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Multiple-Precision Correctly rounded Newton-Cotes quadrature

Laurent Fousse (2007)

RAIRO - Theoretical Informatics and Applications

Numerical integration is an important operation for scientific computations. Although the different quadrature methods have been well studied from a mathematical point of view, the analysis of the actual error when performing the quadrature on a computer is often neglected. This step is however required for certified arithmetics.
We study the Newton-Cotes quadrature scheme in the context of multiple-precision arithmetic and give enough details on the algorithms and the error bounds to enable software...

Multiplicative Schwarz Methods for Discontinuous Galerkin Approximations of Elliptic Problems

Paola F. Antonietti, Blanca Ayuso (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we introduce and analyze some non-overlapping multiplicative Schwarz methods for discontinuous Galerkin (DG) approximations of elliptic problems. The construction of the Schwarz preconditioners is presented in a unified framework for a wide class of DG methods. For symmetric DG approximations we provide optimal convergence bounds for the corresponding error propagation operator, and we show that the resulting methods can be accelerated by using suitable Krylov space solvers. A discussion...

Multiscale analysis of wave propagation in random media. Application to correlation-based imaging

Josselin Garnier (2013/2014)

Séminaire Laurent Schwartz — EDP et applications

We consider sensor array imaging with the purpose to image reflectors embedded in a medium. Array imaging consists in two steps. In the first step waves emitted by an array of sources probe the medium to be imaged and are recorded by an array of receivers. In the second step the recorded signals are processed to form an image of the medium. Array imaging in a scattering medium is limited because coherent signals recorded at the receiver array and coming from a reflector to be imaged are weak and...

Multiscale expansion and numerical approximation for surface defects⋆

V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux, G. Vial (2011)

ESAIM: Proceedings

This paper is a survey of articles [5, 6, 8, 9, 13, 17, 18]. We are interested in the influence of small geometrical perturbations on the solution of elliptic problems. The cases of a single inclusion or several well-separated inclusions have been deeply studied. We recall here techniques to construct an asymptotic expansion. Then we consider moderately close inclusions, i.e. the distance between the inclusions tends to zero more slowly than their characteristic size. We provide a complete asymptotic...

Multiscale Finite Element approach for “weakly” random problems and related issues

Claude Le Bris, Frédéric Legoll, Florian Thomines (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We address multiscale elliptic problems with random coefficients that are a perturbation of multiscale deterministic problems. Our approach consists in taking benefit of the perturbative context to suitably modify the classical Finite Element basis into a deterministic multiscale Finite Element basis. The latter essentially shares the same approximation properties as a multiscale Finite Element basis directly generated on the random problem. The specific reference method that we use is the Multiscale...

Multiscale finite element coarse spaces for the application to linear elasticity

Marco Buck, Oleg Iliev, Heiko Andrä (2013)

Open Mathematics

We extend the multiscale finite element method (MsFEM) as formulated by Hou and Wu in [Hou T.Y., Wu X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 1997, 134(1), 169–189] to the PDE system of linear elasticity. The application, motivated by the multiscale analysis of highly heterogeneous composite materials, is twofold. Resolving the heterogeneities on the finest scale, we utilize the linear MsFEM basis for the construction...

Multiscale modelling of sound propagation through the lung parenchyma

Paul Cazeaux, Jan S. Hesthaven (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we develop and study numerically a model to describe some aspects of sound propagation in the human lung, considered as a deformable and viscoelastic porous medium (the parenchyma) with millions of alveoli filled with air. Transmission of sound through the lung above 1 kHz is known to be highly frequency-dependent. We pursue the key idea that the viscoelastic parenchyma structure is highly heterogeneous on the small scale ε and use two-scale homogenization techniques to derive effective...

Multivariate multiple comparisons with a control in elliptical populations

Naoya Okamoto, Takashi Seo (2013)

Discussiones Mathematicae Probability and Statistics

The approximate upper percentile of Hotelling's T²-type statistic is derived in order to construct simultaneous confidence intervals for comparisons with a control under elliptical populations with unequal sample sizes. Accuracy and conservativeness of Bonferroni approximations are evaluated via a Monte Carlo simulation study. Finally, we explain the real data analysis using procedures derived in this paper.

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