138 citations · 788 across the 67 of their papers we have counts for
9 papers · 2 filters
Numerical integration in -Korobov and -cosine spaces
Josef Dick, Peter Kritzer, Gunther Leobacher +1
QMC rules are equal weight quadrature rules for approximating integrals over . One line of research studies the integration error of functions in the unit ball of so-calle…
Higher Order Quasi Monte-Carlo Integration in Uncertainty Quantification
Josef Dick, Quoc Thong Le Gia, Christoph Schwab
We review recent results on dimension-robust higher order convergence rates of Quasi-Monte Carlo Petrov-Galerkin approximations for response functionals of infinite-dimensional, pa…
Higher order Quasi-Monte Carlo integration for holomorphic, parametric operator equations
Josef Dick, Quoc T. Le Gia, Christoph Schwab
We analyze the convergence of higher order Quasi-Monte Carlo (QMC) quadratures of solution-functionals to countably-parametric, nonlinear operator equations with distributed uncert…
Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling
Johann S. Brauchart, Josef Dick, Lou Fang
In this paper we introduce a reproducing kernel Hilbert space defined on as the tensor product of a reproducing kernel defined on the unit sphere $\mathbb{S}^{d}…
The inverse of the star-discrepancy problem and the generation of pseudo-random numbers
Josef Dick, Friedrich Pillichshammer
The inverse of the star-discrepancy problem asks for point sets of size in the -dimensional unit cube whose star-discrepancy satisfies…
Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces
Johann S. Brauchart, Josef Dick, Edward B. Saff +3
We prove that the covering radius of an -point subset of the unit sphere is bounded above by a power of the worst-case error for equal weight cubatur…
Multi-level higher order QMC Galerkin discretization for affine parametric operator equations
Josef Dick, Frances Kuo, Quoc Thong Le Gia +1
We develop a convergence analysis of a multi-level algorithm combining higher order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin discretizations of countably af…
A reduced fast component-by-component construction of lattice points for integration in weighted spaces with fast decreasing weights
Josef Dick, Peter Kritzer, Gunther Leobacher +1
Lattice rules and polynomial lattice rules are quadrature rules for approximating integrals over the -dimensional unit cube. Since no explicit constructions of such quadrature m…
Proof Techniques in Quasi-Monte Carlo Theory
Josef Dick, Aicke Hinrichs, Friedrich Pillichshammer
In this survey paper we discuss some tools and methods which are of use in quasi-Monte Carlo (QMC) theory. We group them in chapters on Numerical Analysis, Harmonic Analysis, Algeb…