Discussiones Mathematicae. Differential Inclusions, Control and Optimization 

About: Discussiones Mathematicae. Differential Inclusions, Control and Optimization is an academic journal published by Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora. The journal publishes majorly in the area(s): Differential inclusion & Banach space. It has an ISSN identifier of 1509-9407. Over the lifetime, 256 publications have been published receiving 1312 citations. The journal is also known as: DMDICO.

Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems

Abstract: Proper orthogonal decomposition (POD) is a powerful technique for model reduction of linear and non-linear systems. It is based on a Galerkin type discretization with basis elements created from the system itself. In this work, error estimates for Galerkin POD methods for linear elliptic, parameter-dependent systems are proved. The resulting error bounds depend on the number of POD basis functions and on the parameter grid that is used to generate the snapshots and to compute the POD basis. The error estimates also hold for semi-linear elliptic problems with monotone nonlinearity. Numerical examples are included.

Boundary value problems for differential inclusions with fractional order

Abstract: In this paper, we shall establish sufficient conditions for the existence of solutions for a boundary value problem for fractional differential inclusions. Both cases of convex valued and nonconvex valued right hand sides are considered.

Boundary integral representations of second derivatives in shape optimization

Abstract: For a shape optimization problem second derivatives are investigated, obtained by a special approach for the description of the boundary variation and the use of a potential ansatz for the state. The natural embedding of the problem in a Banach space allows the application of a standard dierential calculus in order to get second derivatives by a straight forward ”repetition of dierentiation”. Moreover, by using boundary value characerizations for more regular data, a complete boundary integral representation of the second derivative of the objective is possible. Basing on this, one easily obtains that the second derivative contains only normal components for stationary domains, i.e. for domains, satisfying the first order necessary condition for a free optimum. Moreover, the nature of the second derivative is discussed, which is helpful for the investigation of sucient optimality

Generalized Newton and NCP-methods: convergence, regularity, actions

Abstract: Solutions of several problems can be modelled as solutions of nonsmooth equations. Then, Newton-type methods for solving such equations induce particular iteration steps (actions) and regularity requirements in the original problems. We study these actions and requirements for nonlinear complementarity problems (NCP’s) and Karush–Kuhn–Tucker systems (KKT) of optimization models. We demonstrate their dependence on the applied Newton techniques and the corresponding reformulations. In this way, connections to SQPmethods, to penalty-barrier methods and to general properties of so-called NCP-functions are shown. Moreover, direct comparisons of the hypotheses and actions in terms of the original problems become possible. Besides, we point out the possibilities and bounds of such methods in dependence of smoothness.