arXiv subjects

Christian Meyer

Publications and source records attributed to Christian Meyer.

At least 19 recordsLinked to original sources

arXiv2026

No-gap second-order conditions for optimization problems involving transport distances

We consider optimization problems in the space of measures. As a regularization term, the problem includes the transport distance to a given prior measure. For the derivation of second-order optimality conditions of no-gap type, the theory of weak-$\star$ second subderivatives is used which will lead to an equivalence with quadratic growth under additional assumptions on the smooth part of the objective and on the Kantorovich potential, i.e., the solution of the dual transport problem. Further, the weak-$\star$ second subderivative is calculated and weak-$\star$ epidifferentiability is proven. Finally, the results are applied to optimal control problems in measure space.

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arXiv2026

Switching Point Optimization for Abstract Parabolic Equations

This work is concerned with a switching point optimization problem governed by a semilinear parabolic equation in abstract function spaces. It is shown that the switching-point-to-control mapping is continuously Fr\'echet-differentiable when considered with values in the dual of H\"older continuous functions in time. By treating the state equation in weak form based on the concept of maximal parabolic regularity, one can then show that the reduced objective is continuously differentiable w.r.t. the switching points, which allows to use gradient-based methods like the proximal gradient method for its minimization. Numerical experiments confirm our theoretical findings, but also illustrate that such a method will in general not be able to solve the problem to global optimality due to the non-convex nature of the switching-point-to-control map. We therefore give a precise characterization of the convex hull of set of feasible switching functions in terms of an extended formulation. The latter might be useful for a branch-and-bound approach for the computation of global minimizers, but this is subject to future research.

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arXiv2026

Viscous Approximation of Optimal Control Problems Governed by Rate-Independent Systems with Non-Convex Energies

We consider an optimal control problem governed by a rate-inde\-pendent system with non-convex energy. The state equation is approximated by means of viscous regularization w.r.t.\ to hierarchy of two different Hilbert spaces. The regularized problem corresponds to an optimal control problem subject to a non-smooth ODE in Hilbert space, which is substantially easier to solve than the original optimal control problem. The convergence properties of the viscous regularization are investigated. It is shown that every sequence of globally optimal solutions of the viscous problems admits a (weakly) converging subsequence whose limit is a globally optimal solution of the original problem, provided that the latter admits at least one optimal solution with an optimal state that is continuous in time.

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arXiv2025

Optimal control of the Poisson equation with transport regularization: Properties of optimal transport plans and transport map

An optimal control problem in the space of Borel measures governed by the Poisson equation is investigated. The characteristic feature of the problem under consideration is the Tikhonov regularization term in form of the transportation distance of the control to a given prior. Existence of optimal solutions is shown and first-order necessary optimality conditions are derived. The latter are used to deduce structural a priori information about the optimal control and its support based on properties of the associated optimal transport plan.

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arXiv2025

Dual Regularization and Outer Approximation of Optimal Control Problems in BV

This paper is concerned with an elliptic optimal control problem with total variation (TV) restriction on the control in the constraints. We introduce a regularized optimal control problem by applying a quadratic regularization of the dual representation of the TV-seminorm. The regularized optimal control problem can be solved by means of an outer approximation algorithm. Convergence of the regularization for vanishing regularization parameter as well as convergence of the outer approximation algorithm is proven. Moreover, we derive necessary and sufficient optimality conditions for the original unregularized optimal control problem and use these to construct an exact solution that we use in our numerical experiments to confirm our theoretical results.

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arXiv2024

Parabolic optimal control problems with combinatorial switching constraints -- Part III: Branch-and-bound algorithm

We present a branch-and-bound algorithm for globally solving parabolic optimal control problems with binary switches that have bounded variation and possibly need to satisfy further combinatorial constraints. More precisely, for a given tolerance $\varepsilon>0$, we show how to compute in finite time an $\varepsilon$-optimal solution in function space, independently of any prior discretization. The main ingredients in our approach are an appropriate branching strategy in infinite dimension, an a posteriori error estimation in order to obtain safe dual bounds, and an adaptive refinement strategy in order to allow arbitrary switching points in the limit. The performance of our approach is demonstrated by extensive experimental results.

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arXiv2023

Parabolic optimal control problems with combinatorial switching constraints -- Part I: Convex relaxations

We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon, they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. While such combinatorial constraints are often seen as an additional complication that is treated in a heuristic postprocessing, the core of our approach is to investigate the convex hull of all feasible switching patterns in order to define a tight convex relaxation of the control problem. The convex relaxation is built by cutting planes derived from finite-dimensional projections, which can be studied by means of polyhedral combinatorics. A numerical example for the case of a bounded number of switchings shows that our approach can significantly improve the dual bounds given by the straightforward continuous relaxation, which is obtained by relaxing binarity constraints.

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arXiv2023

Parabolic optimal control problems with combinatorial switching constraints -- Part II: Outer approximation algorithm

We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon, they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. In a companion paper [arXiv:2203.07121], we describe the $L^p$-closure of the convex hull of feasible switching patterns as intersection of convex sets derived from finite-dimensional projections. In this paper, the resulting outer description is used for the construction of an outer approximation algorithm in function space, whose iterates are proven to converge strongly in $L^2$ to the global minimizer of the convexified optimal control problem. The linear-quadratic subproblems arising in each iteration of the outer approximation algorithm are solved by means of a semi-smooth Newton method. A numerical example in two spatial dimensions illustrates the efficiency of the overall algorithm.

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arXiv2023

On a Lack of Stability of Parametrized BV Solutions to Rate-Independent Systems with Non-Convex Energies and Discontinuous Loads

We consider a rate-independent system with nonconvex energy under discontinuous external loading. The underlying space is finite dimensional and the loads are functions in $BV([0,T];\mathbb{R}^d)$. We investigate the stability of various solution concepts w.r.t. a sequence of loads converging weakly$*$ in $BV([0,T];\mathbb{R}^d)$ with a particular emphasis on the so-called normalized, $\mathfrak{p}$-parametrized balanced viscosity solutions. By means of two counterexamples, it is shown that common solution concepts are not stable w.r.t. weak$*$ convergence of loads in the sense that a limit of a sequence of solutions associated with these loads need not be a solution corresponding to the load in the limit. We moreover introduce a new solution concept, which is stable in this sense, but our examples show that this concept necessarily allows "solutions" that are physically meaningless.

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arXiv2023

Finite Element Approximation of Data-Driven Problems in Conductivity

This paper is concerned with the finite element discretization of the data driven approach according to arXiv:1510.04232 for the solution of PDEs with a material law arising from measurement data. To simplify the setting, we focus on a scalar diffusion problem instead of a problem in elasticity. It is proven that the data convergence analysis from arXiv:1708.02880 carries over to the finite element discretization as long as $H(\mathrm{div})$-conforming finite elements such as the Raviart-Thomas element are used. As a corollary, minimizers of the discretized problems converge in data in the sense of arXiv:1708.02880, as the mesh size tends to zero and the approximation of the local material data set gets more and more accurate. We moreover present several heuristics for the solution of the discretized data driven problems, which is equivalent to a quadratic semi-assignment problem and therefore NP-hard. We test these heuristics by means of two examples and it turns out that the "classical" alternating projection method according to arXiv:1510.04232 is superior w.r.t. the ratio of accuracy and computational time.

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arXiv2022

Bilevel Optimization of the Kantorovich Problem and its Quadratic Regularization Part II: Convergence Analysis

This paper is concerned with an optimization problem that is constrained by the Kantorovich optimal transportation problem. This bilevel optimization problem can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness induced by the complementarity relations, problems of this type are frequently regularized. Here we apply a quadratic regularization of the Kantorovich problem. As the title indicates, this is the second part in a series of three papers. While the existence of optimal solutions to both the bilevel Kantorovich problem and its regularized counterpart were shown in the first part, this paper deals with the (weak-*) convergence of solutions to the regularized bilevel problem to solutions of the original bilevel Kantorovich problem.

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arXiv2022

Bilevel Optimization of the Kantorovich Problem and its Quadratic Regularization Part I: Existence Results

This paper is concerned with an optimization problem governed by the Kantorovich optimal transportation problem. This gives rise to a bilevel optimization problem, which can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness induced by the complementarity relations, problems of this type are frequently regularized. Here we apply a quadratic regularization of the Kantorovich problem. As the title indicates, this is the first part in a series of three papers. It addresses the existence of optimal solutions to the bilevel Kantorovich problem and its quadratic regularization, whereas part II and III are dedicated to the convergence analysis for vanishing regularization.

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arXiv2022

An Adaptive Time Stepping Scheme for Rate-Independent Systems with Non-Convex Energy

We investigate a local incremental stationary scheme for the numerical solution of rate-independent systems. Such systems are characterized by a (possibly) non-convex energy and a dissipation potential, which is positively homogeneous of degree one. Due to the non-convexity of the energy, the system does in general not admit a time-continuous solution. In order to resolve these potential discontinuities, the algorithm produces a sequence of state variables and physical time points as functions of a curve parameter. The main novelty of our approach in comparison to existing methods is an adaptive choice of the step size for the update of the curve parameter depending on a prescribed tolerance for the residua in the energy-dissipation balance and in a complementarity relation concerning the so-called local stability condition. It is proven that, for tolerance tending to zero, the piecewise affine approximations generated by the algorithm converge (weakly) to a so-called $\mathbb{V}$-parametrized balanced viscosity solution. Numerical experiments illustrate the theoretical findings and show that an adaptive choice of the step size indeed pays off as they lead to a significant increase of the step size during sticking and in viscous jumps.

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arXiv2022

$L^α$-Regularization of the Beckmann Problem

We investigate the problem of optimal transport in the so-called Beckmann form, i.e. given two Radon measures on a compact set, we seek an optimal flow field which is a vector valued Radon measure on the same set that describes a flow between these two measures and minimizes a certain linear cost function. We consider $L^α$ regularization of the problem, which guarantees uniqueness and forces the solution to be an integrable function rather than a Radon measure. This regularization naturally gives rise to a semi-smooth Newton scheme that can be used to solve the problem numerically. Besides motivating and developing the numerical scheme, we also include approximation results for vanishing regularization in the continuous setting.

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arXiv2021

Estimation of inter-sector asset correlations

Asset correlations are an intuitive and therefore popular way to incorporate event dependence into event risk, e.g., default risk, modeling. In this paper we study the case of estimation of inter-sector asset correlations by separation of cross-sectional dimension and time dimension.

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arXiv2021

Model Risk in Credit Portfolio Models

Model risk in credit portfolio models is a serious issue for banks but has so far not been tackled comprehensively. We will demonstrate how to deal with uncertainty in all model parameters in an all-embracing, yet easy-to-implement way.

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arXiv2020

A Priori Error Analysis for an Optimal Control Problem Governed by a Variational Inequality of the Second Kind

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.

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arXiv2020

Optimal control of an abstract evolution variational inequality with application to homogenized plasticity

The paper is concerned with an optimal control problem governed by a state equation in form of a generalized abstract operator differential equation involving a maximal monotone operator. The state equation is uniquely solvable, but the associated solution operator is in general not Gâteaux-differentiable. In order to derive optimality conditions, we therefore regularize the state equation and its solution operator, respectively, by means of a (smoothed) Yosida approximation. We show convergence of global minimizers for regularization parameter tending to zero and derive necessary and sufficient optimality conditions for the regularized problems. The paper ends with an application of the abstract theory to optimal control of homogenized quasi-static elastoplasticity.

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