QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights

Herrmann, Lukas; Schwab, Christoph

doi:10.1007/s00211-018-0991-1

Cited by 25 publications

(72 citation statements)

References 26 publications

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“…While retaining linear scaling w.r. to the spatial resolution of the approximate GRF in D, the hierarchical nature of multiresolution analyses (MRAs) naturally enables multilevel QMC (MLQMC) algorithms with a discretization level dependent resolution of GRF and QMC integration. In addition, as observed by us recently in [20,31], the localization of the supports of the representation system in D allows us to use QMC quadrature with product weights. This, in turn, is known to afford linear scaling of the work with respect to the parameter dimension s to compute the QMC generating vectors (see [14,42] and the references there).…”

mentioning

confidence: 86%

“…The present results go in several respects beyond those in [33]. We consider, in particular, MLQMC-FE discretizations, and use sharper bounds than those in [33] on the error caused by truncating the expansion of the GRF, from our single-level analysis in [31]. We also generalize, based on [31], the QMC error analysis by admitting Gaussian type weight functions in the anisotropic QMC norms, as opposed to the exponential weights used in [26,36].…”

mentioning

confidence: 97%

“…We consider, in particular, MLQMC-FE discretizations, and use sharper bounds than those in [33] on the error caused by truncating the expansion of the GRF, from our single-level analysis in [31]. We also generalize, based on [31], the QMC error analysis by admitting Gaussian type weight functions in the anisotropic QMC norms, as opposed to the exponential weights used in [26,36]. We prove that this extends the summability range of Karhunen-Loève and wavelet expansions of the admissible GRFs for the applicability of QMC with product weights, in addition to obviating stationarity, as compared to [26,36].…”

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confidence: 99%

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Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

Herrmann

Schwab

2019

ESAIM: M2AN

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We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domain D ⊂ R d . The multilevel algorithm Q * L which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuo et al. (Found. Comput. Math. 15 (2015) 411-449). The random coefficient is assumed to admit a representation with locally supported coefficient functions, as arise for example in spline-or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math. 141 (2019) 63-102). It also extends the MLQMC error analysis in Kuo et al. (Math. Comput. 86 (2017) 2827-2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) in D, and to product weights in QMC integration. In particular, in polytopal domains D ⊂ R d , d = 2, 3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices of D. This allows for non-stationary GRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators in D with boundary conditions on ∂D. In the weighted function spaces in D, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations of D yield optimal asymptotic convergence rates. Comparison of the ε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of Karhunen-Loève type. Our analysis yields general bounds for the ε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.Mathematics Subject Classification. 65D30, 65N30, 60G60.

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confidence: 86%

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confidence: 97%

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confidence: 99%

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Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

Herrmann

Schwab

2019

ESAIM: M2AN

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“…In contrast to our paper, they treat the truncation error in a general setting, and as for the QMC integration error, they consider both the exponential weight functions and the Gaussian weight function for the weighted Sobolev space. As for the exponential weight function, the current paper and [17] impose essentially the same assumptions (Assumption B below), and show the same convergence rate. However, our proof strategy is different, which turns out to result in different (product) weights, (and a different constant, although it does not seem to be easy to say which is bigger).…”

Section: Introductionmentioning

confidence: 93%

“…Herrmann and Schwab [17] develops a theory under the setting essentially the same as ours. In contrast to our paper, they treat the truncation error in a general setting, and as for the QMC integration error, they consider both the exponential weight functions and the Gaussian weight function for the weighted Sobolev space.…”

Section: Introductionmentioning

confidence: 99%

Quasi–Monte Carlo integration with product weights for elliptic PDEs with log-normal coefficients

Kazashi

2018

IMA Journal of Numerical Analysis

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Quasi-Monte Carlo (QMC) integration of output functionals of solutions of the diffusion problem with a log-normal random coefficient is considered. The random coefficient is assumed to be given by an exponential of a Gaussian random field that is represented by a series expansion of some system of functions. Graham et al. [16] developed a latticebased QMC theory for this problem and established a quadrature error decay rate ≈ 1 with respect to the number of quadrature points. The key assumption there was a suitable summability condition on the aforementioned system of functions. As a consequence, product-order-dependent (POD) weights were used to construct the lattice rule. In this paper, a different assumption on the system is considered. This assumption, originally considered by Bachmayr et al. [3] to utilise the locality of support of basis functions in the context of polynomial approximations applied to the same type of the diffusion problem, is shown to work well in the same lattice-based QMC method considered by Graham et al.: the assumption leads us to product weights, which enables the construction of the QMC method with a smaller computational cost than Graham et al. A quadrature error decay rate ≈ 1 is established, and the theory developed here is applied to a wavelet stochastic model. By a characterisation of the Besov smoothness, it is shown that a wide class of path smoothness can be treated with this framework.

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Constructing QMC Finite Element Methods for Elliptic PDEs with Random Coefficients by a Reduced CBC Construction

Ebert

Kritzer

Nuyens

2020

Springer Proceedings in Mathematics &Amp; Statistics

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QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights

Cited by 25 publications

References 26 publications

Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

Quasi–Monte Carlo integration with product weights for elliptic PDEs with log-normal coefficients

Constructing QMC Finite Element Methods for Elliptic PDEs with Random Coefficients by a Reduced CBC Construction

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