On Multilevel Quadrature for Elliptic Stochastic Partial Differential Equations

Harbrecht, Helmut; Peters, Michael; Siebenmorgen, Markus

doi:10.1007/978-3-642-31703-3_8

Cited by 32 publications

(74 citation statements)

References 17 publications

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“…Figure 1 exemplifies these ideas for various levels of electron correlation, basis sets, and molecular training set sizes. The sparse grid combination technique known for highdimensional approximation [79][80][81][82][83][84][85][86][87][88][89] and quadrature / uncertainty quantification 90,91 in numerical analysis corresponds to a rigorous means to generate QML models constructed on a combination of sets of different subspaces. The general idea is to combine the subspaces such that only very few very expensive training samples are needed at target accuracy (e.g.…”

Section: B the Cqml Approachmentioning

confidence: 99%

Boosting Quantum Machine Learning Models with a Multilevel Combination Technique: Pople Diagrams Revisited

Zaspel

Huang

Harbrecht

et al. 2018

J. Chem. Theory Comput.

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184

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Inspired by Pople diagrams popular in quantum chemistry, we introduce a hierarchical scheme, based on the multi-level combination (C) technique, to combine various levels of approximations made when calculating molecular energies within quantum chemistry. When combined with quantum machine learning (QML) models, the resulting CQML model is a generalized unified recursive kernel ridge regression which exploits correlations implicitly encoded in training data comprised of multiple levels in multiple dimensions. Here, we have investigated up to three dimensions: Chemical space, basis set, and electron correlation treatment. Numerical results have been obtained for atomization energies of a set of ∼7'000 organic molecules with up to 7 atoms (not counting hydrogens) containing CHONFClS, as well as for ∼6'000 constitutional isomers of C 7 H 10 O 2 . CQML learning curves for atomization energies suggest a dramatic reduction in necessary training samples calculated with the most accurate and costly method. In order to generate milli-second estimates of CCSD(T)/cc-pvdz atomization energies with prediction errors reaching chemical accuracy (∼1 kcal/mol), the CQML model requires only ∼100 training instances at CCSD(T)/cc-pvdz level, rather than thousands within conventional QML, while more training molecules are required at lower levels. Our results suggest a possibly favorable trade-off between various hierarchical approximations whose computational cost scales differently with electron number.

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Section: B the Cqml Approachmentioning

confidence: 99%

Boosting Quantum Machine Learning Models with a Multilevel Combination Technique: Pople Diagrams Revisited

Zaspel

Huang

Harbrecht

et al. 2018

J. Chem. Theory Comput.

Self Cite

104

184

View full text Add to dashboard Cite

show abstract

“…[8,20,42]. This idea has already been proposed for different quadrature strategies in case of uniformly elliptic diffusion coefficients in [23]. The well-known Multilevel Monte Carlo Method (MLMC), as introduced in [3,15,16,26,27], and also the Randomized Multilevel Quasi-Monte Carlo Method, as introduced in [30], only provide probabilistic error estimates in the mean-square sense.…”

mentioning

confidence: 99%

“…The well-known Multilevel Monte Carlo Method (MLMC), as introduced in [3,15,16,26,27], and also the Randomized Multilevel Quasi-Monte Carlo Method, as introduced in [30], only provide probabilistic error estimates in the mean-square sense. To avoid this drawback, two fully deterministic methods have been proposed in [23], namely the Multilevel Quasi-Monte Carlo Method (MLQMC) and the Multilevel Polynomial Chaos Method (MLPC).The multilevel Monte Carlo method has been considered at first for a log-normal diffusion coefficient in [12] and further been analyzed in [11,40]. However, for deterministic quadrature methods, the log-normal case is much more involved due to the unboundedness of the domain of integration, i.e.…”

mentioning

confidence: 99%

Multilevel Accelerated Quadrature for PDEs with Log-Normally Distributed Diffusion Coefficient

Harbrecht¹,

Peters²,

Siebenmorgen³

2016

SIAM/ASA J. Uncertainty Quantification

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Abstract. This article is dedicated to multilevel quadrature methods for the rapid solution of stochastic partial differential equations with a log-normally distributed diffusion coefficient. The key idea of such approaches is a sparse-grid approximation of the occurring product space between the stochastic and the spatial variable. We develop the mathematical theory and present error estimates for the computation of the solution's moments with focus on the mean and the variance. Especially, the present framework covers the multilevel Monte Carlo method and the multilevel quasi-Monte Carlo method as special cases. The theoretical findings are supplemented by numerical experiments.Key words. multilevel quadrature, PDEs with stochastic data, log-normal diffusion, Karhunen-Loève expansion, finite element method AMS subject classifications. 65N30, 65D32, 60H15, 60H351. Introduction. In this article, we consider multilevel quadrature methods to compute the moments of the solution to elliptic partial differential equations with log-normally distributed diffusion coefficient. The basic idea of the multilevel quadrature is a sparse-grid-like discretization of the underlying Bochner space L 2 P Ω; H 1 0 (D) . The spatial variable is discretized by a classical finite element method whereas the stochastic variable is treated by an appropriately chosen quadrature rule, which naturally leads to a non-intrusive method. Since the problem's solution provides the necessary mixed Sobolev regularity, the approximation errors on the different levels of resolution can be equilibrated in a sparse-grid-like fashion, cf. [8,20,42]. This idea has already been proposed for different quadrature strategies in case of uniformly elliptic diffusion coefficients in [23]. The well-known Multilevel Monte Carlo Method (MLMC), as introduced in [3,15,16,26,27], and also the Randomized Multilevel Quasi-Monte Carlo Method, as introduced in [30], only provide probabilistic error estimates in the mean-square sense. To avoid this drawback, two fully deterministic methods have been proposed in [23], namely the Multilevel Quasi-Monte Carlo Method (MLQMC) and the Multilevel Polynomial Chaos Method (MLPC).The multilevel Monte Carlo method has been considered at first for a log-normal diffusion coefficient in [12] and further been analyzed in [11,40]. However, for deterministic quadrature methods, the log-normal case is much more involved due to the unboundedness of the domain of integration, i.e. R m for some m ∈ N, in combination with the stronger regularity requirements on the integrand. This makes the analysis of the quadrature error difficult. In particular, special regularity results are required which extend those of [2,10,29].For a finite stochastic dimension, we show that the multilevel quasi-Monte Carlo quadrature is feasible also for a log-normal diffusion coefficients if an auxiliary density is introduced.

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“…Indeed, this is the sparse grid combination technique as introduced in [15], see also [14,20]. It thus follows that…”

Section: Pymentioning

confidence: 97%

Multilevel methods for uncertainty quantification of elliptic PDEs with random anisotropic diffusion

Harbrecht

Schmidlin

2019

Stoch PDE: Anal Comp

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We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen-Loève expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem. We show that, given regularity of the elliptic diffusion problem, the decay of the Karhunen-Loève expansion entirely determines the regularity of the solution's dependence on the random parameter, also when considering this higher spatial regularity. This result then implies that multilevel collocation and multilevel quadrature methods may be used to lessen the computation complexity when approximating quantities of interest, like the solution's mean or its second moment, while still yielding the expected rates of convergence. Numerical examples in three spatial dimensions are provided to validate the presented theory.2010 Mathematics Subject Classification. Primary 35R60, 65N30, 60H35.

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On Multilevel Quadrature for Elliptic Stochastic Partial Differential Equations

Cited by 32 publications

References 17 publications

Boosting Quantum Machine Learning Models with a Multilevel Combination Technique: Pople Diagrams Revisited

Boosting Quantum Machine Learning Models with a Multilevel Combination Technique: Pople Diagrams Revisited

Multilevel Accelerated Quadrature for PDEs with Log-Normally Distributed Diffusion Coefficient

Multilevel methods for uncertainty quantification of elliptic PDEs with random anisotropic diffusion

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