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Finite-differences discretizations of the mumford-shah functional

Antonin Chambolle (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

About two years ago, Gobbino [21] gave a proof of a De Giorgi's conjecture on the approximation of the Mumford-Shah energy by means of finite-differences based non-local functionals. In this work, we introduce a discretized version of De Giorgi's approximation, that may be seen as a generalization of Blake and Zisserman's “weak membrane” energy (first introduced in the image segmentation framework). A simple adaptation of Gobbino's results allows us to compute the Γ-limit of this discrete functional...

Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

Karl Kunisch, Georg Stadler (2005)

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

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented lagrangians allows to apply an infinite-dimensional...

Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

Karl Kunisch, Georg Stadler (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an...

Globalization of SQP-methods in control of the instationary Navier-Stokes equations

Michael Hintermüller, Michael Hinze (2002)

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

A numerically inexpensive globalization strategy of sequential quadratic programming methods (SQP-methods) for control of the instationary Navier Stokes equations is investigated. Based on the proper functional analytic setting a convergence analysis for the globalized method is given. It is argued that the a priori formidable SQP-step can be decomposed into linear primal and linear adjoint systems, which is amenable for existing CFL-software. A report on a numerical test demonstrates the feasibility...

Globalization of SQP-Methods in Control of the Instationary Navier-Stokes Equations

Michael Hintermüller, Michael Hinze (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A numerically inexpensive globalization strategy of sequential quadratic programming methods (SQP-methods) for control of the instationary Navier Stokes equations is investigated. Based on the proper functional analytic setting a convergence analysis for the globalized method is given. It is argued that the a priori formidable SQP-step can be decomposed into linear primal and linear adjoint systems, which is amenable for existing CFL-software. A report on a numerical test demonstrates the feasibility...

Guaranteed and computable bounds of the limit load for variational problems with linear growth energy functionals

Jaroslav Haslinger, Sergey Repin, Stanislav Sysala (2016)

Applications of Mathematics

The paper is concerned with guaranteed and computable bounds of the limit (or safety) load, which is one of the most important quantitative characteristics of mathematical models associated with linear growth functionals. We suggest a new method for getting such bounds and illustrate its performance. First, the main ideas are demonstrated with the paradigm of a simple variational problem with a linear growth functional defined on a set of scalar valued functions. Then, the method is extended to...

Homotopy method for minimum consumption orbit transfer problem

Joseph Gergaud, Thomas Haberkorn (2006)

ESAIM: Control, Optimisation and Calculus of Variations

The numerical resolution of the low thrust orbital transfer problem around the Earth with the maximization of the final mass or minimization of the consumption is investigated. This problem is difficult to solve by shooting method because the optimal control is discontinuous and a homotopic method is proposed to deal with these difficulties for which convergence properties are established. For a thrust of 0.1 Newton and a final time 50% greater than the minimum one, we obtain 1786 switching times....

Identification of critical curves. II. Discretization and numerical realization

Jaroslav Haslinger, Václav Horák, Pekka Neittaanmäki, Kimmo Salmenjoki (1991)

Applications of Mathematics

We consider the finite element approximation of the identification problem, where one wishes to identify a curve along which a given solution of the boundary value problem possesses some specific property. We prove the convergence of FE-approximation and give some results of numerical tests.

Image Compression with Schauder Bases

Zbigniew Ciesielski (2001)

Applicationes Mathematicae

As is known, color images are represented as multiple, channels, i.e. integer-valued functions on a discrete rectangle, corresponding to pixels on the screen. Thus, image compression, can be reduced to investigating suitable properties of such, functions. Each channel is compressed independently. We are, representing each such function by means of multi-dimensional, Haar and diamond bases so that the functions can be remembered, by their basis coefficients without loss of information. For, each...

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