Weighted Tensor Product Algorithms for Linear Multivariate Problems

Wasilkowski, Grzegorz W.; Woźniakowski, Henryk

doi:10.1006/jcom.1999.0512

Cited by 93 publications

(134 citation statements)

References 10 publications

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“…Instead, we will allow more general index sets [18,28,39] in the summation of (1) and try to choose them properly. To this end, we will consider the selection of the whole index set as an optimization problem, i.e.…”

Section: Dimension-adaptive Quadraturementioning

confidence: 99%

“…, k d } ≤ l}) as special cases. Unfortunately, little is known about error bounds of quadrature formulas associated to general index sets I (see [28,39]). However, by a careful construction of the index sets I we can hope that the error for generalized sparse grid quadrature formulas is at least as good as in the case of conventional sparse grids.…”

Section: Generalized Sparse Gridsmentioning

confidence: 99%

“…Numerical methods could concentrate on the more important dimensions and spend more work for these dimensions than for the unimportant ones. Interestingly, also complexity theory reveals that integration problems with weighted dimensions can become tractable even if the unweighted problem is not [39]. Unfortunately, classical adaptive numerical integration algorithms [13,35] cannot be applied to high-dimensional problems since the work overhead in order to find and adaptively refine in important dimensions would be too large.…”

Section: Introductionmentioning

confidence: 99%

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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: Dimension-adaptive Quadraturementioning

confidence: 99%

Section: Generalized Sparse Gridsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dimension?Adaptive Tensor?Product Quadrature

2003

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“…Clearly this holds when h = 0 or k = 0. When h = 0 and k = 0, we have We now discuss the approximation problem in the weighted Hilbert space H s following closely the discussions from [11,21] for the weighted Korobov space, see also [25,28,29] for general results. Without loss of generality (see, e.g., [25]), we approximate f by a linear algorithm of the form…”

Section: /2mentioning

confidence: 99%

Approximation of Functions Using Digital Nets

Dick

Kritzer

Kuo

2008

Monte Carlo and Quasi-Monte Carlo Methods 2006

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Summary. In analogy to the recent paper [11] which studied lattice rule algorithms for approximation in weighted Korobov spaces, we consider the approximation problem in a weighted Hilbert space of Walsh series. Our approximation uses a truncated Walsh series with Walsh coefficients approximated by numerical integration using digital nets. We show that digital nets (or more precisely, polynomial lattices) tailored specially for the approximation problem lead to better error bounds. The error bounds can be independent of the dimension s, or depend only polynomially on s, under certain conditions on the weights defining the function space.

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“…Again this result is not constructive. There is a constructive proof that p * = 1 in Wasilkowski and Woźniakowski [22] under the more restrictive assumption that…”

Section: Introductionmentioning

confidence: 99%

Strong tractability of multivariate integration using quasi–Monte Carlo algorithms

Wang

2002

Math. Comp.

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Abstract. We study quasi-Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of ε depends on ε −1 and the dimension s. Strong tractability means that it does not depend on s and is bounded by a polynomial in ε −1 . The least possible value of the power of ε −1 is called the ε-exponent of strong tractability. Sloan and Woźniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the ε-exponent of strong tractability is between 1 and 2. However, their proof is not constructive.In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with ε-exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Woźniakowski's assumption. We show that quasi-Monte Carlo algorithms using Niederreiter's (t, s)-sequences and Sobol sequences achieve the optimal convergence order O(N −1+δ ) for any δ > 0 independent of the dimension with a worst case deterministic guarantee (where N is the number of function evaluations). This implies that strong tractability with the best ε-exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter's (t, s)-sequences and Sobol sequences.

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Weighted Tensor Product Algorithms for Linear Multivariate Problems

Cited by 93 publications

References 10 publications

Dimension?Adaptive Tensor?Product Quadrature

Dimension?Adaptive Tensor?Product Quadrature

Approximation of Functions Using Digital Nets

Strong tractability of multivariate integration using quasi–Monte Carlo algorithms

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