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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 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.

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...

Quasi-static evolution for fatigue debonding

Alessandro Ferriero (2008)

ESAIM: Control, Optimisation and Calculus of Variations

The propagation of fractures in a solid undergoing cyclic loadings is known as the fatigue phenomenon. In this paper, we present a time continuous model for fatigue, in the special situation of the debonding of thin layers, coming from a time discretized version recently proposed by Jaubert and Marigo [C. R. Mecanique333 (2005) 550–556]. Under very general assumptions on the surface energy density and on the applied displacement, we discuss the well-posedness of our problem and we give the main...

Random perturbation of the projected variable metric method for nonsmooth nonconvex optimization problems with linear constraints

Abdelkrim El Mouatasim, Rachid Ellaia, Eduardo Souza de Cursi (2011)

International Journal of Applied Mathematics and Computer Science

We present a random perturbation of the projected variable metric method for solving linearly constrained nonsmooth (i.e., nondifferentiable) nonconvex optimization problems, and we establish the convergence to a global minimum for a locally Lipschitz continuous objective function which may be nondifferentiable on a countable set of points. Numerical results show the effectiveness of the proposed approach.

Receding horizon optimal control for infinite dimensional systems

Kazufumi Ito, Karl Kunisch (2002)

ESAIM: Control, Optimisation and Calculus of Variations

The receding horizon control strategy for dynamical systems posed in infinite dimensional spaces is analysed. Its stabilising property is verified provided control Lyapunov functionals are used as terminal penalty functions. For closed loop dissipative systems the terminal penalty can be chosen as quadratic functional. Applications to the Navier–Stokes equations, semilinear wave equations and reaction diffusion systems are given.

Receding horizon optimal control for infinite dimensional systems

Kazufumi Ito, Karl Kunisch (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The receding horizon control strategy for dynamical systems posed in infinite dimensional spaces is analysed. Its stabilising property is verified provided control Lyapunov functionals are used as terminal penalty functions. For closed loop dissipative systems the terminal penalty can be chosen as quadratic functional. Applications to the Navier–Stokes equations, semilinear wave equations and reaction diffusion systems are given.

Recursive form of general limited memory variable metric methods

Ladislav Lukšan, Jan Vlček (2013)

Kybernetika

In this report we propose a new recursive matrix formulation of limited memory variable metric methods. This approach can be used for an arbitrary update from the Broyden class (and some other updates) and also for the approximation of both the Hessian matrix and its inverse. The new recursive formulation requires approximately 4 m n multiplications and additions per iteration, so it is comparable with other efficient limited memory variable metric methods. Numerical experiments concerning Algorithm...

Regularization parameter selection in discrete ill-posed problems - the use of the U-curve

Dorota Krawczyk-Stańdo, Marek Rudnicki (2007)

International Journal of Applied Mathematics and Computer Science

To obtain smooth solutions to ill-posed problems, the standard Tikhonov regularization method is most often used. For the practical choice of the regularization parameter α we can then employ the well-known L-curve criterion, based on the L-curve which is a plot of the norm of the regularized solution versus the norm of the corresponding residual for all valid regularization parameters. This paper proposes a new criterion for choosing the regularization parameter α, based on the so-called U-curve....

Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes

Uldis Raitums (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the weak closure W Z of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems where Ω 𝐑 n is a bounded Lipschitz domain, F s are strictly convex smooth functions with quadratic growth and S = { σ m e a s u r a b l e σ s ( x ) = 0 or 1 , s = 1 , , s 0 , σ 1 ( x ) + + σ s 0 ( x ) = 1 } . We show that W Z is the zero level set for an integral functional with the integrand Q being the 𝐀 -quasiconvex envelope for a certain function and the operator 𝐀 = ( curl,div ) m . If the functions F s are isotropic, then on the characteristic cone Λ (defined by the operator 𝐀 ) Q coincides...

Relaxation of Quasilinear Elliptic Systems via A-quasiconvex Envelopes

Uldis Raitums (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the weak closure WZ of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems div s = 1 s 0 σ s ( x ) F s ' ( u ( x ) + g ( x ) ) - f ( x ) = 0 in Ω , u = ( u 1 , , u m ) H 0 1 ( Ω ; 𝐑 m ) , σ = ( σ 1 , , σ s 0 ) S , where Ω ⊂ Rn is a bounded Lipschitz domain, Fs are strictly convex smooth functions with quadratic growth and S = { σ m e a s u r a b l e σ s ( x ) = 0 or 1 , s = 1 , , s 0 , σ 1 ( x ) + + σ s 0 ( x ) = 1 } . We show that WZ is the zero level set for an integral functional with the integrand Q being the A-quasiconvex envelope for a certain function and the operator A = (curl,div)m. If the functions Fs are isotropic, then on the characteristic cone...

Reliable computation and local mesh adaptivity in limit analysis

Sysala, Stanislav, Haslinger, Jaroslav, Repin, Sergey (2019)

Programs and Algorithms of Numerical Mathematics

The contribution is devoted to computations of the limit load for a perfectly plastic model with the von Mises yield criterion. The limit factor of a prescribed load is defined by a specific variational problem, the so-called limit analysis problem. This problem is solved in terms of deformation fields by a penalization, the finite element and the semismooth Newton methods. From the numerical solution, we derive a guaranteed upper bound of the limit factor. To achieve more accurate results, a local...

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