Weighted Tensor Product Algorithms for Linear Multivariate Problems

@article{Wasilkowski1999WeightedTP,
  title={Weighted Tensor Product Algorithms for Linear Multivariate Problems},
  author={Grzegorz W. Wasilkowski and Henryk Wozniakowski},
  journal={J. Complex.},
  year={1999},
  volume={15},
  pages={402-447}
}
Abstract We study the e -approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces of functions f of d variables. A class of weighted tensor product (WTP) algorithms is defined which depends on a number of parameters. Two classes of permissible information are studied. Λ all consists of all linear functionals while Λ std consists of evaluations of f or its derivatives. We show that these multivariate problems are sometimes tractable even with a worst-case… 
Tractability of Approximation and Integration for Weighted Tensor Product Problems over Unbounded Domains
We study tractability and strong tractability of multivariate approximation and integration in the worst case deterministic setting. Tractability means that the number of functional evaluations
Tractability of Tensor Product Linear Operators in Weighted Hilbert Spaces
Abstract We study tractability in the worst case setting of tensor product linear operators defined over weighted tensor product Hilbert spaces. Tractability means that the minimal number of
Finite-order weights imply tractability of linear multivariate problems
TLDR
It is proved that finite-order weights imply strong tractability or tractability of linear multivariate problems, depending on a certain condition on the reproducing kernel of the space.
Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Worst Case Setting
TLDR
This work provides a construction of polynomial-time algorithms Ad,ε for the general d-variate problem with the number of evaluations bounded roughly by ε−pdq* to achieve an error ε in the worst case setting.
Tractability of Approximation and Integrationfor Weighted Tensor Product Problems overUnbounded
We study tractability and strong tractability of multivariate approximation and integration in the worst case deterministic setting. Tractability means that the number of functional evaluations
Tractability of approximating multivariate linear functionals
We review selected tractability results for approximating linear tensor product functionals defined over reproducing kernel Hilbert spaces. This review is based on Volume II of our book Tractability
Complexity of Weighted Approximation over R
TLDR
This work studies approximation of univariate functions defined over the reals and provides necessary and sufficient conditions in terms of the weights @j and @r, as well as the parameters r, p, and q for the weighted approximation problem to have finite complexity.
Integration and approximation in arbitrary dimensions
TLDR
It is proved that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure.
...
...

References

SHOWING 1-10 OF 26 REFERENCES
Tractability of Tensor Product Linear Operators
TLDR
The worst case setting for approximating multivariate tensor product linear operators defined over Hilbert spaces is dealt with, and it is proved that tractability of linear functionals depends on the given space of functions.
Strong tractability of weighted tensor products
TLDR
It is proved that strong tractability holds independently of the sequences f ig and f ig In particular if i i the authors get tensor product problems which are strongly tractable the strong exponent in the average or probabilistic setting is always smaller than the strong multiplier in the worst case setting.
Explicit Cost Bounds of Algorithms for Multivariate Tensor Product Problems
TLDR
An upper bound is obtained, which is independent of d, for the number, n(?, d), of points for which discrepancy is at most ?, n (?, d) ? 7.26??2.454, ?d, ? ? 1.
Tractability and Strong Tractability of Linear Multivariate Problems
TLDR
This work provides necessary and sufficient conditions for linear multivariate problems to be tractable or strongly tractable in the worst case, average case, randomized, and probabilistic settings, and considers linearMultivariate problems over reproducing kernel Hilbert spaces, showing that such problems are strong tractable even in the best case setting.
High dimensional integration of smooth functions over cubes
TLDR
A new algorithm for the numerical integration of functions that are defined on a d-dimensional cube is constructed based on the Clenshaw-Curtis rule for d=1 and on Smolyak's construction to make the best use of the smoothness properties of any (nonperiodic) function.
When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals?
TLDR
It is proved that the minimalworst case error of quasi-Monte Carlo algorithms does not depend on the dimensiondiff the sum of the weights is finite, and the minimal number of function values in the worst case setting needed to reduce the initial error by ? is bounded byC??p, where the exponentp? 1, 2], andCdepends exponentially on thesum of weights.
Faster Valuation of Financial Derivatives
TLDR
Numerical testing which compares low-discrepancy and Monte Carlo algorithms on the evaluation of financial derivatives concludes that for this CMO the Sobol algorithm is always superior to the other algorithms.
Theory of Reproducing Kernels.
Abstract : The present paper may be considered as a sequel to our previous paper in the Proceedings of the Cambridge Philosophical Society, Theorie generale de noyaux reproduisants-Premiere partie
New methodologies for valuing derivatives
TLDR
Numerical testing which compares low discrepancy and Monte Carlo algorithms on the evaluation of financial derivatives concludes that for this CMO the Sobol algorithm is always superior to the other algorithms.
...
...