2,117 research outputs found
Total variation regularization of multi-material topology optimization
This work is concerned with the determination of the diffusion coefficient
from distributed data of the state. This problem is related to homogenization
theory on the one hand and to regularization theory on the other hand. An
approach is proposed which involves total variation regularization combined
with a suitably chosen cost functional that promotes the diffusion coefficient
assuming prespecified values at each point of the domain. The main difficulty
lies in the delicate functional-analytic structure of the resulting
nondifferentiable optimization problem with pointwise constraints for functions
of bounded variation, which makes the derivation of useful pointwise optimality
conditions challenging. To cope with this difficulty, a novel reparametrization
technique is introduced. Numerical examples using a regularized semismooth
Newton method illustrate the structure of the obtained diffusion coefficient.
Optimal Control of the Thermistor Problem in Three Spatial Dimensions
This paper is concerned with the state-constrained optimal control of the
three-dimensional thermistor problem, a fully quasilinear coupled system of a
parabolic and elliptic PDE with mixed boundary conditions. This system models
the heating of a conducting material by means of direct current. Local
existence, uniqueness and continuity for the state system are derived by
employing maximal parabolic regularity in the fundamental theorem of Pr\"uss.
Global solutions are addressed, which includes analysis of the linearized state
system via maximal parabolic regularity, and existence of optimal controls is
shown if the temperature gradient is under control. The adjoint system
involving measures is investigated using a duality argument. These results
allow to derive first-order necessary conditions for the optimal control
problem in form of a qualified optimality system. The theoretical findings are
illustrated by numerical results
SUFFICIENT OPTIMALITY CONDITIONS FOR THE MOREAU-YOSIDA TYPE REGULARIZATION CONCEPT APPLIED TO SEMILINEAR ELLIPTIC OPTIMAL CONTROL PROBLEMS WITH POINTWISE STATE CONSTRAINTS
We develop sufficient optimality conditions for a Moreau-Yosidaregularized optimal control problem governed by a semilinear ellipticPDE with pointwise constraints on the state and the control. We makeuse of the equivalence of a setting of Moreau-Yosida regularization to a special setting of the virtual control concept,for which standard second order sufficient conditions have been shown. Moreover, we present a numerical example,solving a Moreau-Yosida regularized model problem with an SQP method
Regularization error estimates for semilinear elliptic optimal control problems with pointwise state and control constraints
In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. A sufficient second order optimality condition and uniqueness of the dual variables are assumed for that problem. Sufficient second order optimality conditions are shown for regularized problems with small regularization parameter. Moreover, error estimates with respect to the regularization parameter are derived
A priori error analysis for state constrained boundary control problems. Part II: Full discretization
This is the second of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [26] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems. The theoretical results are illustrated by numerical computations
Constrained Nonsmooth Problems of the Calculus of Variations
The paper is devoted to an analysis of optimality conditions for nonsmooth
multidimensional problems of the calculus of variations with various types of
constraints, such as additional constraints at the boundary and isoperimetric
constraints. To derive optimality conditions, we study generalised concepts of
differentiability of nonsmooth functions called codifferentiability and
quasidifferentiability. Under some natural and easily verifiable assumptions we
prove that a nonsmooth integral functional defined on the Sobolev space is
continuously codifferentiable and compute its codifferential and
quasidifferential. Then we apply general optimality conditions for nonsmooth
optimisation problems in Banach spaces to obtain optimality conditions for
nonsmooth problems of the calculus of variations. Through a series of simple
examples we demonstrate that our optimality conditions are sometimes better
than existing ones in terms of various subdifferentials, in the sense that our
optimality conditions can detect the non-optimality of a given point, when
subdifferential-based optimality conditions fail to disqualify this point as
non-optimal.Comment: A number of small mistakes and typos was corrected in the second
version of the paper. Moreover, the paper was significantly shortened.
Extended and improved versions of the deleted sections on nonsmooth Noether
equations and nonsmooth variational problems with nonholonomic constraints
will be published in separate submission
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