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Barthelmann, Volker; Novak, Erich; Ritter, Klaus

doi:10.1023/a:1018977404843

Cited by 536 publications

(87 citation statements)

References 20 publications

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“…When one uses the full potential, and therefore multidimensional quadrature, the size of the quadrature grid is an important problem. By using ideas based on those of Smolyak [55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73], it is, nevertheless, possible to compute accurate energy levels using grids that are many orders of magnitude smaller than direct product grids.…”

Section: Mathematical Methods For High-dimensional Problemsmentioning

confidence: 99%

Mathematical Methods in Quantum Molecular Dynamics

2016

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Section: Mathematical Methods For High-dimensional Problemsmentioning

confidence: 99%

Mathematical Methods in Quantum Molecular Dynamics

2016

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“…To begin, we describe how to construct sparse grids on the parameter domain Γ from combinations of sets of interpolation points in one dimension. The idea was first introduced by Smolyak in 1963 [46], but we follow [4]. Assuming that the M underlying random variables in the problem are independent, we have a parameter domain of the form Γ = …”

Section: Reduced Basis Collocationmentioning

confidence: 99%

Efficient Reduced Basis Methods for Saddle Point Problems with Applications in Groundwater Flow

Newsum¹,

Powell²

2017

SIAM/ASA J. Uncertainty Quantification

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Abstract. Reduced basis methods (RBMs) are recommended to reduce the computational cost of solving parameter-dependent PDEs in scenarios where many choices of parameters need to be considered, for example in uncertainty quantification (UQ). A reduced basis is constructed during a computationally demanding offline (or setup) stage that allows the user to obtain cheap approximations for parameters choices of interest, online. In this paper we consider RBMs for parameter-dependent saddle point problems, in particular the one that arises in the mixed formulation of the Darcy flow problem in groundwater flow modelling. We apply a discrete empirical interpolation method (DEIM) to approximate the inverse of the diffusion coefficient, which depends non-affinely on the system parameters. We develop an efficient RBM that exploits the DEIM approximation and combine it with a sparse grid stochastic collocation mixed finite element method (SCMFEM) to construct a surrogate solution, which then allows for efficient forward UQ. Through numerical experiments we demonstrate that significant computational savings can be made when we use the RB-DEIM-SCMFEM scheme over standard high fidelity methods. For groundwater flow problems, we provide a thorough cost assessment of the new method and show how the size of the reduced basis, and hence, the extent of the savings, depends on the statistical properties of the input parameters.

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“…One alternative is to use well-nested cubature rules in combination with a sparse-grid sampling method. Tompkins et al (2011b) outline Smolyak sparse-grid sampling (Barthelmann, 2000) for solving a posterior interpolation problem based on highly nested Clenshaw-Curtis grids, whereas Waldvogel (2003) formulates the corresponding sparse cubature rules based on the same grids (termed Fejér cubature). While Smolyak sparse grids are only loosely dependent on the problem dimension (M ∼ N const ∕const: for const: > 1), and can be adapted to any degree accuracy desired, there are still practical limits to using them to solve equation 2 when stochastic dimensions are large (>50).…”

Section: Sparse Cubaturementioning

confidence: 99%

Efficient estimation of nonlinear posterior model covariances using maximally sparse cubature rules

Tompkins

2012

GEOPHYSICS

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We estimated posterior model covariances by randomly perturbing data and start model vectors with samples drawn from prior probability distributions, solving the corresponding deterministic inversion for each sample, and integrating the resulting model vectors over all realizations. The computational burden of this method is the number of samples required for the integration because each sample involves a solution to a nonlinear inverse problem. For certain classes of smooth prior probability functions, we present a new approach to the model covariance estimation problem using grids based on maximally sparse cubature integration rules. The N þ 1 (degree-two) and 2N (degree-three) cubature rules exist to exactly integrate symmetric Gaussian distributions in N stochastic dimensions. These represent the two sparsest integration rules possible and thus provide the most efficient estimation method for model covariances using integration. In contrast to Monte-Carlo (MC) integration, which requires hundreds of random samples to estimate covariances and has ill-defined convergence, sparse-cubature (quadrature) rules have determined numbers of samples and definite convergence. The trade-off is that they also have limited degrees of accuracy. Using a simple 1D electromagnetic example with 67 stochastic variables, we demonstrated that this sparse integration is three to five times more efficient than MC methods in terms of numbers of required samples.

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Cited by 536 publications

References 20 publications

Mathematical Methods in Quantum Molecular Dynamics

Mathematical Methods in Quantum Molecular Dynamics

Efficient Reduced Basis Methods for Saddle Point Problems with Applications in Groundwater Flow

Efficient estimation of nonlinear posterior model covariances using maximally sparse cubature rules

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