The paper is concerned with an optimal control problem governed by the rate-independent system of quasi-static perfect elasto-plasticity. The objective is to optimize the stress field by controlling the displacement at prescribed parts of the boundary. The control thus enters the system in the Dirichlet boundary conditions. Therefore, the safe load condition is automatically fulfilled so that the system admits a solution, whose stress field is unique. This gives rise to a well defined control-to-state operator, which is continuous but not Gâteaux differentiable. The control-to-state map is therefore regularized, first by means of the Yosida regularization resp. viscous approximation and then by a second smoothing in order to obtain a smooth problem. The approximation of global minimizers of the original non-smooth optimal control problem is shown and optimality conditions for the regularized problem are established. A numerical example illustrates the feasibility of the smoothing approach. © 2022, American Institute of Mathematical Sciences. All rights reserved.

This chapter provides a survey on the analysis, simulation, and optimal control of a class of non-smooth evolution systems that appears in the modeling of dissipative solids. Our focus is on models that include internal constraints, such as a flow rule in plasticity, and that account for the temperature dependence of the respective materials. We discuss here two cases, namely purely rate-independent models and viscously regularized models coupled to the temperature equation. © 2022, Springer Nature Switzerland AG.

This chapter is concerned with necessary optimality conditions for optimal control problems governed by variational inequalities of the second kind. The so-called strong stationarity conditions are derived in an abstract framework. Strong stationarity conditions are regarded as the most rigorous ones, since they imply all other types of stationarity concepts and are equivalent to purely primal optimality conditions. The abstract framework is afterward applied to four application-driven examples. © 2022, Springer Nature Switzerland AG.

We consider an optimal control problem governed by an elliptic variational inequality of the second kind. The problem is discretized by linear finite elements for the state and a variational discrete approach for the control. Based on a quadratic growth condition we derive nearly optimal a priori error estimates. Moreover, we establish second order sufficient optimality conditions that ensure a quadratic growth condition. These conditions are rather restrictive, but allow us to construct a one-dimensional locally optimal solution with reduced regularity, which serves as an exact solution for numerical experiments. © 2021 Taylor & Francis Group, LLC.

The paper is concerned with an optimal control problem governed by the rateindependent system of quasi-static perfect elasto-plasticity. The objective is optimize the displacement field in the domain occupied by the body by means of prescribed Dirichlet boundary data, which serve as control variables. The arising optimization problem is nonsmooth for several reasons, in particular, since the control-to-state mapping is not single-valued. We therefore apply a Yosida-regularization (resp., vanishing viscosity approach) to obtain a single-valued control-to-state operator. Beside the existence of optimal solutions, their approximation by means of this regularization approach is the main subject of this work. It turns out that a so-called reverse approximation guaranteeing the existence of a suitable recovery sequence can only be shown under an additional smoothness assumption on at least one optimal solution. © 2021 Society for Industrial and Applied Mathematics.

We propose a general trust-region method for the minimization of nonsmooth and nonconvex, locally Lipschitz continuous functions that can be applied, e.g., to optimization problems constrained by elliptic variational inequalities. The convergence of the considered algorithm to Cstationary points is verified in an abstract setting and under suitable assumptions on the involved model functions. For a special instance of a variational inequality constrained problem, we are able to properly characterize the Bouligand subdifferential of the reduced cost function, and, based on this characterization result, we construct a computable trust-region model which satisfies all hypotheses of our general convergence analysis. The article concludes with numerical experiments that illustrate the main properties of the proposed algorithm. © 2020 Society for Industrial and Applied Mathematics.

This paper is concerned with a priori error estimates for the local incremental minimization scheme, which is an implicit time discretization method for the approximation of rate-independent systems with nonconvex energies. We first show by means of a counterexample that one cannot expect global convergence of the scheme without any further assumptions on the energy. For the class of uniformly convex energies, we derive error estimates of optimal order, provided that the Lipschitz constant of the load is sufficiently small. Afterwards, we extend this result to the case of an energy, which is only locally uniformly convex in a neighborhood of a given solution trajectory. For the latter case, the local incremental minimization scheme turns out to be superior compared to its global counterpart, as a numerical example demonstrates. © 2020 Society for Industrial and Applied Mathematics

The paper deals with a viscous damage model including two damage variables, a local and a non-local one, which are coupled through a penalty term in the free energy functional. Under certain regularity conditions for linear elasticity equations, existence and uniqueness of the solution is proven, provided that the penalization parameter is chosen sufficiently large. Moreover, the regularity of the unique solution is investigated, in particular the differentiability w.r.t. time. © European Mathematical Society.

This paper is concerned with a space-time discretization of a rate-independent evolution governed by a non-smooth dissipation and a non-convex energy functional. For the time discretization, we apply the local minimization scheme introduced in Efendiev and Mielke (J Convex Anal 13(1):151–167, 2006), which is known to resolve time discontinuities, which may show up due to the non-convex energy. The spatial discretization is performed by classical linear finite elements. We show that accumulation points of the sequence of discrete solutions for mesh size tending to zero exist and are so-called parametrized solutions of the continuous problem. The discrete problems are solved by means of a mass lumping scheme for the non-smooth dissipation functional in combination with a semi-smooth Newton method. A numerical test indicates the efficiency of this approach. In addition, we compared the local minimization scheme with a time stepping scheme for global energetic solutions, which shows that both schemes yield different solutions with differing time discontinuities. © 2019, Istituto di Informatica e Telematica (IIT).

We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive two algorithms to solve the dual problem of the regularized problem: A Gauss–Seidel method and a semismooth quasi-Newton method and investigate both methods numerically. Our experiments show that the methods perform well even for small regularization parameters. Quadratic regularization is of interest since the resulting optimal transport plans are sparse, i.e. they have a small support (which is not the case for the often used entropic regularization where the optimal transport plan always has full measure). © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

We study the stability of solutions to H01-elliptic variational inequalities of the second kind that contain a non-differentiable Nemytskii operator. The local Lipschitz continuity of the solution map with respect to perturbations of the right-hand side and perturbations of the coefficient of the Nemytskii operator is proved for a large class of problems, and Hadamard directional differentiability results are obtained under comparatively mild structural assumptions. It is further shown that the directional derivatives of the solution map are typically characterized by elliptic variational inequalities in weighted Sobolev spaces whose bilinear forms contain surface integrals. © 2018, Springer Nature B.V.

This paper is concerned with a priori error estimates for the piecewise linear finite element approximation of the classical obstacle problem. We demonstrate by means of two one-dimensional counterexamples that the L2-error between the exact solution u and the finite element approximation uh is typically not of order two even if the exact solution is in H2(Ω) and an estimate of the form ‖u-uh‖H1≤Ch holds true. This shows that the classical Aubin–Nitsche trick which yields a doubling of the order of convergence when passing over from the H1- to the L2-norm cannot be generalized to the obstacle problem. © 2017, Springer-Verlag GmbH Germany, part of Springer Nature.

Optimal control problems (OCPs) containing both integrality and partial differential equation (PDE) constraints are very challenging in practice. The most wide-spread solution approach is to first discretize the problem, which results in huge and typically nonconvex mixed-integer optimization problems that can be solved to proven optimality only in very small dimensions. In this paper, we propose a novel outer approximation approach to efficiently solve such OCPs in the case of certain semilinear elliptic PDEs with static integer controls over arbitrary combinatorial structures, where we assume the nonlinear part of the PDE to be non-decreasing and convex. The basic idea is to decompose the OCP into an integer linear programming (ILP) master problem and a subproblem for calculating linear cutting planes. These cutting planes rely on the pointwise concavity or submodularity of the PDE solution with respect to the control variables. The decomposition allows us to use standard solution techniques for ILPs as well as for PDEs. We further benefit from reoptimization strategies for the PDE solution due to the iterative structure of the algorithm. Experimental results show that the new approach is capable of solving the combinatorial OCP of a semilinear Poisson equation with up to 180 binary controls to global optimality within a 5 h time limit. In the case of the screened Poisson equation, which yields semi-infinite integer linear programs, problems with as many as 1400 binary controls are solved. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

In this vision paper, we argue that current solutions to data analytics are not suitable for complex tasks from the humanities, as they are agnostic of the user and focused on static, predefined tasks with large-scale benchmarks. Instead, we believe that the human must be put into the loop to address small data scenarios that require expert domain knowledge and fluid, incrementally defined tasks, which are common for many humanities use cases. Besides the main challenges, we discuss existing and urgently required solutions to interactive data acquisition, model development, model interpretation, and system support for interactive data analytics. In the envisioned interactive systems, human users not only provide annotations to a machine learner, but train a model by using the system and demonstrating the task. The learning system will actively query the user for feedback, refine its model in real-time, and is able to explain its decisions. Our vision links natural language processing research with recent advances in machine learning, computer vision, and data management systems, as realizing this vision relies on combining expertise from all of these scientific fields. © Springer Nature Switzerland AG 2018.

This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method. © 2018, American Institute of Mathematical Sciences. All rights reserved.

We present an empirical study addressing the question whether, and to which extent, lexicographic writing aids improve text revision results. German university students were asked to optimise two German texts using (1) no aids at all, (2) highlighted problems, or (3) highlighted problems accompanied by lexicographic resources that could be used to solve the specific problems. We found that participants from the third group corrected the largest number of problems and introduced the fewest semantic distortions during revision. Also, they reached the highest overall score and were most efficient (as measured in points per time). The second group with highlighted problems lies between the two other groups in almost every measure we analysed. We discuss these findings in the scope of intelligent writing environments, the effectiveness of writing aids in practical usage situations and teaching dictionary skills. © 2017 Oxford University Press. All rights reserved.

This paper is concerned with a priori error estimates for the piecewise linear finite element approximation of the classical obstacle problem. We demonstrate by means of two one-dimensional counterexamples that the (Formula presented.)-error between the exact solution u and the finite element approximation (Formula presented.) is typically not of order two even if the exact solution is in (Formula presented.) and an estimate of the form (Formula presented.) holds true. This shows that the classical Aubin–Nitsche trick which yields a doubling of the order of convergence when passing over from the (Formula presented.)- to the (Formula presented.)-norm cannot be generalized to the obstacle problem. © 2017 Springer-Verlag GmbH Germany, part of Springer Nature

A quasistatic, thermoviscoplastic model at small strains with linear kinematic hardening, von Mises yield condition and mixed boundary conditions is considered. The existence of a unique weak solution is proved by means of a fixed-point argument, and by employing maximal parabolic regularity theory. The weak continuity of the solution operator is also shown. As an application, the existence of a global minimizer of a class of optimal control problems is proved. © 2016 Elsevier Ltd

This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable but not Gateaux-differentiable. By employing the limited differentiability properties of the control-to-state map, first-order necessary optimality conditions in qualified form are established, which are equivalent to the purely primal condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. The paper ends with the application of the general results to a semilinear heat equation. © 2017 Society for Industrial and Applied Mathematics.

This paper is concerned with the state-constrained optimal control of the threedimensional 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 as well as existence of optimal solutions, admitting global-in-time solutions, to the optimization problem were shown in the the companion paper of this work. In this part, we address further properties of the set of controls whose associated solutions exist globally, such as openness, which includes analysis of the linearized state system via maximal parabolic regularity. The adjoint system involving measures is investigated using a duality argument. These results allow us to derive first-order necessary conditions for the optimal control problem in the form of a qualified optimality system in which we do not need to refer to the set of controls admitting global solutions. The theoretical findings are illustrated by numerical results. This work is the second of two papers on the three-dimensional thermistor problem. © 2017 Society for Industrial and Applied Mathematics.

An optimal control problem of static plasticity with linear kinematic hardening and von Mises yield condition is studied. The problem is treated in its primal formulation, where the state system is a variational inequality of the second kind. First-order necessary optimality conditions are obtained by means of an approximation by a family of control problems with state system regularized by Huber-type smoothing, and a subsequent limit analysis. The equivalence of the optimality conditions with the C-stationarity system for the equivalent dual formulation of the problem is proved. Numerical experiments are presented, which demonstrate the viability of the Huber-type smoothing approach. © 2016 Society for Industrial and Applied Mathematics.

This note is concerned with an optimal control problem governed by the relativistic Maxwell-Newton-Lorentz equations, which describe the motion of charged particles in electromagnetic fields and consists of a hyperbolic PDE system coupled with a nonlinear ODE. An external magnetic field acts as control variable. Additional control constraints are incorporated by introducing a scalar magnetic potential which leads to an additional state equation in the form of a very weak elliptic PDE. Existence and uniqueness for the state equation is shown and the existence of a global optimal control is established. Moreover, first-order necessary optimality conditions in the form of Karush-Kuhn-Tucker conditions are derived. A numerical test illustrates the theoretical findings. © 2016 Society for Industrial and Applied Mathematics.

We investigate optimality conditions for optimization problems constrained by a class of variational inequalities of the second kind. Based on a nonsmooth primal–dual reformulation of the governing inequality, the differentiability of the solution map is studied. Directional differentiability is proved both for finite-dimensional problems and for problems in function spaces, under suitable assumptions on the active set. A characterization of Bouligand and strong stationary points is obtained thereafter. Finally, based on the obtained first-order information, a trust-region algorithm is proposed for the solution of the optimization problems. © 2015, Springer Science+Business Media New York.

This article is concerned with the derivation of a posteriori error estimates for optimization problems subject to an obstacle problem. To circumvent the nondifferentiability inherent to this type of problem, we introduce a sequence of penalized but differentiable problems. We show differentiability of the central path and derive separate a posteriori dual weighted residual estimates for the errors due to penalization, discretization, and iterative solution of the discrete problems. The effectivity of the derived estimates and of the adaptive algorithm is demonstrated on two numerical examples. © 2015 Society for Industrial and Applied Mathematics.

The paper is concerned with the optimal control of static elastoplasticity with linear kinematic hardening. This leads to an optimal control problem governed by an elliptic variational inequality (VI) of first kind in mixed form. Based on Lp-regularity results for the state equation, it is shown that the control-to-state operator is Bouligand differentiable. This enables to establish second-order sufficient optimality conditions by means of a Taylor expansion of a particularly chosen Lagrange function. © EDP Sciences, SMAI 2014.

An optimal control problem governed by a unilateral obstacle problem is considered. The problem is discretized using linear finite elements for the state and the obstacle and a variational discrete approach for the control. Based on strong stationarity and a quadratic growth condition we establish a priori error estimates which turn out to be quasi-optimal under additional assumptions on the data. The theoretical findings are illustrated by two numerical tests. © 2013 Society for Industrial and Applied Mathematics.

We report on various algorithms used for the nonlinear optimization of RBE-weighted dose in particle therapy. Concerning the dose calculation carbon ions are considered and biological effects are calculated by the Local Effect Model. Taking biological effects fully into account requires iterative methods to solve the optimization problem. We implemented several additional algorithms into GSI's treatment planning system TRiP98, like the BFGS-algorithm and the method of conjugated gradients, in order to investigate their computational performance. We modified textbook iteration procedures to improve the convergence speed. The performance of the algorithms is presented by convergence in terms of iterations and computation time. We found that the Fletcher-Reeves variant of the method of conjugated gradients is the algorithm with the best computational performance. With this algorithm we could speed up computation times by a factor of 4 compared to the method of steepest descent, which was used before. With our new methods it is possible to optimize complex treatment plans in a few minutes leading to good dose distributions. At the end we discuss future goals concerning dose optimization issues in particle therapy which might benefit from fast optimization solvers. © 2013 Institute of Physics and Engineering in Medicine.

Optimal control problems for the variational inequality of static elastoplasticity with linear kinematic hardening are considered. The control-to-state map is shown to be weakly directionally differentiable, and local optimal controls are proved to verify an optimality system of B-stationary type. For a modified problem, local minimizers are shown to even satisfy an optimality system of strongly stationary type. © 2013 Society for Industrial and Applied Mathematics.

Existence of the plastic multiplier with L1 spatial regularity for quasistatic and static plasticity is proved for arbitrary continuous and convex yield functions and linear hardening laws. L2 regularity is shown in the particular cases of kinematic hardening, or combined kinematic and isotropic hardening. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Equations of linear and nonlinear infinitesimal elasticity with mixed boundary conditions are considered. The bounded domain is assumed to have a Lipschitz boundary and to satisfy additional regularity assumptions. W1,p regularity for the displacements and Lp regularity for the stresses are proved for some p>2. © 2011 Elsevier Inc.

An optimal control problem for the static problem of infinitesimal elastoplasticity with linear kinematic hardening is considered. The variational inequality arising on the lower-level is regularized using a Yosida-type approach, and an optimal control problem for the so-called viscoplastic model is obtained. Existence of a global optimizer is proved for both the regularized and original problems, and strong convergence of the solutions is established. An optimal control problem for the static problem of infinitesimal elastoplasticity with linear kinematic hardening is considered. The variational inequality arising on the lower-level is regularized using a Yosida-type approach, and an optimal control problem for the so-called viscoplastic model is obtained. Existence of a global optimizer is proved for both the regularized and original problems, and strong convergence of the solutions is established. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The article considers linear elliptic equations with regular Borel measures as inhomogeneity. Such equations frequently appear in state-constrained optimal control problems. By a counter example of Serrin [18], it is known that, in the presence of non-smooth data, a standard weak formulation does not ensure uniqueness for such equations. Therefore several notions of solution have been developed that guarantee uniqueness. In this note, we compare different definitions of solutions, namely the ones of Stampacchia [19] and Boccardo-Galout [4] and the two notions of solutions of [2, 7], and show that they are equivalent. As side results, we reformulate the solution in the sense of [19], and prove the existence of solutions in the sense of [2, 4, 7] in case of mixed boundary conditions. Copyright © Taylor & Francis Group, LLC.

In this paper we consider a state-constrained optimal control problem with boundary control, where the state constraints are imposed only in an interior subdomain. Our goal is to derive a priori error estimates for a finite element discretization with and without additional regularization. We will show that the separation of the set where the control acts and the set where the state constraints are given improves the approximation rates significantly. The theoretical results are illustrated by numerical computations. © 2010 Society for Industrial and Applied Mathematics.