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Numerical analysis for optimal shape design in elliptic boundary value problems

Zdeněk Kestřánek (1988)

Aplikace matematiky

Shape optimization problems are optimal design problems in which the shape of the boundary plays the role of a design, i.e. the unknown part of the problem. Such problems arise in structural mechanics, acoustics, electrostatics, fluid flow and other areas of engineering and applied science. The mathematical theory of such kind of problems has been developed during the last twelve years. Recently the theory has been extended to cover also situations in which the behaviour of the system is governed...

Numerical analysis of history-dependent quasivariational inequalities with applications in contact mechanics

Kamran Kazmi, Mikael Barboteu, Weimin Han, Mircea Sofonea (2014)

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

A new class of history-dependent quasivariational inequalities was recently studied in [M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22 (2011) 471–491]. Existence, uniqueness and regularity results were proved and used in the study of several mathematical models which describe the contact between a deformable body and an obstacle. The aim of this paper is to provide numerical analysis of the quasivariational inequalities...

Numerical behavior of the method of projection onto an acute cone with level control in convex minimization

Robert Dylewski (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We present the numerical behavior of a projection method for convex minimization problems which was studied by Cegielski [1]. The method is a modification of the Polyak subgradient projection method [6] and of variable target value subgradient method of Kim, Ahn and Cho [2]. In each iteration of the method an obtuse cone is constructed. The obtuse cone is generated by a linearly independent system of subgradients. The next approximation of a solution is the projection onto a translated acute cone...

Numerical considerations of a hybrid proximal projection algorithm for solving variational inequalities

Christina Jager (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, some ideas for the numerical realization of the hybrid proximal projection algorithm from Solodov and Svaiter [22] are presented. An example is given which shows that this hybrid algorithm does not generate a Fejér-monotone sequence. Further, a strategy is suggested for the computation of inexact solutions of the auxiliary problems with a certain tolerance. For that purpose, ε-subdifferentials of the auxiliary functions and the bundle trust region method from Schramm and Zowe [20]...

Numerical identification of a coefficient in a parabolic quasilinear equation

Jan Neumann (1985)

Aplikace matematiky

In the article the following optimal control problem is studied: to determine a certain coefficient in a quasilinear partial differential equation of parabolic type so that the solution of a boundary value problem for this equation would minimise a given integral functional. In addition to the design and analysis of a numerical method the paper contains the solution of the fundamental problems connected with the formulation of the problem in question (existence and uniqueness of the solution of...

Numerical minimization of eigenmodes of a membrane with respect to the domain

Édouard Oudet (2004)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we introduce a numerical approach adapted to the minimization of the eigenmodes of a membrane with respect to the domain. This method is based on the combination of the Level Set method of S. Osher and J.A. Sethian with the relaxed approach. This algorithm enables both changing the topology and working on a fixed regular grid.

Numerical minimization of eigenmodes of a membrane with respect to the domain

Édouard Oudet (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we introduce a numerical approach adapted to the minimization of the eigenmodes of a membrane with respect to the domain. This method is based on the combination of the Level Set method of S. Osher and J.A. Sethian with the relaxed approach. This algorithm enables both changing the topology and working on a fixed regular grid.

Numerical modelling of semi-coercive beam problem with unilateral elastic subsoil of Winkler's type

Stanislav Sysala (2010)

Applications of Mathematics

A non-linear semi-coercive beam problem is solved in this article. Suitable numerical methods are presented and their uniform convergence properties with respect to the finite element discretization parameter are proved here. The methods are based on the minimization of the total energy functional, where the descent directions of the functional are searched by solving the linear problems with a beam on bilateral elastic ``springs''. The influence of external loads on the convergence properties is...

Numerical optimization of parameters in systems of differential equations

Martínek, Josef, Kučera, Václav (2023)

Programs and Algorithms of Numerical Mathematics

We present results on the estimation of unknown parameters in systems of ordinary differential equations in order to fit the output of models to real data. The numerical method is based on the nonlinear least squares problem along with the solution of sensitivity equations corresponding to the differential equations. We will present the performance of the method on the problem of fitting the output of basic compartmental epidemic models to data from the Covid-19 epidemic. This allows us to draw...

Numerical procedure to approximate a singular optimal control problem

Silvia C. Di Marco, Roberto L.V. González (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work we deal with the numerical solution of a Hamilton-Jacobi-Bellman (HJB) equation with infinitely many solutions. To compute the maximal solution – the optimal cost of the original optimal control problem – we present a complete discrete method based on the use of some finite elements and penalization techniques.

Numerical resolution of an “unbalanced” mass transport problem

Jean-David Benamou (2003)

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

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.

Numerical resolution of an “unbalanced” mass transport problem

Jean-David Benamou (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented Lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.

Numerical solutions of the mass transfer problem

Serge Dubuc, Issa Kagabo (2006)

RAIRO - Operations Research

Let μ and ν be two probability measures on the real line and let c be a lower semicontinuous function on the plane. The mass transfer problem consists in determining a measure ξ whose marginals coincide with μ and ν, and whose total cost ∫∫ c(x,y)dξ(x,y) is minimum. In this paper we present three algorithms to solve numerically this Monge-Kantorovitch problem when the commodity being shipped is one-dimensional and not necessarily confined to a bounded interval. We illustrate these numerical...

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