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Person# Diane Sylvie Guignard

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Finite element method

The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the tr

Maximum a posteriori estimation

In Bayesian statistics, a maximum a posteriori probability (MAP) estimate is an estimate of an unknown quantity, that equals the mode of the posterior distribution. The MAP can be used to obtain a po

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathema

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Diane Sylvie Guignard, Fabio Nobile

In this work, we consider an elliptic partial differential equation (PDE) with a random coefficient solved with the stochastic collocation finite element method (SC-FEM). The random diffusion coefficient is assumed to depend in an affine way on independent random variables. We derive a residual-based a posteriori error estimate that is constituted of two parts controlling the SC error and the FE error, respectively. The SC error estimator is then used to drive an adaptive sparse grid algorithm. Several numerical examples are given to illustrate the efficiency of the error estimator and the performance of the adaptive algorithm.

2018Diane Sylvie Guignard, Fabio Nobile

In this work, we consider an elliptic partial differential equation with a random coefficient solved with the stochastic collocation finite element method. The random diffusion coefficient is assumed to depend in an affine way on independent random variables. We derive a residual-based a posteriori error estimate that is constituted of two parts controlling the stochastic collocation (SC) and the finite element (FE) errors, respectively. The SC error estimator is then used to drive an adaptive sparse grid algorithm. Several numerical examples are given to illustrate the efficiency of the error estimator and the performance of the adaptive algorithm.

Diane Sylvie Guignard, Fabio Nobile, Marco Picasso

We consider finite element error approximations of the steady incompressible Navier-Stokes equations defined on a randomly perturbed domain, the perturbation being small. Introducing a random mapping, these equations are transformed into PDEs on a fixed reference domain with random coefficients. Under suitable assumptions on the random mapping and the input data, in particular the so-called small data assumption, we prove the well-posedness of the problem. We assume then that the mapping depends affinely on L independent random variables and adopt a perturbation approach expanding the solution with respect to a small parameter ε that controls the amount of randomness in the problem. We perform an a posteriori error analysis for the first order approximation error, namely the error between the exact (random) solution and the finite element approximation of the first term in the expansion with respect to ε. Numerical results are given to illustrate the theoretical results and the effectiveness of the error estimators.