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Low rank Tucker-type tensor approximation to classical potentials

B. Khoromskij, V. Khoromskaia (2007)

Open Mathematics

This paper investigates best rank-(r 1,..., r d) Tucker tensor approximation of higher-order tensors arising from the discretization of linear operators and functions in ℝd. Super-convergence of the best rank-(r 1,..., r d) Tucker-type decomposition with respect to the relative Frobenius norm is proven. Dimensionality reduction by the two-level Tucker-to-canonical approximation is discussed. Tensor-product representation of basic multi-linear algebra operations is considered, including inner, outer...

Machine Computation Using the Exponentially Convergent Multiscale Spectral Generalized Finite Element Method

Ivo Babuška, Xu Huang, Robert Lipton (2014)

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

A multiscale spectral generalized finite element method (MS-GFEM) is presented for the solution of large two and three dimensional stress analysis problems inside heterogeneous media. It can be employed to solve problems too large to be solved directly with FE techniques and is designed for implementation on massively parallel machines. The method is multiscale in nature and uses an optimal family of spectrally defined local basis functions over a coarse grid. It is proved that the method has an...

Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows

Jean-Luc Guermond, Serge Prudhomme (2003)

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

This paper presents a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows. One particularity of this model is that the hyperviscosity is active only at the short velocity scales, a feature which is reminiscent of Large Eddy Simulation models. We propose a Fourier–Galerkin approximation of the perturbed Navier–Stokes equations and we show that, as the cutoff wavenumber goes to infinity, the solution of the model converges (up to subsequences) to a weak solution...

Mathematical analysis of a spectral hyperviscosity LES model for the simulation of turbulent flows

Jean-Luc Guermond, Serge Prudhomme (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper presents a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows. One particularity of this model is that the hyperviscosity is active only at the short velocity scales, a feature which is reminiscent of Large Eddy Simulation models. We propose a Fourier–Galerkin approximation of the perturbed Navier–Stokes equations and we show that, as the cutoff wavenumber goes to infinity, the solution of the model converges (up to subsequences) to a weak...

Mathematical and numerical analysis of radiative heat transfer in semi-transparent media

Yao-Chuang Han, Yu-Feng Nie, Zhan-Bin Yuan (2019)

Applications of Mathematics

This paper is concerned with mathematical and numerical analysis of the system of radiative integral transfer equations. The existence and uniqueness of solution to the integral system is proved by establishing the boundedness of the radiative integral operators and proving the invertibility of the operator matrix associated with the system. A collocation-boundary element method is developed to discretize the differential-integral system. For the non-convex geometries, an element-subdivision algorithm...

Mathematical Modeling of Atmospheric Flow and Computation of Convex Envelopes

A. Caboussat (2011)

Mathematical Modelling of Natural Phenomena

Atmospheric flow equations govern the time evolution of chemical concentrations in the atmosphere. When considering gas and particle phases, the underlying partial differential equations involve advection and diffusion operators, coagulation effects, and evaporation and condensation phenomena between the aerosol particles and the gas phase. Operator splitting techniques are generally used in global air quality models. When considering organic aerosol...

Mathematical modeling of semiconductor quantum dots based on the nonparabolic effective-mass approximation

Jinn-Liang Liu (2012)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

Within the effective mass and nonparabolic band theory, a general framework of mathematical models and numerical methods is developed for theoretical studies of semiconductor quantum dots. It includes single-electron models and many-electron models of Hartree-Fock, configuration interaction, and current-spin density functional theory approaches. These models result in nonlinear eigenvalue problems from a suitable discretization. Cubic and quintic Jacobi-Davidson methods of block or nonblock version...

Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition

Eric Luneville, Jean-Francois Mercier (2014)

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

We study the time-harmonic acoustic scattering in a duct in presence of a flow and of a discontinuous impedance boundary condition. Unlike a continuous impedance, a discontinuous one leads to still open modeling questions, as in particular the singularity of the solution at the abrupt transition and the choice of the right unknown to formulate the scattering problem. To address these questions we propose a mathematical approach based on variational formulations set in weighted Sobolev spaces. Considering...

Mathematical modelling of rock bolt reinforcement

Runt, David, Novotný, Jaroslav, Pruška, Jan (2017)

Programs and Algorithms of Numerical Mathematics

Rock bolts as construction elements are often used in underground civil engineering projects. This work deals with their numerical modelling. Aydan special finite elements for the description of rock bolts and hexahedral quadratic finite elements for the description of rock massif were used. A code for the computation of stiffness matrices and right hand sides of these elements was developed. The code was tested on several simple test examples and their results were compared with the analytical...

Medical image – based computational model of pulsatile flow in saccular aneurisms

Stéphanie Salmon, Marc Thiriet, Jean-Frédéric Gerbeau (2003)

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

Saccular aneurisms, swelling of a blood vessel, are investigated in order (i) to estimate the development risk of the wall lesion, before and after intravascular treatment, assuming that the pressure is the major factor, and (ii) to better plan medical interventions. Numerical simulations, using the finite element method, are performed in three-dimensional aneurisms. Computational meshes are derived from medical imaging data to take into account both between-subject and within-subject anatomical...

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