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Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations

Nicolas Bacaër (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Using the min-plus version of the spectral radius formula, one proves: 1) that the unique eigenvalue of a min-plus eigenvalue problem depends continuously on parameters involved in the kernel defining the problem; 2) that the numerical method introduced by Chou and Griffiths to compute this eigenvalue converges. A toolbox recently developed at I.n.r.i.a. helps to illustrate these results. Frenkel-Kontorova models serve as example. The analogy with homogenization of Hamilton-Jacobi equations...

Degeneration and complexity of bilinear maps: Some asymptotic spectra.

V. Strassen (1991)

Journal für die reine und angewandte Mathematik

Determination of the Invariant Interval for Positive Linear Operators by a Nonstationary Iterative Procedure

Katarina Surla (1981)

Publications de l'Institut Mathématique

Diskrete Approximation von Eigenwertproblemen. I. Qualitative Konvergenz.

R.D. Grigorieff (1975)

Numerische Mathematik

Diskrete Approximation von Eigenwertproblemen. II. Konvergenzordnung.

R.D. Grigorieff (1975)

Numerische Mathematik

Diskrete Approximation von Eigenwertproblemen. III. Asymptotische Entwicklungen. - Discrete Approximation of Eigenvalue-Problems. III. Asymptotic Expansions.

R.D. Grigorieff (1975/1976)

Numerische Mathematik

Div-curl lemma revisited: Applications in electromagnetism

Marián Slodička, Ján Jr. Buša (2010)

Kybernetika

Two new time-dependent versions of div-curl results in a bounded domain $Ω \subset ℝ^{3}$ are presented. We study a limit of the product $v_{k} w_{k}$ , where the sequences $v_{k}$ and $w_{k}$ belong to $Ł_{2} (Ω)$ . In Theorem 2.1 we assume that $\nabla \times v_{k}$ is bounded in the $L_{p}$ -norm and $\nabla \cdot w_{k}$ is controlled in the $L_{r}$ -norm. In Theorem 2.2 we suppose that $\nabla \times w_{k}$ is bounded in the $L_{p}$ -norm and $\nabla \cdot w_{k}$ is controlled in the $L_{r}$ -norm. The time derivative of $w_{k}$ is bounded in both cases in the norm of $- 1 (Ω)$ . The convergence (in the sense of distributions) of $v_{k} w_{k}$ to the product $v w$ of weak limits...

Double greedy algorithms: Reduced basis methods for transport dominated problems

Wolfgang Dahmen, Christian Plesken, Gerrit Welper (2014)

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

The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible “tight” surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated...