Calcul de lʼespérance de la solution dʼune EDP stochastique unidimensionnelle à lʼaide dʼune base réduite

Erhel, Jocelyne; Mghazli, Zoubida; Oumouni, Mestapha

doi:10.1016/j.crma.2011.07.015

Cited by 3 publications

(9 citation statements)

References 8 publications

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“…The Standard sparse grid method is based on the Smolyak algorithm [18,19] which provides an effective way to approach multivariate functions by a polynomial interpolation. It is given by linear combinations of product formulas (8) where in each dimension, a small order of interpolation is used, this provides a significant reduction of the full interpolation complexity and the curse of dimensionality is reduced.…”

Section: Standard Sparse Grid Methodsmentioning

confidence: 99%

“…In [8], for one-dimensional elliptic problem, we have computed the Karhunen-Loève expansion for k −1 , and use its random variables as a basis to build a projection of u which gives a good approximation of the expectation E[u]. In this work, we propose to use a similar methodology to approximate u in L 2 (Γ).…”

Section: Motivationmentioning

confidence: 99%

“…In this work, we propose to use a similar methodology to approximate u in L 2 (Γ). We expand the indicator λ on the multi-dimensional Lagrange polynomials basis defined in (8) or (10,12), then we use this basis to approximate the solution u.…”

Section: Motivationmentioning

confidence: 99%

“…In this work, we discuss the anisotropic sparse grid method with adaptive approach inspired from the works [3,16,8]. In [8], we have introduced the Kahrunen-Loève expansion of the inverse of the diffusion parameter where we have used the random variables basis of this expansion to compute the average solution of one dimensional elliptic problem. Motivated by this work, we use this inverse of the diffusion to construct an error indicator to carefully choosing the parameter of the collocation method which are the level and the anisotropic weights.…”

Section: Introductionmentioning

confidence: 99%

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An adaptive sparse grid method for elliptic PDEs with stochastic coefficients

Erhel

Mghazli

Oumouni

2015

Computer Methods in Applied Mechanics and Engineering

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To cite this version:Jocelyne Erhel, Zoubida Mghazli, Mestapha Oumouni. An adaptive sparse grid method for elliptic PDEs with stochastic coefficients. AbstractThe stochastic collocation method based on the anisotropic sparse grid has become a significant tool to solve partial differential equations with stochastic inputs. The aim is to seek a vector weight and a convenient level for the method. The classical approach uses a posteriori approach on the solution, yielding to an additional prohibitive cost with a large stochastic dimension.In this work, we discuss an adaptive approach of this method to calculate the statistics of the solution. It is based on an adaptive approximation of the inverse diffusion parameter. We construct an efficient error indicator which is an upper bound of the error on the solution. In the case of unbounded variables, we use an appropriate error estimation to compute suitable weights of the method. Numerical examples are presented to confirm the efficiency of the approach and showing that the cost is considerably reduced without loss of accuracy.

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Section: Standard Sparse Grid Methodsmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An adaptive sparse grid method for elliptic PDEs with stochastic coefficients

Erhel

Mghazli

Oumouni

2015

Computer Methods in Applied Mechanics and Engineering

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“…We choose the Karhunen-Loève (K-L) expansion [1,8,19] to approximate the field G and then a: a ≈ a N = e GN , where…”

Section: Approximation Of the Permeabilitymentioning

confidence: 99%

A combined collocation and Monte Carlo method for advection-diffusion equation of a solute in random porous media

Erhel

Mghazli²,

Oumouni

2014

ESAIM: ProcS

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Abstract. In this work, we present a numerical analysis of a method which combines a deterministic and a probabilistic approaches to quantify the migration of a contaminant, under the presence of uncertainty on the permeability of the porous medium. More precisely, we consider the flow equation in a random porous medium coupled with the advection-diffusion equation. Quantities of interest are the mean spread and the mean dispersion of the solute. The means are approximated by a quadrature rule, based on a sparse grid defined by a truncated Karhunen-Loève expansion and a stochastic collocation method. For each grid point, the flow model is solved with a mixed finite element method in the physical space and the advection-diffusion equation is solved with a probabilistic Lagrangian method. The spread and the dispersion are expressed as functions of a stochastic process. A priori error estimates are established on the mean of the spread and the dispersion. Keywords: Uncertainty quantification, elliptic PDE with random coefficients, advection-diffusion equation, collocation techniques, anisotropic sparse grids, Monte Carlo method, Euler scheme for SDE. IntroductionMathematical modeling and numerical simulation are important tools in the prediction of pollutant transport in groundwater and in the evaluation of potential risks of contaminants. The main constraint to the development of these models is the limited knowledge of the geological characteristics and the natural heterogeneity which implies uncertainty in the parameters and data. Stochastic approaches have been developed to deal with this uncertainty, where the permeability field is modeled as a random field a = e G , where G is a correlated Gaussian field [2, 13]. Our objective is to quantify the migration of a contaminant by computing statistics of interest defined by the mean of the spread and of the dispersion of the solute. The permeability a is discretized using a Karhunen-Loève (K-L) truncation up to a suitable and moderately large order.The Monte Carlo method is the most widely used approach to deal with uncertainty. This method is used in [2,4,7,12,13] to approximate the mean spread and dispersion.Recently, stochastic collocation (see [1,9,18,19]), based on sparse tensor product approximation, has gained much attention since it is very effective and accurate for computing statistics from solutions of PDEs with random input data. In this paper, we use a truncated Karhunen-Loève expansion of the random data and an anisotropic sparse grid with Gaussian knots. Then we propose to approximate the mean values by a quadrature rule using this sparse grid. 329The cost is proportional to the number of samples in the Monte Carlo method and to the sparse grid size in the collocation method. Each sample or grid point requires solving a flow PDE and a transport PDE; this can be done in a non intrusive and parallel way. In this paper, we approximate the steady-state flow problem by a mixed finite element method in the physical space to get the velocity field. The tra...

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Convergence analysis of macro spreading in 3D heterogeneous porous media

et al. 2013

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Models of hydrogeology must deal with both heterogeneity and lack of data. We consider in this paper a flow and transport model for an inert solute. The conductivity is a random field following a stationary log normal distribution with an exponential or Gaussian covariance function, with a very small correlation length. The quantities of interest studied here are the expectation of the spatial mean velocity, the equivalent permeability and the macro spreading. In particular, the asymptotic behavior of the plume is characterized, leading to large simulation times, consequently to large physical domains. Uncertainty is dealt with a classical Monte Carlo method, which turns out to be very efficient, thanks to the ergodicity of the conductivity field and to the very large domain. These large scale simulations are achieved by means of high performance computing algorithms and tools. * This work was partially supported by ANR-CIS, with the MICAS project and ANR-MN, with the H2MNO4 project. * * Some computations were done with the Grid'5000 facility.

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Calcul de lʼespérance de la solution dʼune EDP stochastique unidimensionnelle à lʼaide dʼune base réduite

Cited by 3 publications

References 8 publications

An adaptive sparse grid method for elliptic PDEs with stochastic coefficients

An adaptive sparse grid method for elliptic PDEs with stochastic coefficients

A combined collocation and Monte Carlo method for advection-diffusion equation of a solute in random porous media

Convergence analysis of macro spreading in 3D heterogeneous porous media

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