Related papers: Homogenization problems in the Calculus of Variati…
We develop a numerical homogenization method for fourth-order singular perturbation problems within the framework of heterogeneous multiscale method. These problems arise from heterogeneous strain gradient elasticity and elasticity models…
The purpose of this paper is to collect the homotopical methods used in the development of the theory of flows initialized by author's paper ``A model category for the homotopy theory of concurrency''. It is presented generalizations of the…
We present a general formula for the particular solution of an inhomogeneous linear difference equation with variable coefficients. The answer is expressed as a weighted sum of fundamental solutions of the associated linear difference…
The homogenization of a metamaterial made of a collection of scatterers periodically disposed is studied from three different points of view. Specifically tools for multiple scattering theory, functional analysis, differential geometry and…
In this paper, we are concerned with the sign-changing solutions of variational inequality problems. In order to give the existence results of the sign-changing solutions for variational inequality problems, we first construct a suitable…
This article is concerned with numerical methods to approximate effective coefficients in stochastic homogenization of discrete linear elliptic equations, and their numerical analysis --- which has been made possible by recent contributions…
We study quantitative homogenization of the eigenvalues for elliptic systems with periodically distributed inclusions, where the conductivity of inclusions are strongly contrast to that of the matrix. We propose a quantitative version of…
This mini-course provides a presentation of the method of characteristics to initial/boundary-value problems for systems of first-order partial differential equations and to Hamilton-Jacobi variational inequalities. In particular, these…
This paper is a survey of methods for solving smooth (strongly) monotone stochastic variational inequalities. To begin with, we give the deterministic foundation from which the stochastic methods eventually evolved. Then we review methods…
We analyze a recent application of homotopy perturbation method to some heat-like and wave-like models and show that its main results are merely the Taylor expansions of exponential and hyperbolic functions. Besides, the authors require…
This paper rediscovers a classical homogenization result for a prototypical linear elliptic boundary value problem with periodically oscillating diffusion coefficient. Unlike classical analytical approaches such as asymptotic analysis,…
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods…
We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a…
We consider hierarchical variational inequality problems, or more generally, variational inequalities defined over the set of zeros of a monotone operator. This framework includes convex optimization over equilibrium constraints and…
Standandard Hamiltonian mechanics in its homogeneous formulation is applied to the study of discontinuities representing rapid changes of Hamiltonians. Different formulations of Hamiltonian mechanics are reviewed. An original representation…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we…
This paper studies the existence and uniqueness of a classical solution to a type of Robin boundary problems on the nonnegative orthant. We propose a new decomposition-homogenization method for the Robin boundary problem based on…
Divergence-form operators with random coefficients homogenize over large scales. Over the last decade, an intensive research effort focused on turning this asymptotic statement into quantitative estimates. The goal of this note is to review…