 4, Issue 1, pp. 48 - 79

We analyze the convergence of higher order quasi-Monte Carlo (QMC) quadratures of solution functionals to countably parametric, nonlinear operator equations...

Infinite-dimensional quadrature | Uncertainty quantification | SPOD weights | Parametric operator equations | Lattice rules | Quasi-Monte Carlo | CBC construction | Digital nets | uncertainty quantification | SMOOTH FUNCTIONS | digital nets | ANALYTIC REGULARITY | PHYSICS, MATHEMATICAL | infinite-dimensional quadrature | MULTIVARIATE INTEGRATION | WEIGHTED SOBOLEV SPACES | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | POLYNOMIAL LATTICE RULES | NONLINEAR PROBLEMS | PARTIAL-DIFFERENTIAL-EQUATIONS | parametric operator equations | ARBITRARY HIGH-ORDER | quasi-Monte Carlo | lattice rules | BY-COMPONENT CONSTRUCTION | STRONG TRACTABILITY

Infinite-dimensional quadrature | Uncertainty quantification | SPOD weights | Parametric operator equations | Lattice rules | Quasi-Monte Carlo | CBC construction | Digital nets | uncertainty quantification | SMOOTH FUNCTIONS | digital nets | ANALYTIC REGULARITY | PHYSICS, MATHEMATICAL | infinite-dimensional quadrature | MULTIVARIATE INTEGRATION | WEIGHTED SOBOLEV SPACES | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | POLYNOMIAL LATTICE RULES | NONLINEAR PROBLEMS | PARTIAL-DIFFERENTIAL-EQUATIONS | parametric operator equations | ARBITRARY HIGH-ORDER | quasi-Monte Carlo | lattice rules | BY-COMPONENT CONSTRUCTION | STRONG TRACTABILITY

Journal Article

INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING, ISSN 0219-6913, 11/2019, Volume 17, Issue 6

Recently, quasi-Monte Carlo (QMC) rules for multivariate integration in some weighted Sobolev spaces of functions defined over unit cubes [0, 1]m, products of...

COMPUTER SCIENCE, SOFTWARE ENGINEERING | HIGH-DIMENSIONAL INTEGRATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | multivariate integration | Quasi-Monte Carlo rules | worst case error | tractability | EFFICIENT | product of balls | information complexity

COMPUTER SCIENCE, SOFTWARE ENGINEERING | HIGH-DIMENSIONAL INTEGRATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | multivariate integration | Quasi-Monte Carlo rules | worst case error | tractability | EFFICIENT | product of balls | information complexity

Journal Article

IMA Journal of Numerical Analysis, ISSN 0272-4979, 01/2017, Volume 37, Issue 1, pp. 505 - 518

We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness alpha is an element of N,...

quasi-Monte Carlo | numerical integration | Sobolev space | higher order digital nets | MATHEMATICS, APPLIED | SEQUENCES | DIMENSION | LOW-DISCREPANCY | RULES | Mathematics - Numerical Analysis

quasi-Monte Carlo | numerical integration | Sobolev space | higher order digital nets | MATHEMATICS, APPLIED | SEQUENCES | DIMENSION | LOW-DISCREPANCY | RULES | Mathematics - Numerical Analysis

Journal Article