Scrambled Polynomial Lattice Rules for Infinite-Dimensional Integration

Baldeaux, Jan

doi:10.1007/978-3-642-27440-4_11

Cited by 12 publications

(28 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Obviously, the assumption (A1) and (A2) are satisfied, and we have equivalence of the norms · and · H , given by (2). It follows that H, equipped with · H , is a reproducing kernel Hilbert space, too.…”

Section: Embedding Results and Norm Estimatesmentioning

confidence: 88%

Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration

Gnewuch

Hefter

Hinrichs

et al. 2017

Journal of Approximation Theory

View full text Add to dashboard Cite

We study embeddings and norm estimates for tensor products of weighted reproducing kernel Hilbert spaces. These results lead to a transfer principle that is directly applicable to tractability studies of multivariate problems as integration and approximation, and to their infinite-dimensional counterparts. In an application we consider weighted tensor product Sobolev spaces of mixed smoothness of any integer order, equipped with the classical, the anchored, or the ANOVA norm. Here we derive new results for multivariate and infinite-dimensional integration.

show abstract

Section: Embedding Results and Norm Estimatesmentioning

confidence: 88%

Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration

Gnewuch

Hefter

Hinrichs

et al. 2017

Journal of Approximation Theory

View full text Add to dashboard Cite

show abstract

“…In a series of papers, see e.g. Gnewuch (2012aGnewuch ( , 2012b, Hickernell et al (2010), Niu et al (2011), Baldeaux (2012b, Baldeaux and Gnewuch (2012), it was shown how to combine biased qMC rules using a multilevel approach to recover the optimal qMC rate. We refer the reader to these references for details.…”

Section: Multilevel Quasi-monte Carlo Methodsmentioning

confidence: 99%

Functionals of Multidimensional Diffusions with Applications to Finance

Baldeaux

Platen

2013

Bocconi &Amp; Springer Series

Self Cite

View full text Add to dashboard Cite

“…The Multivariate Decomposition Method (MDM) proposed in [35,43,44] is a generalisation of the Changing Dimension Algorithm in [28,34]. Here too it is assumed that an expansion of the form (1) exists, and that values of f u (x u ), while not available explicitly, can be obtained by a modest number of evaluations of f (x). In Section 5 we give specific examples in which values of f u (x u ) can be obtained from at most 2 |u| evaluations of f (x) -an acceptably small number if the cardinality |u| is small.…”

Section: Introductionmentioning

confidence: 99%

“…To approximate the integral (5), the MDM uses the decomposition of f given in (1). Indeed, assuming that the partial sums in (3) converge dominantly to f , we have…”

Section: Introductionmentioning

confidence: 99%

Infinite-dimensional integration and the multivariate decomposition method

Kuo

Nuyens

Plaskota

et al. 2017

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

We further develop the Multivariate Decomposition Method (MDM) for the Lebesgue integration of functions of infinitely many variables x 1 , x 2 , x 3 , . . . with respect to a corresponding product of a one dimensional probability measure. The method is designed for functions that admit a dominantly convergent decomposition f = u f u , where u runs over all finite subsets of positive integers, and for each u = {i 1 , . . . , i k } the function f u depends only on x i1 , . . . , x i k .Although a number of concepts of infinite-dimensional integrals have been used in the literature, questions of uniqueness and compatibility have mostly not been studied. We show that, under appropriate convergence conditions, the Lebesgue integral equals the 'anchored' integral, independently of the anchor.For approximating the integral, the MDM assumes that point values of f u are available for important subsets u, at some known cost. In this paper we introduce a new setting, in which it is assumed that each f u belongs to a normed space F u , and that bounds B u on f u Fu are known. This contrasts with the assumption in many papers that weights γ u , appearing in the norm of the infinite-dimensional function space, are somehow known. Often such weights γ u were determined by minimizing an error bound depending on the B u , the γ u and the chosen algorithm, resulting in weights that depend on the algorithm. In contrast, in this paper only the bounds B u are assumed known. We give two examples in which we specialize the MDM: in the first case F u is the |u|-fold tensor product of an anchored reproducing kernel Hilbert space, and in the second case it is a particular non-Hilbert space for integration over an unbounded domain. arXiv:1501.05445v3 [math.NA]

show abstract

Scrambled Polynomial Lattice Rules for Infinite-Dimensional Integration

Cited by 12 publications

References 16 publications

Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration

Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration

Functionals of Multidimensional Diffusions with Applications to Finance

Infinite-dimensional integration and the multivariate decomposition method

Contact Info

Product

Resources

About