arXiv subjects

Stefan Volkwein

Publications and source records attributed to Stefan Volkwein.

At least 19 recordsLinked to original sources

arXiv2026

Differentiability of the value function in control-constrained parabolic problems

Along the optimal trajectory of an optimal control problem constrained by a semilinear parabolic partial differential equation, we prove the differentiability of the value function with respect to the initial condition and, under additional assumptions on the solution of the state equation, the differentiability of the value function with respect to the time variable. In our proof, we rely on local growth assumptions commonly associated with the study of second-order sufficient conditions. These assumptions are generally applicable to a wide range of problems, including, for instance, certain tracking-type problems. Finally, we discuss the differentiability of the value function in a neighborhood of the optimal trajectory when a growth condition for optimal controls is used.

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arXiv2026

Optimality-Based Control Space Reduction for Infinite-Dimensional Control Spaces

We consider linear model reduction in both the control and state variables for unconstrained linear-quadratic optimal control problems subject to time-varying parabolic PDEs. The first-order optimality condition for a state-space reduced model naturally leads to a reduced structure of the optimal control. Thus, we consider a control- and state-reduced problem that admits the same minimizer as the solely state-reduced problem. Lower and upper \emph{a posteriori} error bounds for the optimal control and a representation for the error in the optimal function value are provided. These bounds are used in an adaptive algorithm to solve the control problem. We prove its convergence and numerically demonstrate the advantage of combined control and state space reduction.

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arXiv2026

Certified Model Predictive Control for Switched Evolution Equations using Model Order Reduction

We present a model predictive control (MPC) framework for linear switched evolution equations arising from a parabolic partial differential equation (PDE). First-order optimality conditions for the resulting finite-horizon optimal control problems are derived. The analysis allows for the incorporation of convex control constraints and sparse regularization. Then, to mitigate the computational burden of the MPC procedure, we employ Galerkin reduced-order modeling (ROM) techniques to obtain a low-dimensional surrogate for the state-adjoint systems. We derive recursive a-posteriori estimates for the ROM feedback law and the ROM-MPC closed-loop state and show that the ROM-MPC trajectory evolves within a neighborhood of the true MPC trajectory, whose size can be explicitly computed and is controlled by the quality of the ROM. Such estimates are then used to formulate two ROM-MPC algorithms with closed-loop certification.

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arXiv2026

Stabilization of Parabolic Time-Varying PDEs using Certified Reduced-Order Receding Horizon Control

We address the stabilization of linear, time-varying parabolic PDEs using finite-dimensional receding horizon controls (RHCs) derived from reduced-order models (ROMs). We first prove exponential stability and suboptimality of the continuous-time full-order model (FOM) RHC scheme in Hilbert spaces. A Galerkin model reduction is then introduced, along with a rigorous a posteriori error analysis for the associated finite-horizon optimal control problems. This results in a ROM-based RHC algorithm that adaptively constructs reduced-order controls, ensuring exponential stability of the FOM closed-loop state and providing computable performance bounds with respect to the infinite-horizon FOM control problem. Numerical experiments with a non-smooth cost functional involving the squared l1-norm confirm the methods effectiveness, even for exponentially unstable systems.

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arXiv2026

Reconstructing the system coefficients for coupled harmonic oscillators

Physical models often contain unknown functions and relations. In order to gain more insights into the nature of physical processes, these unknown functions have to be identified or reconstructed. Mathematically, we can formulate this research question within the framework of inverse problems. In this work, we consider optimization techniques to solve the inverse problem using Tikhonov regularization and data from laboratory experiments. We propose an iterative strategy that eliminates the need for further laboratory experiments. Our method is applied to identify the coupling and damping coefficients in a system of oscillators, ensuring an efficient and experiment-free approach. We present our results and compare them with those obtained from an alternative, purely experimental approach. By employing our proposed strategy, we demonstrate a significant reduction in the number of laboratory experiments required.

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arXiv2025

Adaptive Reduced Basis Trust Region Methods for Parabolic Inverse Problems

We consider nonlinear inverse problems arising in the context of parameter identification for parabolic partial differential equations (PDEs). For stable reconstructions, regularization methods such as the iteratively regularized Gauss-Newton method (IRGNM) are commonly used, but their application is computationally demanding due to the high-dimensional nature of PDE discretizations. To address this bottleneck, we propose a reduced-order modeling approach that accelerates both the state and adjoint evaluations required for derivative-based optimization. Our method builds on the recent contribution [Kartmann et al. Adaptive reduced basis trust region methods for parameter identification problems. Comput. Sci. Eng. 1, 3 (2024)] for elliptic forward operators and constructs the reduced forward operator adaptively in an online fashion, combining both parameter and state space reduction. To ensure reliability, we embed the IRGNM iteration within an adaptive, error-aware trust-region framework that certifies the accuracy of the reduced-order approximations. We demonstrate the effectiveness of the proposed approach through numerical results for both time-dependent and time-independent parameter identification problems in dynamic reaction-diffusion systems. The implementation is made available for reproducibility and further use.

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arXiv2025

Error estimates for a multiobjective optimal control of a pointwise tracking problem

We analyze a pointwise tracking multiobjective optimal control problem subject to the Poisson problem and bilateral control constraints. To approximate Pareto optimal points and the Pareto front numerically, we consider two different finite element-based scalarization techniques, namely the weighted-sum method and the reference point method, where in both methods many scalar-constrained optimization problems have to be solved. We prove a priori error estimates for both scalarizations. The underlying subproblems of either method are solved with a Barzilai-Borwein gradient method. Numerical experiments illustrate the accuracy of the proposed method.

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arXiv2025

Reconstruction of unknown monotone nonlinear operators in semilinear elliptic models using optimal inputs

Physical models often contain unknown functions and relations. The goal of our work is to answer the question of how one should excite or control a system under consideration in an appropriate way to be able to reconstruct an unknown nonlinear relation. To answer this question, we propose a greedy reconstruction algorithm within an offline-online strategy. We apply this strategy to a two-dimensional semilinear elliptic model. Our identification is based on the application of several space-dependent excitations (also called controls). These specific controls are designed by the algorithm in order to obtain a deeper insight into the underlying physical problem and a more precise reconstruction of the unknown relation. We perform numerical simulations that demonstrate the effectiveness of our approach which is not limited to the current type of equation. Since our algorithm provides not only a way to determine unknown operators by existing data but also protocols for new experiments, it is a holistic concept to tackle the problem of improving physical models.

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arXiv2024

Adjoint-based calibration of nonlinear stochastic differential equations

To study the nonlinear properties of complex natural phenomena, the evolution of the quantity of interest can be often represented by systems of coupled nonlinear stochastic differential equations (SDEs). These SDEs typically contain several parameters which have to be chosen carefully to match the experimental data and to validate the effectiveness of the model. In the present paper the calibration of these parameters is described by nonlinear SDE-constrained optimization problems. In the optimize-before-discretize setting a rigorous analysis is carried out to ensure the existence of optimal solutions and to derive necessary first-order optimality conditions. For the numerical solution a Monte-Carlo method is applied using parallelization strategies to compensate for the high computational time. In the numerical examples an Ornstein-Uhlenbeck and a stochastic Prandtl-Tomlinson bath model are considered.

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arXiv2024

Adaptive Reduced Basis Trust Region Methods for Parameter Identification Problems

In this contribution, we are concerned with model order reduction in the context of iterative regularization methods for the solution of inverse problems arising from parameter identification in elliptic partial differential equations. Such methods typically require a large number of forward solutions, which makes the use of the reduced basis method attractive to reduce computational complexity. However, the considered inverse problems are typically ill-posed due to their infinite-dimensional parameter space. Moreover, the infinite-dimensional parameter space makes it impossible to build and certify classical reduced-order models efficiently in a so-called "offline phase". We thus propose a new algorithm that adaptively builds a reduced parameter space in the online phase. The enrichment of the reduced parameter space is naturally inherited from the Tikhonov regularization within an iteratively regularized Gauß-Newton method. Finally, the adaptive parameter space reduction is combined with a certified reduced basis state space reduction within an adaptive error-aware trust region framework. Numerical experiments are presented to show the efficiency of the combined parameter and state space reduction for inverse parameter identification problems with distributed reaction or diffusion coefficients.

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arXiv2024

A non-intrusive neural-network based BFGS algorithm for parameter estimation in non-stationary elasticity

We present a non-intrusive gradient and a non-intrusive BFGS algorithm for parameter estimation problems in non-stationary elasticity. To avoid multiple (and potentially expensive) solutions of the underlying partial differential equation (PDE), we approximate the PDE solver by a neural network within the algorithms. The network is trained offline for a given set of parameters. The algorithms are applied to an unsteady linear-elastic contact problem; their convergence and approximation properties are investigated numerically.

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arXiv2024

Optimal Experimental Design for Large-Scale Inverse Problems via Multi-PDE-constrained Optimization

Accurate parameter dependent electro-chemical numerical models for lithium-ion batteries are essential in industrial application. The exact parameters of each battery cell are unknown and a process of estimation is necessary to infer them. The parameter estimation generates an accurate model able to reproduce real cell data. The field of optimal input/experimental design deals with creating the experimental settings facilitating the estimation problem. Here we apply two different input design algorithms that aim at maximizing the observability of the true, unknown parameters: in the first algorithm, we design the applied current and the starting voltage. This lets the algorithm collect information on different states of charge, but requires long experimental times (60 000 s). In the second algorithm, we generate a continuous current, composed of concatenated optimal intervals. In this case, the experimental time is shorter (7000 s) and numerical experiments with virtual data give an even better accuracy results, but experiments with real battery data reveal that the accuracy could decrease hundredfold. As the design algorithms are built independent of the model, the same results and motivation are applicable to more complex battery cell models and, moreover, to other applications.

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arXiv2024

POD Suboptimal Control of Evolution Problems: Theory and Applications

The work is organized as follows. First an introduction is given in Chapter 1. In Chapter 2 we introduce the POD method in finite and infinite-dimensional Hilbert spaces and discuss various applications. Chapter 3 is devoted to to POD-based Galerkin schemes for evolution problems. Mainly, we study linear problems taking different discretization methods into account. We provide a certified a-priori and a-posteriori error analysis. Furthermore, the numerical realizations are explained and illustrated by test examples. Quadratic programming problems governed by liner evolution problems are investigated in Chapter 4. As in Chapter 3 we present a certified a-priori and a-posteriori error analysis. Moreover, we discuss basis update strategies. In Chapter 5 we give an outlook to further directions in reduced-order modeling in optimal control and optimization. More precisely, a nonlinear optimal control problem is studied. Moreover, state-constrained optimization problems are solved by a tailored combination of primal-dual active set methods and POD-aesed reduced-order modeling. Furthermore, POD Galerkin methods for multiobjective optimal control problems are investigated. Finally, some required theoretical foundations are summarized in the appendix.

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arXiv2024

A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces

The efficient optimization method for locally Lipschitz continuous multiobjective optimization problems from [1] is extended from finite-dimensional problems to general Hilbert spaces. The method iteratively computes Pareto critical points, where in each iteration, an approximation of the subdifferential is computed in an efficient manner and then used to compute a common descent direction for all objective functions. To prove convergence, we present some new optimality results for nonsmooth multiobjective optimization problems in Hilbert spaces. Using these, we can show that every accumulation point of the sequence generated by our algorithm is Pareto critical under common assumptions. Computational efficiency for finding Pareto critical points is numerically demonstrated for multiobjective optimal control of an obstacle problem.

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arXiv2024

Finite-Dimensional RHC Control of Linear Time-Varying Parabolic PDEs: Stability Analysis and Model-Order Reduction

This chapter deals with the stabilization of a class of linear time-varying parabolic partial differential equations employing receding horizon control (RHC). Here, RHC is finite-dimensional, i.e., it enters as a time-depending linear combination of finitely many indicator functions whose total supports cover only a small part of the spatial domain. Further, we consider the squared l1-norm as the control cost. This leads to a nonsmooth infinite-horizon problem which allows a stabilizing optimal control with a low number of active actuators over time. First, the stabilizability of RHC is investigated. Then, to speed-up numerical computation, the data-driven model-order reduction (MOR) approaches are adequately incorporated within the RHC framework. Numerical experiments are also reported which illustrate the advantages of our MOR approaches.

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arXiv2023

Optimal Control of Dynamic District Heating Networks

In the present paper an optimal control problem for a system of differential-algebraic equations (DAEs) is considered. This problem arises in the dynamic optimization of unsteady district heating networks. Based on the Carathéodory theory existence of a unique solution to the DAE system is proved using specific properties of the district heating network model. Moreover, it is shown that the optimal control problem possesses optimal solutions. For the numerical experiments different networks are considered including also data from a real district heating network.

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arXiv2023

Modelling and Simulation of District Heating Networks

In the present paper a detailed mathematical model is derived for district heating networks. After semidiscretization of the convective heat equation and introducing coupling conditions at the nodes of the network one gets a high-dimensional system of differential-algebraic equations (DAEs). Neglecting temporal changes of the water velocity in the pipes, the numerical solutions do not change significantly and the DAEs have index one. Numerical experiments illustrate that the model describes the real situation very well.

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arXiv2023

Multiobjective Optimization of Non-Smooth PDE-Constrained Problems

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".

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