Efficient algorithms for multivariate and ∞-variate integration with exponential weight

Plaskota, Leszek; Wasilkowski, Grzegorz W.

doi:10.1007/s11075-013-9798-4

Cited by 6 publications

(10 citation statements)

References 24 publications

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“…We see more speedup in the case of Smolyak MDM compared with QMC MDM. This is as expected, because for QMC MDM there is extra work in managing the different positions that a nonempty set can originate from as a subset of another set in the active set: we need to cope with a more complicated data structure for (19) compared with (13) or (27), and we need more function evaluations. Additionally, we expect the QMC algorithms to be much more efficient when the truncation dimension goes up (i.e., when ε goes down), since the sizes of the Smolyak grids based on trapezoidal rules then increase faster than the powers of 2 of the QMC algorithms.…”

Section: Timing Resultsmentioning

confidence: 61%

“…Note that the same set v = (1, 7) can originate from the position w = (1, 3) of different supersets u: for example, u = (1, 6, 7), u = (1,4,7,13), and many others. We can make use of this repetition to save on computational cost.…”

Section: Quadrature Rules Based On Quasi-monte Carlo Methodsmentioning

confidence: 99%

“…where T > 0 is a "threshold" parameter that depends on the overall error demand ε > 0 and possibly on all of w(u). For example, w(u) can be related to the weight parameters from a weighted function space setting (as in [16,17,13,4]), or it can be related to the bounds on the norm of f u (as in [7]). In this section we will treat T and w(u) as input parameters (ignoring the mathematical details of where they come from), and focus on the efficient implementation of the active set given these parameters.…”

Section: Constructing the Active Setmentioning

confidence: 99%

“…The formulation (25)-(27) is very similar to the formulation (12)- (13), and the computation of the values c(v, m) can also be done while constructing the extended active set. This is shown in Pseudocode 2A , which works in a similar way to Pseudocode 2A.…”

Section: Smolyak Quadrature Via the Combination Techniquementioning

confidence: 99%

“…The Multivariate Decomposition Method (MDM) is an algorithm for approximating the integral of an ∞-variate function f defined on some domain D N with D ⊆ R, and this paper presents the first results on the implementation of the MDM. The general idea of the MDM, see [4,7,13,16,17] (as well as [9,12] under the name of Changing Dimension Algorithm), goes as follows. Assume that f admits a decomposition…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Efficient Implementations of the Multivariate Decomposition Method for Approximating Infinite-Variate Integrals

Gilbert¹,

Kuo²,

Nuyens³

et al. 2018

SIAM J. Sci. Comput.

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In this paper we focus on efficient implementations of the Multivariate Decomposition Method (MDM) for approximating integrals of ∞-variate functions. Such ∞-variate integrals occur for example as expectations in uncertainty quantification. Starting with the anchored decomposition f = u⊂N f u , where the sum is over all finite subsets of N and each f u depends only on the variables x j with j ∈ u, our MDM algorithm approximates the integral of f by first truncating the sum to some 'active set' and then approximating the integral of the remaining functions f u term-by-term using Smolyak or (randomized) quasi-Monte Carlo (QMC) quadratures. The anchored decomposition allows us to compute f u explicitly by function evaluations of f . Given the specification of the active set and theoretically derived parameters of the quadrature rules, we exploit structures in both the formula for computing f u and the quadrature rules to develop computationally efficient strategies to implement the MDM in various scenarios. In particular, we avoid repeated function evaluations at the same point. We provide numerical results for a test function to demonstrate the effectiveness of the algorithm.

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Section: Timing Resultsmentioning

confidence: 61%

Section: Quadrature Rules Based On Quasi-monte Carlo Methodsmentioning

confidence: 99%

Section: Constructing the Active Setmentioning

confidence: 99%

Section: Smolyak Quadrature Via the Combination Techniquementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Efficient Implementations of the Multivariate Decomposition Method for Approximating Infinite-Variate Integrals

Gilbert¹,

Kuo²,

Nuyens³

et al. 2018

SIAM J. Sci. Comput.

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Integral-type functionals of first hitting times for continuous-time Markov chains

Liu

Song

2018

Front. Math. China

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Although there doesn't exist the Lebesgue measure in the ball M of C[0, 1] with p−norm, the average values (expectation) EY and variance DY of some functionals Y on M can still be defined through the procedure of limitation from finite dimension to infinite dimension. In particular, the probability densities of coordinates of points in the ball M exist and are derived out even though the density of points in M doesn't exist. These densities include high order normal distribution, high order exponent distribution. This also can be considered as the geometrical origins of these probability distributions. Further, the exact values (which is represented in terms of finite dimensional integral) of a kind of infinite-dimensional functional integrals are obtained, and specially the variance DY is proven to be zero, and then the nonlinear exchange formulas of average values of functionals are also given. Instead of measure, the variance is used to measure the deviation of functional from its average value. DY = 0 means that a functional takes its average on a ball with probability 1 by using the language of probability theory, and this is just the concentration without measure. In addition, we prove that the average value depends on the discretization.

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Infinite-dimensional integration and the multivariate decomposition method

Kuo

Nuyens

Plaskota

et al. 2017

Journal of Computational and Applied Mathematics

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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]

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Efficient algorithms for multivariate and ∞-variate integration with exponential weight

Abstract: Using the Multivariate Decomposition Method (MDM), we develop an efficient algorithm for approximating the ∞-variate integral

Cited by 6 publications

References 24 publications

Efficient Implementations of the Multivariate Decomposition Method for Approximating Infinite-Variate Integrals

Efficient Implementations of the Multivariate Decomposition Method for Approximating Infinite-Variate Integrals

Integral-type functionals of first hitting times for continuous-time Markov chains

Infinite-dimensional integration and the multivariate decomposition method

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