Explicit Cost Bounds of Algorithms for Multivariate Tensor Product Problems

Wasilkowski, Grzegorz W.; Woźniakowski, Henryk

doi:10.1006/jcom.1995.1001

Cited by 295 publications

(241 citation statements)

References 28 publications

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“…for f ∈ W r d [38]. We see that the convergence rate depends only weakly on the dimension but strongly on the smoothness r. However, the conventional sparse grid method treats all dimensions equally (because this is also true for unit simplex) and thus the dependence of the quadrature error on the dimension in its logarithmic term will cause problems for high-dimensional integrands.…”

Section: Sparse Gridsmentioning

confidence: 94%

Dimension?Adaptive Tensor?Product Quadrature

2003

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We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.The dimension-adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low-and moderatedimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm.The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.

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Section: Sparse Gridsmentioning

confidence: 94%

Dimension?Adaptive Tensor?Product Quadrature

2003

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“…2.30 can be written as [82] A (w, n) = w+1≤|i|≤w+n (−1) Nonlinear growth rules are used for fully nested rules (e.g., Clenshaw-Curtis is closed fully nested and GaussPatterson is open fully nested), and linear growth rules are best for standard Gauss rules that take advantage of, at most, "weak" nesting (e.g., reuse of the center point).…”

Section: Smolyak Sparse Gridsmentioning

confidence: 99%

DAKOTA : a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis.

Eldred

Vigil

Dalbey

et al. 2011

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The DAKOTA (Design Analysis Kit for Optimization and Terascale Applications) toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. DAKOTA contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the DAKOTA toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers.This report serves as a theoretical manual for selected algorithms implemented within the DAKOTA software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of DAKOTA-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of DAKOTA's iterative analysis capabilities.4

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“…, d as its components. It is known that the error of Smolyak's algorithm is of order n −q d times a logarithmic factor log n raised to a power that is linear in d − 1; see, e.g., [27,Remark 2] and the literature cited therein. (Further details on various aspects of Smolyak's algorithm can be found in [6,7,11,12,21].)…”

Section: Nonuniversal Algorithmsmentioning

confidence: 99%

On the optimal convergence rate of universal and nonuniversal algorithms for multivariate integration and approximation

Griebel

Woźniakowski

2006

Math. Comp.

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Abstract. We study the maximal rate of convergence (mrc) of algorithms for (multivariate) integration and approximation of d-variate functions from reproducing kernel Hilbert spaces H(K d ). Here K d is an arbitrary kernel all of whose partial derivatives up to order r satisfy a Hölder-type condition with exponent 2β. Algorithms use n function values and we analyze their rate of convergence as n tends to infinity. We focus on universal algorithms which depend on d, r, and β but not on the specific kernel K d , and nonuniversal algorithms which may depend additionally on K d .For universal algorithms the mrc is (r + β)/d for both integration and approximation, and for nonuniversal algorithms it is 1/2+(r+β)/d for integration and a + (r + β)/d with a ∈ [1/(4 + 4(r + β)/d), 1/2] for approximation. Hence, the mrc for universal algorithms suffers from the curse of dimensionality if d is large relative to r + β, whereas the mrc for nonuniversal algorithms does not since it is always at least 1/2 for integration, and 1/4 for approximation. This is the price we have to pay for using universal algorithms. On the other hand, if r + β is large relative to d, then the mrc for universal and nonuniversal algorithms is approximately the same.We also consider the case when we have the additional knowledge that the kernel K d has product structure,Here K r j ,β j are some univariate kernels whose all derivatives up to order r j satisfy a Hölder-type condition with exponent 2β j . Then the mrc for universal algorithms is q := min j=1,2,...,d (r j + β j ) for both integration and approximation, and for nonuniversal algorithms it is 1/2 + q for integration and a + q with a ∈ [1/(4 + 4q), 1/2] for approximation. If r j ≥ 1 or β j ≥ β > 0 for all j, then the mrc is at least min(1, β), and the curse of dimensionality is not present. Hence, the product form of reproducing kernels breaks the curse of dimensionality even for universal algorithms.

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Explicit Cost Bounds of Algorithms for Multivariate Tensor Product Problems

Cited by 295 publications

References 28 publications

Dimension?Adaptive Tensor?Product Quadrature

Dimension?Adaptive Tensor?Product Quadrature

DAKOTA : a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis.

On the optimal convergence rate of universal and nonuniversal algorithms for multivariate integration and approximation

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