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Proper orthogonal decomposition for optimality systems

Karl Kunisch, Stefan Volkwein (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

Proper orthogonal decomposition (POD) is a powerful technique for model reduction of non-linear systems. It is based on a Galerkin type discretization with basis elements created from the dynamical system itself. In the context of optimal control this approach may suffer from the fact that the basis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. A method is proposed which avoids this problem of unmodelled...

Properties of a quasi-uniformly monotone operator and its application to the electromagnetic $p$-$\text {curl}$ systems

Chang-Ho Song, Yong-Gon Ri, Cholmin Sin (2022)

Applications of Mathematics

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation $Au=b$. We prove that if $A$ is a quasi-uniformly monotone and hemi-continuous operator, then $A^{-1}$ is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence...

Properties of projection and penalty methods for discretized elliptic control problems

Andrzej Cegielski, Christian Grossmann (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, properties of projection and penalty methods are studied in connection with control problems and their discretizations. In particular, the convergence of an interior-exterior penalty method applied to simple state constraints as well as the contraction behavior of projection mappings are analyzed. In this study, the focus is on the application of these methods to discretized control problem.

Properties of triangulations obtained by the longest-edge bisection

Francisco Perdomo, Ángel Plaza (2014)

Open Mathematics

The Longest-Edge (LE) bisection of a triangle is obtained by joining the midpoint of its longest edge with the opposite vertex. Here two properties of the longest-edge bisection scheme for triangles are proved. For any triangle, the number of distinct triangles (up to similarity) generated by longest-edge bisection is finite. In addition, if LE-bisection is iteratively applied to an initial triangle, then minimum angle of the resulting triangles is greater or equal than a half of the minimum angle...

Proposition de préconditionneurs pseudo-différentiels pour l’équation CFIE de l’électromagnétisme

David P. Levadoux (2005)

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

We present a weak parametrix of the operator of the CFIE equation. An interesting feature of this parametrix is that it is compatible with different discretization strategies and hence allows for the construction of efficient preconditioners dedicated to the CFIE. Furthermore, one shows that the underlying operator of the CFIE verifies an uniform discrete Inf-Sup condition which allows to predict an original convergence result of the numerical solution of the CFIE to the exact one.

Proposition de préconditionneurs pseudo-différentiels pour l'équation CFIE de l'électromagnétisme

David P. Levadoux (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

On exhibe dans cette note une paramétrix (au sens faible) de l'opérateur sous-jacent à l'équation CFIE de l'électromagnétisme. L'intérêt de cette paramétrix est de se prêter à différentes stratégies de discrétisation et ainsi de pouvoir être utilisée comme préconditionneur de la CFIE. On montre aussi que l'opérateur sous-jacent à la CFIE satisfait une condition Inf-Sup discrète uniforme, applicable aux espaces de discrétisation usuellement rencontrés en électromagnétisme, et qui permet d'établir...

Prox-regularization and solution of ill-posed elliptic variational inequalities

Alexander Kaplan, Rainer Tichatschke (1997)

Applications of Mathematics

In this paper new methods for solving elliptic variational inequalities with weakly coercive operators are considered. The use of the iterative prox-regularization coupled with a successive discretization of the variational inequality by means of a finite element method ensures well-posedness of the auxiliary problems and strong convergence of their approximate solutions to a solution of the original problem. In particular, regularization on the kernel of the differential operator and regularization...

Psi-series of quadratic vector fields on the plane.

Amadeu Delshams, Arnau Mir (1997)

Publicacions Matemàtiques

Psi-series (i.e., logarithmic series) for the solutions of quadratic vector fields on the plane are considered. Its existence and convergence is studied, and an algorithm for the location of logarithmic singularities is developed. Moreover, the relationship between psi-series and non-integrability is stressed and in particular it is proved that quadratic systems with psi-series that are not Laurent series do not have an algebraic first integral. Besides, a criterion about non-existence of an analytic...

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