Quasi--Monte Carlo Integration for Affine-Parametric, Elliptic PDEs: Local Supports and Product Weights

Gantner, Robert N.; Herrmann, Lukas; Schwab, Christoph

doi:10.1137/16m1082597

Cited by 29 publications

(20 citation statements)

References 16 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).…”

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confidence: 86%

“…This may also be achieved by exponentially decaying ψ j , which are not compactly supported, see [8]. An assumption of the type of (A1) in the case of so called affine-parametric coefficients in conjunction with the application of QMC with product weights was already discussed by us in [20]. In the present work, we extend our analysis of [31] to a MLQMC-FE algorithm with log-Gaussian inputs to reduce the overall work.…”

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confidence: 90%

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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: 90%

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

Herrmann

Schwab

2019

ESAIM: M2AN

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“…Similarly, the sequence D α (S) can be identified with a digital sequence over To construct an interlaced finite point set D α (P ), one can use polynomial lattice point sets in dimension αs instead of digital (t, m, αs)-nets. The resulting point set D α (P ) is called an interlaced polynomial lattice point set, and has been often used in applications of HoQMC methods, see [18,17,20,11,30,31,32,12]. Here we need to find good generating vectors q = (q 1 , .…”

Section: Digit Interlacing Constructionmentioning

confidence: 99%

4. Recent advances in higher order quasi-Monte Carlo methods

Goda¹,

Suzuki²

2020

Discrepancy Theory

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In this article we review some of recent results on higher order quasi-Monte Carlo (HoQMC) methods. After a seminal work by Dick (2007Dick ( , 2008 who originally introduced the concept of HoQMC, there have been significant theoretical progresses on HoQMC in terms of discrepancy as well as multivariate numerical integration. Moreover, several successful and promising applications of HoQMC to partial differential equations with random coefficients and Bayesian estimation/inversion problems have been reported recently. In this article we start with standard quasi-Monte Carlo methods based on digital nets and sequences in the sense of Niederreiter, and then move onto their higher order version due to Dick. The Walsh analysis of smooth functions plays a crucial role in developing the theory of HoQMC, and the aim of this article is to provide a unified picture on how the Walsh analysis enables recent developments of HoQMC both for discrepancy and numerical integration. P ⊂ [0, 1] s be a finite multiset, that is, if an element occurs multiple times, it is counted according to its multiplicity. Then I(f ) is approximated bywhere |P | denotes the cardinality of P . It is obvious that this algorithm is exact for any choice of P if f is a constant function, but except for such a trivial case, a careful design of P and the accompanying theoretical analysis are required to show that the algorithm works well for various functions f . One fairly easy but sensible approach is to choose each point x independently and uniformly from [0, 1] s . This is widely known under the name of Monte Carlo methods [39]. Many fundamental results in probability theory, including the law of large numbers and the central limit theorem, apply to this approach. Looking at I(f ; P ) as a random variable (with P being the underlying stochastic variable), we have E[I(f ; P )] = I(f ) and V[I(f ; P )] = V[f ] |P | ,

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“…We note, however, that the assumption of product weights is rather limiting. For instance, for integration problems involving parameterized PDEs, the best convergence rates known up to now are obtained by considering weighted space for the parameter-to-solution map with (S)POD weights [9,12,17], whereas weighted spaces with product weights lead to the best known rates only for special models [7,11,14]. In this paper we extend these results to the case of kernel approximation (as opposed to integration) of the parameterto-solution map, and we are able to show dimension-independent convergence rates using (S)POD weights in the general case.…”

Section: Introductionmentioning

confidence: 99%

Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification

et al. 2021

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This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice—a setting that, as pointed out by Zeng et al. (Monte Carlo and Quasi-Monte Carlo Methods 2004, Springer, New York, 2006) and Zeng et al. (Constr. Approx. 30: 529–555, 2009), allows fast evaluation by fast Fourier transform, so avoiding the need for a linear solver. The main contribution of the paper is the application to the approximation problem for uncertainty quantification of elliptic partial differential equations, with the diffusion coefficient given by a random field that is periodic in the stochastic variables, in the model proposed recently by Kaarnioja et al. (SIAM J Numer Anal 58(2): 1068–1091, 2020). The paper gives a full error analysis, and full details of the construction of lattices needed to ensure a good (but inevitably not optimal) rate of convergence and an error bound independent of dimension. Numerical experiments support the theory.

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Quasi--Monte Carlo Integration for Affine-Parametric, Elliptic PDEs: Local Supports and Product Weights

Cited by 29 publications

References 16 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

4. Recent advances in higher order quasi-Monte Carlo methods

Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification

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