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Approximation of a Martensitic Laminate with Varying Volume Fractions

Bo Li, Mitchell Luskin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We give results for the approximation of a laminate with varying volume fractions for multi-well energy minimization problems modeling martensitic crystals that can undergo either an orthorhombic to monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfy the corresponding boundary condition, and we establish a series of error bounds in terms of the elastic energy for the approximation of the limiting macroscopic deformation and...

Approximation of a solidification problem

Rajae Aboulaïch, Ilham Haggouch, Ali Souissi (2001)

International Journal of Applied Mathematics and Computer Science

A two-dimensional Stefan problem is usually introduced as a model of solidification, melting or sublimation phenomena. The two-phase Stefan problem has been studied as a direct problem, where the free boundary separating the two regions is eliminated using a variational inequality (Baiocchi, 1977; Baiocchi et al., 1973; Rodrigues, 1980; Saguez, 1980; Srunk and Friedman, 1994), the enthalpy function (Ciavaldini, 1972; Lions, 1969; Nochetto et al., 1991; Saguez, 1980), or a control problem (El Bagdouri,...

Approximation of control problems involving ordinary and impulsive controls

Fabio Camilli, Maurizio Falcone (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study an approximation scheme for a class of control problems involving an ordinary control v, an impulsive control u and its derivative u ˙ . Adopting a space-time reparametrization of the problem which adds one variable to the state space we overcome some difficulties connected to the presence of u ˙ . We construct an approximation scheme for that augmented system, prove that it converges to the value function of the augmented problem and establish an error estimates in L∞ for this approximation....

Approximation of maximal Cheeger sets by projection

Guillaume Carlier, Myriam Comte, Gabriel Peyré (2009)

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

This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of ℝ d . This problem is motivated by landslide modelling as well as by the continuous maximal flow problem. Using the fact that the maximal Cheeger set can be approximated by solving a rather simple projection problem, we propose a numerical strategy to compute maximal Cheeger sets and Cheeger constants.

Approximation of maximal Cheeger sets by projection

Guillaume Carlier, Myriam Comte, Gabriel Peyré (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of ℝ d . This problem is motivated by landslide modelling as well as by the continuous maximal flow problem. Using the fact that the maximal Cheeger set can be approximated by solving a rather simple projection problem, we propose a numerical strategy to compute maximal Cheeger sets and Cheeger constants.

Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group

Yves Achdou, Italo Capuzzo-Dolcetta (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like h where h is the mesh step. Such...

Approximation of the pareto optimal set for multiobjective optimal control problems using viability kernels

Alexis Guigue (2014)

ESAIM: Control, Optimisation and Calculus of Variations

This paper provides a convergent numerical approximation of the Pareto optimal set for finite-horizon multiobjective optimal control problems in which the objective space is not necessarily convex. Our approach is based on Viability Theory. We first introduce a set-valued return function V and show that the epigraph of V equals the viability kernel of a certain related augmented dynamical system. We then introduce an approximate set-valued return function with finite set-values as the solution of...

Approximation of the Snell envelope and american options prices in dimension one

Vlad Bally, Bruno Saussereau (2002)

ESAIM: Probability and Statistics

We establish some error estimates for the approximation of an optimal stopping problem along the paths of the Black–Scholes model. This approximation is based on a tree method. Moreover, we give a global approximation result for the related obstacle problem.

Approximation of the Snell Envelope and American Options Prices in dimension one

Vlad Bally, Bruno Saussereau (2010)

ESAIM: Probability and Statistics

We establish some error estimates for the approximation of an optimal stopping problem along the paths of the Black–Scholes model. This approximation is based on a tree method. Moreover, we give a global approximation result for the related obstacle problem.

Approximations by regular sets and Wiener solutions in metric spaces

Anders Björn, Jana Björn (2007)

Commentationes Mathematicae Universitatis Carolinae

Let X be a complete metric space equipped with a doubling Borel measure supporting a weak Poincaré inequality. We show that open subsets of X can be approximated by regular sets. This has applications in nonlinear potential theory on metric spaces. In particular it makes it possible to define Wiener solutions of the Dirichlet problem for p -harmonic functions and to show that they coincide with three other notions of generalized solutions.

Approximations of parabolic variational inequalities

Alexander Ženíšek (1985)

Aplikace matematiky

The paper deals with an initial problem of a parabolic variational inequality whichcontains a nonlinear elliptic form a ( v , w ) having a potential J ( v ) , which is twice G -differentiable at arbitrary v ∈ H 1 ( Ω ) . This property of a ( v , w ) makes it possible to prove convergence of an approximate solution defined by a linearized scheme which is fully discretized - in space by the finite elements method and in time by a one-step finite-difference method. Strong convergence of the approximate solution is proved without any regularity...

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