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An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations

Antoine Gloria, Stefan Neukamm, Felix Otto (2014)

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

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm...

Analysis of a Population Model Structured by the Cells Molecular Content

M. Doumic (2010)

Mathematical Modelling of Natural Phenomena

We study the mathematical properties of a general model of cell division structured with several internal variables. We begin with a simpler and specific model with two variables, we solve the eigenvalue problem with strong or weak assumptions, and deduce from it the long-time convergence. The main difficulty comes from natural degeneracy of birth terms that we overcome with a regularization technique. We then extend the results to the case with several parameters and recall the link between this...

Analysis of pattern formation using numerical continuation

Vladimír Janovský (2022)

Applications of Mathematics

The paper deals with the issue of self-organization in applied sciences. It is particularly related to the emergence of Turing patterns. The goal is to analyze the domain size driven instability: We introduce the parameter L , which scales the size of the domain. We investigate a particular reaction-diffusion model in 1-D for two species. We consider and analyze the steady-state solution. We want to compute the solution branches by numerical continuation. The model in question has certain symmetries....

Analysis of the accuracy and convergence of equation-free projection to a slow manifold

Antonios Zagaris, C. William Gear, Tasso J. Kaper, Yannis G. Kevrekidis (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris, SIAM J. Appl. Dyn. Syst. 4 (2005) 711–732], we developed a class of iterative algorithms within the context of equation-free methods to approximate low-dimensional, attracting, slow manifolds in systems of differential equations with multiple time scales. For user-specified values of a finite number of the observables, the mth member of the class of algorithms ( m = 0 , 1 , ... ) finds iteratively an approximation of the appropriate zero of the (m+1)st...

Analysis of total variation flow and its finite element approximations

Xiaobing Feng, Andreas Prohl (2003)

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

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε , and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our...

Analysis of total variation flow and its finite element approximations

Xiaobing Feng, Andreas Prohl (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since...

Analytic continuation of fundamental solutions to differential equations with constant coefficients

Christer O. Kiselman (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

If P is a polynomial in R n such that 1 / P integrable, then the inverse Fourier transform of 1 / P is a fundamental solution E P to the differential operator P ( D ) . The purpose of the article is to study the dependence of this fundamental solution on the polynomial P . For n = 1 it is shown that E P can be analytically continued to a Riemann space over the set of all polynomials of the same degree as P . The singularities of this extension are studied.

Analytic semigroups generated on a functional extrapolation space by variational elliptic equations

Vincenzo Vespri (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove that any elliptic operator of second order in variational form is the infinitesimal generator of an analytic semigroup in the functional space C - 1 , α ⁢ ( Ω ) consinsting of all derivatives of hölder-continuous functions in Ω where Ω is a domain in ℝ n not necessarily bounded. We characterize, moreover the domain of the operator and the interpolation spaces between this and the space C - 1 , α ⁢ ( Ω ) . We prove also that the spaces C - 1 , α ⁢ ( Ω ) can be considered as extrapolation spaces relative to suitable non-variational operators....

Currently displaying 421 – 440 of 667