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A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems

M. Billaud-Friess, A. Nouy, O. Zahm (2014)

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

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with a measure of the residual corresponding to the error in a specified solution norm. The residual norm can be designed such that the resulting low-rank approximations are optimal with respect to particular norms of interest, thus allowing to take...

A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions

Jamol I. Baltaev, Milan Kučera, Martin Väth (2012)

Applications of Mathematics

We consider a simple reaction-diffusion system exhibiting Turing's diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential...

A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions

Jan Eisner, Milan Kučera, Martin Väth (2016)

Applications of Mathematics

Given a reaction-diffusion system which exhibits Turing's diffusion-driven instability, the influence of unilateral obstacles of opposite sign (source and sink) on bifurcation and critical points is studied. In particular, in some cases it is shown that spatially nonhomogeneous stationary solutions (spatial patterns) bifurcate from a basic spatially homogeneous steady state for an arbitrarily small ratio of diffusions of inhibitor and activator, while a sufficiently large ratio is necessary in the...

Adaptivity and variational stabilization for convection-diffusion equations

Albert Cohen, Wolfgang Dahmen, Gerrit Welper (2012)

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

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies,...

Adaptivity and variational stabilization for convection-diffusion equations∗

Albert Cohen, Wolfgang Dahmen, Gerrit Welper (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies,...

Asymmetric heteroclinic double layers

Michelle Schatzman (2002)

ESAIM: Control, Optimisation and Calculus of Variations

Let W be a non-negative function of class C 3 from ℝ 2 to ℝ , which vanishes exactly at two points 𝐚 and 𝐛 . Let S 1 ( 𝐚 , 𝐛 ) be the set of functions of a real variable which tend to 𝐚 at - ∞ and to 𝐛 at + ∞ and whose one dimensional energy E 1 ( v ) = ∫ ℝ W ( v ) + | v ' | 2 / 2 d x is finite. Assume that there exist two isolated minimizers z + and z - of the energy E 1 over S 1 ( 𝐚 , 𝐛 ) . Under a mild coercivity condition on the potential W and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at z + and z - , it is possible to prove...

Asymmetric heteroclinic double layers

Michelle Schatzman (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Let W be a non-negative function of class C3 from 2 to , which vanishes exactly at two points a and b. Let S1(a, b) be the set of functions of a real variable which tend to a at -∞ and to b at +∞ and whose one dimensional energy E 1 ( v ) = ∫ W ( v ) + | v ' | 2 / 2 x is finite. Assume that there exist two isolated minimizers z+ and z- of the energy E1 over S1(a, b). Under a mild coercivity condition on the potential W and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at z+ and...

Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

Fanghua Lin, Tristan Rivière (1999)

Journal of the European Mathematical Society

There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S 1 -valued function defined on the boundary of a bounded regular domain of R n . When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology...

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