To Be or Not to be Intrusive? The Solution of Parametric and Stochastic Equations---Proper Generalized Decomposition

Giraldi, Loïc; Liu, Dishi; Matthies, Hermann G.; Nouy, Anthony

doi:10.1137/140969063

Cited by 15 publications

(14 citation statements)

References 29 publications

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“…In many situations our method is equivalent to the solution of a regression problem and we are not the first to apply regression techniques for solving PDEs, see e.g. [16]. Nevertheless, it is our intention to present a unified and general theoretical foundation in the chosen framework.…”

Section: Introductionmentioning

confidence: 99%

Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations

et al. 2019

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A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors.

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Section: Introductionmentioning

confidence: 99%

Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations

et al. 2019

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“…The PGD remains intrusive in the sense that, to be implemented, numerous additional developments in a deterministic software are required. However, recently, a method has been proposed to compute an approximation of the solution based on simple evaluations of the residual of the deterministic problem [43].…”

Section: Intrusive Methodsmentioning

confidence: 99%

Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics

Clénet

2016

Scientific Computing in Electrical Engineering

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. To cite this version :Stephane CLÉNET -Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics -2016Any correspondence concerning this service should be sent to the repository Administrator : archiveouverte@ensam.eu Approximation Methods to solve Stochastic Problems in Computational ElectromagneticsStéphane ClénetAbstract To account for uncertainties on model parameters, the stochastic approach can be used. The model parameters as well as the outputs are then random fields or variables. Several methods are available in the literature to solve stochastic models like sampling methods, perturbation methods or approximation methods. In this paper, we propose an overview on the solution of stochastic problems in computational electromagnetics using approximation methods. Some applications will be presented in order to illustrate the possibilities offered by the approximation methods but also their current limitations due to the curse of dimensionality. Finally, recent numerical techniques proposed in the literature to face the curse of dimensionality are presented for non-intrusive and intrusive approaches.

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“…by solving min v∈M ≤r J (v), (5.6) where M ≤r is a low-rank manifold. There is a natural choice of functional for problems where (5.1) corresponds to the stationary condition of a functional J [27]. Also, J (v) can be taken as a certain norm of the residual R(v).…”

Section: Optimization On Low-rank Manifoldsmentioning

confidence: 99%

Low-Rank Tensor Methods for Model Order Reduction

Nouy

2017

Handbook of Uncertainty Quantification

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Parameter-dependent models arise in many contexts such as uncertainty quantification, sensitivity analysis, inverse problems or optimization. Parametric or uncertainty analyses usually require the evaluation of an output of a model for many instances of the input parameters, which may be intractable for complex numerical models. A possible remedy consists in replacing the model by an approximate model with reduced complexity (a so called reduced order model) allowing a fast evaluation of output variables of interest. This chapter provides an overview of low-rank methods for the approximation of functions that are identified either with order-two tensors (for vectorvalued functions) or higher-order tensors (for multivariate functions). Different approaches are presented for the computation of low-rank approximations, either based on samples of the function or on the equations that are satisfied by the function, the latter approaches including projection-based model order reduction methods. For multivariate functions, different notions of ranks and the corresponding low-rank approximation formats are introduced.

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To Be or Not to be Intrusive? The Solution of Parametric and Stochastic Equations---Proper Generalized Decomposition

Cited by 15 publications

References 29 publications

Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations

Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations

Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics

Low-Rank Tensor Methods for Model Order Reduction

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