148 research outputs found
Constructing lattice points for numerical integration by a reduced fast successive coordinate search algorithm
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 Sobolev spaces. The algorithm presented is a reduced fast successive
coordinate search (SCS) algorithm, which is adapted to situations where the
weights in the function space show a sufficiently fast decay. The new SCS
algorithm is designed to work for the construction of lattice points, and, in a
modified version, for polynomial lattice points, and the corresponding
integration rules can be used to treat functions in different kinds of function
spaces. We show that the integration rules constructed by our algorithms
satisfy error bounds of optimal convergence order. Furthermore, we give details
on efficient implementation such that we obtain a considerable speed-up of
previously known SCS algorithms. This improvement is illustrated by numerical
results. The speed-up obtained by our results may be of particular interest in
the context of QMC for PDEs with random coefficients, where both the dimension
and the required numberof points are usually very large. Furthermore, our main
theorems yield previously unknown generalizations of earlier results.Comment: 33 pages, 2 figure
Discrepancy bounds for low-dimensional point sets
The class of -nets and -sequences, introduced in their most
general form by Niederreiter, are important examples of point sets and
sequences that are commonly used in quasi-Monte Carlo algorithms for
integration and approximation. Low-dimensional versions of -nets and
-sequences, such as Hammersley point sets and van der Corput sequences,
form important sub-classes, as they are interesting mathematical objects from a
theoretical point of view, and simultaneously serve as examples that make it
easier to understand the structural properties of -nets and
-sequences in arbitrary dimension. For these reasons, a considerable
number of papers have been written on the properties of low-dimensional nets
and sequences
Tractability of multivariate analytic problems
In the theory of tractability of multivariate problems one usually studies
problems with finite smoothness. Then we want to know which -variate
problems can be approximated to within by using, say,
polynomially many in and function values or arbitrary
linear functionals.
There is a recent stream of work for multivariate analytic problems for which
we want to answer the usual tractability questions with
replaced by . In this vein of research, multivariate
integration and approximation have been studied over Korobov spaces with
exponentially fast decaying Fourier coefficients. This is work of J. Dick, G.
Larcher, and the authors. There is a natural need to analyze more general
analytic problems defined over more general spaces and obtain tractability
results in terms of and .
The goal of this paper is to survey the existing results, present some new
results, and propose further questions for the study of tractability of
multivariate analytic questions
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