arXiv subjects

Morten Vierling

Publications and source records attributed to Morten Vierling.

5 recordsLinked to original sources

arXiv2015

Crank-Nicolson time stepping and variational discretization of control-constrained parabolic optimal control problems

We consider a control constrained parabolic optimal control problem and use variational discretization for its time semi-discretization. The state equation is treated with a Petrov-Galerkin scheme using a piecewise constant Ansatz for the state and piecewise linear, continuous test functions. This results in variants of the Crank-Nicolson scheme for the state and the adjoint state. Exploiting a superconvergence result we prove second order convergence in time of the error in the controls. Moreover, the piecewise linear and continuous parabolic projection of the discrete state on the dual time grid provides a second order convergent approximation of the optimal state without further numerical effort. Numerical experiments confirm our analytical findings.

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arXiv2011

Control-constrained parabolic optimal control problems on evolving surfaces - theory and variational discretization

We consider control-constrained linear-quadratic optimal control problems on evolving surfaces. In order to formulate well-posed problems, we prove existence and uniqueness of weak solutions for the state equation, in the sense of vector-valued distributions. We then carry out and prove convergence of the variational discretization of a distributed optimal control problem. In the process, we investigate the convergence of a fully discrete approximation of the state equation, and obtain optimal orders of convergence under weak regularity assumptions. We conclude with a numerical example.

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arXiv2011

Optimal Control of the Laplace-Beltrami operator on compact surfaces - concept and numerical treatment

We consider optimal control problems of elliptic PDEs on hypersurfaces in 2- or 3-dimensional Euclidean space. The leading part of the PDE is given by the Laplace-Beltrami operator, which is discretized by finite elements on a polyhedral approximation of the surface. The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semismooth Newton algorithm. We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings.

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arXiv2010

An optimal order error estimate for the variational discretization of optimal control problems in the presence of pointwise control and state constraints

We consider the variational discretization of a linear-quadratic optimal control problem with pointwise control and state constraints. In order to allow for a Fréchet smooth norm, the problem is reformulated by means of a reflexive Sobolev space instead of the space of continuous functions. The discretization of the state equation yields a family of perturbed optimal control problems, whose solutions can be computed numerically. We apply an implicit multifunction theorem (IMT) to the first order necessary conditions to proof a bound on the perturbation error for these solutions. In order to verify the abstract regularity condition of the IMT, we compute the Fréchet coderivative of a set-valued representation of the necessary conditions. Applying our results to an elliptic state equation in two dimensions, undergoing a simple finite element discretization, we obtain convergence of order O(h).

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arXiv2009

Variational discretization and semi-smooth Newton methods; implementation, convergence and globalization in pde constrained optimization with control constraints

When combining the numerical concept of variational discretization and semi-smooth Newton methods for the numerical solution of pde constrained optimization with control constraints, special emphasis has to be taken on the implementation, convergence and globalization of the numerical algorithm. In the present work we address all these issues. In particular we prove fast local convergence of the algorithm and propose two different globalization strategies which are applicable in many practically relevant mathematical settings. We illustrate our analytical and algorithmical findings by numerical experiments.

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