Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

Kuo, F. Y.; Schwab, Christoph; Sloan, I. H.

doi:10.21914/anziamj.v53i0.5230

Cited by 34 publications

(56 citation statements)

References 30 publications

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“…The "standard" function spaces are weighted Sobolev spaces consisting of functions whose mixed first derivatives are square integrable, see e.g., [34,35]. In particular, it is known from the standard theory that good randomly shifted lattice rules can be constructed to achieve the optimal rate of convergence close to O(n −1 ), provided that the integrand lies in such a weighted Sobolev space, see e.g., [6,8,11,18,29,30,33]; recent surveys can be found in [9,20]. There are also higher order QMC methods that can achieve better than order one convergence for smooth integrands, see e.g., [9,10].…”

Section: A Suitable Weighted Function Space Setting In R Smentioning

confidence: 99%

Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients

Graham

Kuo²,

Nichols³

et al. 2014

Numer. Math.

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In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in R d (d = 1, 2, 3), with diffusion coefficient a(x, ω) given as a lognormal random field, i.e., a(x, ω) = exp(Z (x, ω)) where x is the spatial variable and Z (x, ·) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 or ∞ over all possible realizations of a. Focusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem, we give a rigorous error analysis for methods constructed from (1) standard continuous and piecewise linear finite element approximation in physical space; (2) truncated Karhunen-Loève expansion for computing realizations of a (leading to a possibly high-dimensional parametrized deterministic diffusion problem); and (3) lattice-based quasi-Monte Carlo (QMC) quadrature rules for computing integrals over parameter space which define the expected values. The paper contains novel error analysis which accounts for the effect of all three types of approximation. The QMC analysis is based

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Section: A Suitable Weighted Function Space Setting In R Smentioning

confidence: 99%

Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients

Graham

Kuo²,

Nichols³

et al. 2014

Numer. Math.

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140

261

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“…A short summary of these results, together with references, can be found in [24,Section 2]. More detailed surveys can be found in [10] or [23]. For the purpose of this paper, we only need the following bound on the root-mean-square error.…”

Section: Qmc Approximationmentioning

confidence: 99%

“…It turns out that the "optimal" weights (in the sense of minimizing an upper bound on the overall error) for the multi-level QMC FE algorithm are again POD weights (5), but they are different from the POD weights for the singlelevel algorithm in [24]. In any case, fast CBC construction algorithms for randomly shifted lattice rules are available for POD weights, see [10] or [23] for recent surveys, as well as [7,9,12,22,28,29,33]. The outline of this paper is as follows.…”

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confidence: 99%

Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients

2015

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This paper is a sequel to our previous work (Kuo et al. in SIAM J Numer Anal, 2012) where quasi-Monte Carlo (QMC) methods (specifically, randomly shifted lattice rules) are applied to finite element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient represented by a countably infinite number of terms. We estimate the expected value of some linear functional of the solution, as an infinite-dimensional integral in the parameter space. Here, the (singlelevel) error analysis of our previous work is generalized to a multi-level scheme, with the number of QMC points depending on the discretization level and with a level-dependent dimension truncation strategy. In some scenarios, it is shown that the overall error (i.e., the root-mean-square error averaged over all shifts) is of order O(h 2 ), where h is the finest FE mesh width, or O(N −1+δ ) for arbitrary δ > 0, where N denotes the maximal number of QMC sampling points in the parameter space. For these scenarios, the total work for all PDE solves in the multi-level QMC FE method is essentially of the order of one single PDE solve at the finest FE discretization level, for spatial dimension d = 2 with linear elements. The analysis exploits regularity of the parametric solution with respect to both the physical variables (the variables in Communicated by Philippe Ciarlet. 123Found Comput Math the physical domain) and the parametric variables (the parameters corresponding to randomness). As in our previous work, families of QMC rules with "POD weights" ("product and order dependent weights") which quantify the relative importance of subsets of the variables are found to be natural for proving convergence rates of QMC errors that are independent of the number of parametric variables.

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“…They consider a sequence of d-dimensional settings in which d → ∞. We will look at their weights restricted to u ⊆ 1:d. We draw on the summary of tractability results given by [20].…”

Section: Tractability and Product Weightsmentioning

confidence: 99%

Effective Dimension of Some Weighted Pre-Sobolev Spaces with Dominating Mixed Partial Derivatives

Owen¹

2019

SIAM J. Numer. Anal.

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This paper considers two notions of effective dimension for quadrature in weighted pre-Sobolev spaces with dominating mixed partial derivatives. We begin by finding a ball in those spaces just barely large enough to contain a function with unit variance. If no function in that ball has more than ε of its variance from ANOVA components involving interactions of order s or more, then the space has effective dimension at most s in the superposition sense. A similar truncation sense notion replaces the cardinality of the ANOVA component by the largest index it contains. Some Poincaré type inequalities are used to bound variance components by multiples of these space's squared norm and those in turn provide bounds on effective dimension. Very low effective dimension in the superposition sense holds for some spaces defined by product weights in which quadrature is strongly tractable. The superposition dimension is O(log(1/ε)/ log(log(1/ε))) just like the superposition dimension used in the multidimensional decomposition method. Surprisingly, even spaces where all subset weights are equal, regardless of their cardinality or included indices, have low superposition dimension in this sense. This paper does not require periodicity of the integrands.

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Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

Cited by 34 publications

References 30 publications

Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients

Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients

Multi-level Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic PDEs with Random Coefficients

Effective Dimension of Some Weighted Pre-Sobolev Spaces with Dominating Mixed Partial Derivatives

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