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Isolating different types of motion in video data is a highly relevant problem in video analysis. Applications can be found, for example, in dynamic medical or biological imaging, where the analysis and further processing of the dynamics of interest is often complicated by additional, unwanted dynamics, such as motion of the measurement subject. In this work, it is empirically shown that a representation of video data via untrained generator networks, together with a specific technique for latent space disentanglement that uses minimal, one-dimensional information on some of the underlying dynamics, allows to efficiently isolate different, highly non-linear motion types. In particular, such a representation allows to freeze any selection of motion types, and to obtain accurate independent representations of other dynamics of interest. Obtaining such a representation does not require any pre-training on a training data set, i.e., all parameters of the generator network are learned directly from a single video.

In this paper, we consider an optimal control problem for the two-dimensional stationary Navier--Stokes system. Looking for sparsity, we take Borel measures as controls. We prove the well-posedness of the control problem and derive necessary and sufficient conditions for local optimality of the controls. Finally, under a second order condition, we prove rates of the optimal states with respect to small perturbations in the data of the control problem., European Research Council 668998, Accepted version

In this paper, we consider an optimal control problem for the two-dimensional stationary Navier--Stokes system. Looking for sparsity, we take Borel measures as controls. We prove the well-posedness of the control problem and derive necessary and sufficient conditions for local optimality of the controls. Finally, under a second order condition, we prove rates of the optimal states with respect to small perturbations in the data of the control problem., European Research Council 668998, Accepted version

We develop an efficient sampling-free approximation scheme for moment-based distributionally robust nonlinear optimization problems. Our approach utilizes a smoothing method that allows the use of gradient-based optimization methods. We apply our scheme to finite-dimensional optimization problems and to optimal control problems with nonlinear partial differential equations. Furthermore, we apply the sample average approximation method to convex risk-neutral optimal control problems posed in Hilbert spaces and derive non-asymptotic error bounds, including exponential tail bounds, for their optimal controls and optimal values. Finally, we establish large deviations for the multilevel Monte Carlo mean estimator., Der Großteil der Arbeit befasst sich mit der Analyse und numerischen Umsetzung eines effizienten Ansatzes zur Approximation von momentenbasierten verteilungsrobusten nichtlinearen Optimierungsproblemen. Wir behandeln sowohl endlich-dimensionale Probleme als auch Steuerungsprobleme, die sich durch nichtlineare partiellen Differentialgleichungen ergeben. Desweiteren approximieren wir konvexe risikoneutrale Optimalsteuerungsprobleme, die in Hilberträumen gestellt sind, mittels empirischer Mittelwerte und leiten nicht-asymptotische Fehlerabschätzungen für optimale Steuerungen und Optimalwerte her. Im letzten Kapitel leiten wir große Abweichungen für Multilevel-Monte-Carlo-Schätzer her.

We derive pointwise best-approximation results for the stationary Stokes problem. For the instationary problem, we show approximation error estimates pointwise in time and in L² in space. To that end, we employ weighted norm and discrete resolvent estimates. We discuss discrete maximal regularity results for the instationary Stokes problem. Furthermore, we present applications of the stationary best-approximation results to optimal control problems with pointwise tracking or sparse control., Wir zeigen punktweise Bestapproximationsraten für das stationäre Stokes Problem. Für das instationäre Problem zeigen wir Approximationsraten punktweise in der Zeit und in L² im Ort. Dabei setzen wir gewichtete Norm- und diskrete Resolventenabschätzungen ein. Wir diskutieren diskrete maximale Regularität für das instationäre Stokes Problem. Des Weiteren präsentieren wir Anwendungen dieser Ergebnisse im stationären Fall für punktuelle Minimierungs- oder sparse Kontrollprobleme.

We derive pointwise best-approximation results for the stationary Stokes problem. For the instationary problem, we show approximation error estimates pointwise in time and in L² in space. To that end, we employ weighted norm and discrete resolvent estimates. We discuss discrete maximal regularity results for the instationary Stokes problem. Furthermore, we present applications of the stationary best-approximation results to optimal control problems with pointwise tracking or sparse control., Wir zeigen punktweise Bestapproximationsraten für das stationäre Stokes Problem. Für das instationäre Problem zeigen wir Approximationsraten punktweise in der Zeit und in L² im Ort. Dabei setzen wir gewichtete Norm- und diskrete Resolventenabschätzungen ein. Wir diskutieren diskrete maximale Regularität für das instationäre Stokes Problem. Des Weiteren präsentieren wir Anwendungen dieser Ergebnisse im stationären Fall für punktuelle Minimierungs- oder sparse Kontrollprobleme.

We develop an efficient sampling-free approximation scheme for moment-based distributionally robust nonlinear optimization problems. Our approach utilizes a smoothing method that allows the use of gradient-based optimization methods. We apply our scheme to finite-dimensional optimization problems and to optimal control problems with nonlinear partial differential equations. Furthermore, we apply the sample average approximation method to convex risk-neutral optimal control problems posed in Hilbert spaces and derive non-asymptotic error bounds, including exponential tail bounds, for their optimal controls and optimal values. Finally, we establish large deviations for the multilevel Monte Carlo mean estimator., Der Großteil der Arbeit befasst sich mit der Analyse und numerischen Umsetzung eines effizienten Ansatzes zur Approximation von momentenbasierten verteilungsrobusten nichtlinearen Optimierungsproblemen. Wir behandeln sowohl endlich-dimensionale Probleme als auch Steuerungsprobleme, die sich durch nichtlineare partiellen Differentialgleichungen ergeben. Desweiteren approximieren wir konvexe risikoneutrale Optimalsteuerungsprobleme, die in Hilberträumen gestellt sind, mittels empirischer Mittelwerte und leiten nicht-asymptotische Fehlerabschätzungen für optimale Steuerungen und Optimalwerte her. Im letzten Kapitel leiten wir große Abweichungen für Multilevel-Monte-Carlo-Schätzer her.

Various problems in computer vision and medical imaging can be cast as inverse problems. A frequent method for solving inverse problems is the variational approach, which amounts to minimizing an energy composed of a data fidelity term and a regularizer. Classically, handcrafted regularizers are used, which are commonly outperformed by state-of-the-art deep learning approaches. In this work, we combine the variational formulation of inverse problems with deep learning by introducing the data-driven general-purpose total deep variation regularizer. In its core, a convolutional neural network extracts local features on multiple scales and in successive blocks. This combination allows for a rigorous mathematical analysis including an optimal control formulation of the training problem in a mean-field setting and a stability analysis with respect to the initial values and the parameters of the regularizer. In addition, we experimentally verify the robustness against adversarial attacks and numerically derive upper bounds for the generalization error. Finally, we achieve state-of-the-art results for numerous imaging tasks., Comment: 30 pages, 12 figures. arXiv admin note: text overlap with arXiv:2001.05005

Policy iteration is a widely used technique to solve the Hamilton Jacobi Bellman (HJB) equation, which arises from nonlinear optimal feedback control theory. Its convergence analysis has attracted much attention in the unconstrained case. Here we analyze the case with control constraints both for the HJB equations which arise in deterministic and in stochastic control cases. The linear equations in each iteration step are solved by an implicit upwind scheme. Numerical examples are conducted to solve the HJB equation with control constraints and comparisons are shown with the unconstrained cases., Comment: 21 pages, 19 figure

Diverse inverse problems in imaging can be cast as variational problems composed of a task-specific data fidelity term and a regularization term. In this paper, we propose a novel learnable general-purpose regularizer exploiting recent architectural design patterns from deep learning. We cast the learning problem as a discrete sampled optimal control problem, for which we derive the adjoint state equations and an optimality condition. By exploiting the variational structure of our approach, we perform a sensitivity analysis with respect to the learned parameters obtained from different training datasets. Moreover, we carry out a nonlinear eigenfunction analysis, which reveals interesting properties of the learned regularizer. We show state-of-the-art performance for classical image restoration and medical image reconstruction problems., Comment: 21 pages, 10 figures

This thesis is concerned with the formulation and analysis of a sparse optimization framework for the optimal placement of measurement sensors in inverse problems. To mitigate the influence of measurement noise we propose to determine the optimal number of sensors and their positions based on the solution of an optimization problem. Therefore we minimize an optimality criterion for the distribution of the sensors which is modeled as a measure on the set of possible candidate locations., Diese Arbeit befasst sich mit der Formulierung und Analysis eines "sparsen" Optimierungsansatzes für die Platzierung von Messsensoren in inversen Problemen. Um den Einfluss von Messfehlern zu verringern wird vorgeschlagen die Anzahl von Sensoren und deren Positionen basierend auf der Lösung eines mathematischen Optimierungsproblems zu bestimmen. Wir minimieren ein Optimalitätskriterium bezüglich der Verteilung der Messensoren welche als Maß auf den möglichen Positionen modelliert wird.

This thesis is concerned with the formulation and analysis of a sparse optimization framework for the optimal placement of measurement sensors in inverse problems. To mitigate the influence of measurement noise we propose to determine the optimal number of sensors and their positions based on the solution of an optimization problem. Therefore we minimize an optimality criterion for the distribution of the sensors which is modeled as a measure on the set of possible candidate locations., Diese Arbeit befasst sich mit der Formulierung und Analysis eines "sparsen" Optimierungsansatzes für die Platzierung von Messsensoren in inversen Problemen. Um den Einfluss von Messfehlern zu verringern wird vorgeschlagen die Anzahl von Sensoren und deren Positionen basierend auf der Lösung eines mathematischen Optimierungsproblems zu bestimmen. Wir minimieren ein Optimalitätskriterium bezüglich der Verteilung der Messensoren welche als Maß auf den möglichen Positionen modelliert wird.

A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton-Jacobi-Bellman equations arising in optimal feedback control of nonlinear dynamics is presented. The method combines a tensor train approximation for the value function together with a Newton-like iterative method for the solution of the resulting nonlinear system. The tensor approximation leads to a polynomial scaling with respect to the dimension, partially circumventing the curse of dimensionality. A convergence analysis for the linear-quadratic case is presented. For nonlinear dynamics, the effectiveness of the high-dimensional control synthesis method is assessed in the optimal feedback stabilization of the Allen-Cahn and Fokker-Planck equations with a hundred of variables., Comment: 26pp, to appear in SIAM J. Sci. Comput

We investigate a well-known phenomenon of variational approaches in image processing, where typically the best image quality is achieved when the gradient flow process is stopped before converging to a stationary point. This paradox originates from a tradeoff between optimization and modelling errors of the underlying variational model and holds true even if deep learning methods are used to learn highly expressive regularizers from data. In this paper, we take advantage of this paradox and introduce an optimal stopping time into the gradient flow process, which in turn is learned from data by means of an optimal control approach. As a result, we obtain highly efficient numerical schemes that achieve competitive results for image denoising and image deblurring. A nonlinear spectral analysis of the gradient of the learned regularizer gives enlightening insights about the different regularization properties., Comment: 14 figures

Measure valued optimal control problems governed by the linear wave equation are analyzed. The space of vector measures $\mathcal{M}(\Omega_c,L 2(I))$ is chosen as control space and the corresponding total variation norm as the control cost functional. The support of the controls (sparsity pattern) is time-independent, which is desired in many applications, e.g., inverse problems or optimal actuator placement. New regularity results for the linear wave equation are proven and used to show the well-posedness of the control problem in all three space dimensions. Furthermore first order optimality conditions are derived and structural properties of the optimal control are investigated. Higher regularity of optimal controls in time is shown on the basis of the regularity results for the state. Finally the optimal control problem is used to solve an inverse source problem.

Motivated by the termination of undesirable arrhythmia, a time optimal control formulation for the monodomain equations is proposed. It is shown that, under certain conditions, the optimal solutions of this problem steer the system into an appropriate stable neighborhood of the resting state. Towards this goal, some new regularity results and asymptotic properties for the monodomain equations with the Rogers−McCulloch ionic model are obtained. For the numerical realization, a monolithic approach, which simultaneously optimizes for the optimal times and optimal controls, is presented and analyzed. Its practical realization is based on a semismooth Newton method. Numerical examples and comparisons are included.

Measure valued optimal control problems governed by the linear wave equation are analyzed. The space of vector measures $\mathcal{M}(\Omega_c,L 2(I))$ is chosen as control space and the corresponding total variation norm as the control cost functional. The support of the controls (sparsity pattern) is time-independent, which is desired in many applications, e.g., inverse problems or optimal actuator placement. New regularity results for the linear wave equation are proven and used to show the well-posedness of the control problem in all three space dimensions. Furthermore first order optimality conditions are derived and structural properties of the optimal control are investigated. Higher regularity of optimal controls in time is shown on the basis of the regularity results for the state. Finally the optimal control problem is used to solve an inverse source problem.

Motivated by the termination of undesirable arrhythmia, a time optimal control formulation for the monodomain equations is proposed. It is shown that, under certain conditions, the optimal solutions of this problem steer the system into an appropriate stable neighborhood of the resting state. Towards this goal, some new regularity results and asymptotic properties for the monodomain equations with the Rogers−McCulloch ionic model are obtained. For the numerical realization, a monolithic approach, which simultaneously optimizes for the optimal times and optimal controls, is presented and analyzed. Its practical realization is based on a semismooth Newton method. Numerical examples and comparisons are included.

This thesis is concerned with the analysis of finite element discretizations for time-optimal control problems subject to linear parabolic partial differential equations and constraints for the state evaluated at the free end time. Necessary and sufficient optimality conditions are provided for the regular case and the case of bang-bang controls. A priori discretization error estimates are proved for different control discretization strategies. Efficient algorithms for the numerical solution are discussed., Diese Arbeit befasst sich mit der Analyse finiter Elemente Diskretisierungen zeitoptimaler Steuerungsprobleme mit linearer, parabolischer, partieller Differentialgleichung und Zustandsrestriktionen am freien Endzeitpunkt. Notwendige und hinreichende Optimalitätsbedingungen werden für den regulären und den bang-bang Fall untersucht. A priori Fehlerabschätzungen für verschiedene Diskretisierungen der Kontrolle werden bewiesen. Effiziente Algorithmen für die numerische Lösung werden diskutiert.

A procedure for the numerical approximation of high-dimensional Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the successive Galerkin approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen.

Stabilization of the wave equation by the receding horizon framework is investigated. Distributed control, Dirichlet boundary control, and Neumann boundary control are considered. Moreover for each of these control actions, the well-posedness of the control system and the exponential stability of Receding Horizon Control (RHC) with respect to a proper functional analytic setting are investigated. Observability conditions are necessary to show the suboptimality and exponential stability of RHC. Numerical experiments are given to illustrate the theoretical results.

Stabilization of the nonlinear Korteweg-de Vries (KdV) equation on a bounded interval by model predictive control (MPC) is investigated. This MPC strategy does not need any terminal cost or terminal constraint to guarantee the stability. The semi-global stabilizability is the key condition. Based on this condition, the suboptimality and exponential stability of the model predictive control are investigated. Finally, numerical experiment is presented which validates the theoretical results.

Optimal control approaches have proved useful in designing RF pulses for large tip-angle applications. A typical challenge for optimal control design is the inclusion of constraints resulting from physiological or technical limitations, that assure the realizability of the optimized pulses. In this work we show how to treat such inequality constraints, in particular, amplitude constraints on the B1 field, the slice-selective gradient and its slew rate, as well as constraints on the slice profile accuracy. For the latter a pointwise profile error and additional phase constraints are prescribed. Here, a penalization method is introduced that corresponds to a higher-order tracking instead of the common quadratic tracking. The order is driven to infinity in the course of the optimization. We jointly optimize for the RF and slice-selective gradient waveform. The amplitude constraints on these control variables are treated efficiently by semismooth Newton or quasi-Newton methods. The method is flexible, adapting to many optimization goals. As an application we reduce the power of refocusing pulses, which is important for spin echo based applications with a short echo spacing. Here, the optimization method is tested in numerical experiments for reducing the pulse power of simultaneous multislice refocusing pulses. The results are validated by phantom and in-vivo experiments. Keywords: RF pulse design, slice-selective, optimal control, physical constraints, inequality constraints.

In this paper local exact controllability to the trajectories for the one-dimensional monodomain equations with the FitzHugh–Nagumo and Rogers–McCulloch ionic models using distributed controls with a moving support is investigated. In a first step a new Carleman inequality for the adjoint of the linearized monodomain equations, under assumptions on the movement of the control region, is presented. It leads to null controllability at any positive time. Subsequently, a local result concerning the exact controllability to the trajectories for the nonlinear monodomain equations is deduced., Dans cet article, la contrôlabilité exacte locale aux trajectoires des équations monodomaines unidimensionnelles à l'aide des modèles ioniques FitzHugh–Nagumo et Rogers–McCulloch en utilisant des contrôles distribués à support mobile est étudiée. Tout d'abord, une nouvelle inégalité de Carleman pour l'adjoint des équations monodomaines linéarisées, sous des hypothèses sur le déplacement de la région de contrôle, est présentée. Cela conduit à la contrôlabilité à zéro en tout temps positif. Un résultat local concernant la contrôlabilité exacte aux trajectoires des équations monodomaines non-linéaires est ensuite déduit.

Purpose: Joint design of minimum duration RF pulses and slice-selective gradient shapes for MRI via time optimal control with strict physical constraints, and its application to simultaneous multislice imaging. Theory and Methods: The minimization of the pulse duration is cast as a time optimal control problem with inequality constraints describing the refocusing quality and physical constraints. It is solved with a bilevel method, where the pulse length is minimized in the upper level, and the constraints are satisfied in the lower level. To address the inherent nonconvexity of the optimization problem, the upper level is enhanced with new heuristics for finding a near global optimizer based on a second optimization problem. Results: A large set of optimized examples shows an average temporal reduction of 87.1% for double diffusion and 74% for turbo spin echo pulses compared to power independent number of slices pulses. The optimized results are validated on a 3T scanner with phantom measurements. Conclusion: The presented design method computes minimum duration RF pulse and slice-selective gradient shapes subject to physical constraints. The shorter pulse duration can be used to decrease the effective echo time in existing echo-planar imaging or echo spacing in turbo spin echo sequences., European Research Council 668998, Accepted version

Stabilization of the wave equation by the receding horizon framework is investigated. Distributed control, Dirichlet boundary control, and Neumann boundary control are considered. Moreover for each of these control actions, the well-posedness of the control system and the exponential stability of Receding Horizon Control (RHC) with respect to a proper functional analytic setting are investigated. Observability conditions are necessary to show the suboptimality and exponential stability of RHC. Numerical experiments are given to illustrate the theoretical results.

Stabilization of the nonlinear Korteweg-de Vries (KdV) equation on a bounded interval by model predictive control (MPC) is investigated. This MPC strategy does not need any terminal cost or terminal constraint to guarantee the stability. The semi-global stabilizability is the key condition. Based on this condition, the suboptimality and exponential stability of the model predictive control are investigated. Finally, numerical experiment is presented which validates the theoretical results.

Optimal control approaches have proved useful in designing RF pulses for large tip-angle applications. A typical challenge for optimal control design is the inclusion of constraints resulting from physiological or technical limitations, that assure the realizability of the optimized pulses. In this work we show how to treat such inequality constraints, in particular, amplitude constraints on the B1 field, the slice-selective gradient and its slew rate, as well as constraints on the slice profile accuracy. For the latter a pointwise profile error and additional phase constraints are prescribed. Here, a penalization method is introduced that corresponds to a higher-order tracking instead of the common quadratic tracking. The order is driven to infinity in the course of the optimization. We jointly optimize for the RF and slice-selective gradient waveform. The amplitude constraints on these control variables are treated efficiently by semismooth Newton or quasi-Newton methods. The method is flexible, adapting to many optimization goals. As an application we reduce the power of refocusing pulses, which is important for spin echo based applications with a short echo spacing. Here, the optimization method is tested in numerical experiments for reducing the pulse power of simultaneous multislice refocusing pulses. The results are validated by phantom and in-vivo experiments. Keywords: RF pulse design, slice-selective, optimal control, physical constraints, inequality constraints.

A procedure for the numerical approximation of high-dimensional Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the successive Galerkin approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen.

This thesis is concerned with the analysis of finite element discretizations for time-optimal control problems subject to linear parabolic partial differential equations and constraints for the state evaluated at the free end time. Necessary and sufficient optimality conditions are provided for the regular case and the case of bang-bang controls. A priori discretization error estimates are proved for different control discretization strategies. Efficient algorithms for the numerical solution are discussed., Diese Arbeit befasst sich mit der Analyse finiter Elemente Diskretisierungen zeitoptimaler Steuerungsprobleme mit linearer, parabolischer, partieller Differentialgleichung und Zustandsrestriktionen am freien Endzeitpunkt. Notwendige und hinreichende Optimalitätsbedingungen werden für den regulären und den bang-bang Fall untersucht. A priori Fehlerabschätzungen für verschiedene Diskretisierungen der Kontrolle werden bewiesen. Effiziente Algorithmen für die numerische Lösung werden diskutiert.

A class of infinite horizon optimal control problems involving mixed quasi-norms of $L^p$-type cost functionals for the controls is discussed. These functionals enhance sparsity and switching properties of the optimal controls. The existence of optimal controls and their structural properties are analyzed on the basis of first order optimality conditions. A dynamic programming approach is used for numerical realization.

An optimal control approach to a simplified reaction diffusion system describing cardiac defibrillation is proposed that allows for joint optimization of shape and duration of defibrillation pulses. Within the framework, optimized multiphasic pulses with low energy, short duration and/or low amplitude can be designed according to specific needs. The approach is based on a novel time optimal control formulation for the monodomain model, which takes into consideration the dynamical system properties of the uncontrolled equation. The highly complex dynamics requires a consistent discretization of first and second order information to guarantee effective optimization schemes leading to successful defibrillation. Numerical examples underline the efficiency of the proposed method.

Time optimal control problems for systems with impulsive controls are investigated. Sufficient conditions for the existence of time optimal controls are given. A dynamical programming principle is derived and Lipschitz continuity of an appropriately defined value functional is established. The value functional satisfies a Hamilton–Jacobi–Bellman equation in the viscosity sense. A numerical example for a rider-swing system is presented and it is shown that the reachable set is enlargered by allowing for impulsive controls, when compared to nonimpulsive controls.

We consider proper orthogonal decomposition (POD) based Galerkin approximations to parabolic systems and establish uniform convergence with respect to forcing functions. The result is used to prove convergence of POD approximations to optimal control problems that automatically update the POD basis in order to avoid problems due to unmodeled dynamics in the POD reduced order system. A numerical example illustrates the results.

We consider proper orthogonal decomposition (POD) based Galerkin approximations to parabolic systems and establish uniform convergence with respect to forcing functions. The result is used to prove convergence of POD approximations to optimal control problems that automatically update the POD basis in order to avoid problems due to unmodeled dynamics in the POD reduced order system. A numerical example illustrates the results.

An optimal control approach to a simplified reaction diffusion system describing cardiac defibrillation is proposed that allows for joint optimization of shape and duration of defibrillation pulses. Within the framework, optimized multiphasic pulses with low energy, short duration and/or low amplitude can be designed according to specific needs. The approach is based on a novel time optimal control formulation for the monodomain model, which takes into consideration the dynamical system properties of the uncontrolled equation. The highly complex dynamics requires a consistent discretization of first and second order information to guarantee effective optimization schemes leading to successful defibrillation. Numerical examples underline the efficiency of the proposed method.

Time optimal control problems for systems with impulsive controls are investigated. Sufficient conditions for the existence of time optimal controls are given. A dynamical programming principle is derived and Lipschitz continuity of an appropriately defined value functional is established. The value functional satisfies a Hamilton–Jacobi–Bellman equation in the viscosity sense. A numerical example for a rider-swing system is presented and it is shown that the reachable set is enlargered by allowing for impulsive controls, when compared to nonimpulsive controls.

The infimal convolution of total (generalized) variation–type functionals and its application as regularization for video and image reconstruction is considered. The definition of this particular type of regularization functional is motivated by the need of suitably combining spatial and temporal regularity requirements for video processing. The proposed functional is defined in an infinite dimensional setting, and important analytical properties are established. As applications, the reconstruction of compressed video data and of noisy still images is considered. The resulting problem settings are posed in function space, and suitable numerical solution schemes are established. Experiments confirm a significant improvement compared to standard total variation–type methods, which originates from the introduction of spatio-temporal and spatial anisotropies.

A Gautschi time-stepping scheme for optimal control of linear second order systems is proposed and analyzed. Convergence rates are proved and shown to be valid in numerical experiments. The temporal discretization is combined with finite element and spectral based spatial discretizations, which are compared among themselves.

The infimal convolution of total (generalized) variation–type functionals and its application as regularization for video and image reconstruction is considered. The definition of this particular type of regularization functional is motivated by the need of suitably combining spatial and temporal regularity requirements for video processing. The proposed functional is defined in an infinite dimensional setting, and important analytical properties are established. As applications, the reconstruction of compressed video data and of noisy still images is considered. The resulting problem settings are posed in function space, and suitable numerical solution schemes are established. Experiments confirm a significant improvement compared to standard total variation–type methods, which originates from the introduction of spatio-temporal and spatial anisotropies.

A Gautschi time-stepping scheme for optimal control of linear second order systems is proposed and analyzed. Convergence rates are proved and shown to be valid in numerical experiments. The temporal discretization is combined with finite element and spectral based spatial discretizations, which are compared among themselves.

This paper is concerned with optimal control problems for parabolic partial differential equations with pointwise in time switching constraints on the control. A standard approach to treat constraints in nonlinear optimization is penalization, in particular using L1-type norms. Applying this approach to the switching constraint leads to a nonsmooth and nonconvex infinite-dimensional minimization problem which is challenging both analytically and numerically. Adding H1 regularization or restricting to a finite-dimensional control space allows showing existence of optimal controls. First-order necessary optimality conditions are then derived using tools of nonsmooth analysis. Their solution can be computed using a combination of Moreau–Yosida regularization and a semismooth Newton method. Numerical examples illustrate the properties of this approach.

Using a projection-based decoupling of the Fokker−Planck equation, control strategies that allow to speed up the convergence to the stationary distribution are investigated. By means of an operator theoretic framework for a bilinear control system, two different feedback control laws are proposed. Projected Riccati and Lyapunov equations are derived and properties of the associated solutions are given. The well-posedness of the closed loop systems is shown and local and global stabilization results, respectively, are obtained. An essential tool in the construction of the controls is the choice of appropriate control shape functions. Results for a two dimensional double well potential illustrate the theoretical findings in a numerical setup.

An optimal control problem with a time-parameter is considered. The functional to be optimized includes the maximum over time-horizon reached by a function of the state variable, and so an L∞-term. In addition to the classical control function, the time at which this maximum is reached is considered as a free parameter. The problem couples the behavior of the state and the control, with this time-parameter. A change of variable is introduced to derive first and second-order optimality conditions. This allows the implementation of a Newton method. Numerical simulations are developed, for selected ordinary differential equations and a partial differential equation, which illustrate the influence of the additional parameter and the original motivation.

Nonsmooth nonconvex optimization problems are considered in infinite dimensional sequence spaces l p with p ∈ (0; 1]. Our starting points are necessary optimality conditions in the form of a complementary system and a monotonically convergent algorithm for a regularized version of the original problem. We propose an algorithm for solving the necessary optimality condition based on a combination of the monotone scheme and an active-set strategy. Numerical results for different test cases are provided, e.g. for optimal control problems and microscopy image reconstruction.

Boundary feedback control for a coupled nonlinear PDE-ODE system (in the two and three dimensional cases) is studied. Particular focus is put on the monodomain equations arising in the context of cardiac electrophysiology. Neumann as well as Dirichlet based boundary control laws are obtained by an algebraic operator Riccati equation associated with the linearized system. Local exponential stability of the nonlinear closed loop system is shown by a fixed-point argument. Numerical examples are given for a finite element discretization of the two dimensional monodomain equations.

Stabilization problems for parabolic equations with polynomial nonlinearities are investigated in the context of an optimal control formulation with a sparsity enhancing cost functional. This formulation allows that the optimal control completely shuts down once the trajectory is sufficiently close to a stable steady state. Such a property is not present for commonly chosen control mechanisms. To establish these results it is necessary to develop a function space framework for a class of optimal control problems posed on infinite time horizons, which is otherwise not available.

A control strategy which allows to speed up the convergence of a bilinear system governed by the Fokker-Planck equation to its stationary distribution is developed. After linearization of the state equation, a linear feedback control is computed by solving the Riccati equation associated with the linearized problem. A reduction method for approximating this feedback is proposed. From a numerical point of view, this method avoids the resolution of a high-dimensional Riccati equation. Numerical results are provided for a double-well potential and the efficiency of the reduction method is demonstrated.

Stabilization problems for parabolic equations with polynomial nonlinearities are investigated in the context of an optimal control formulation with a sparsity enhancing cost functional. This formulation allows that the optimal control completely shuts down once the trajectory is sufficiently close to a stable steady state. Such a property is not present for commonly chosen control mechanisms. To establish these results it is necessary to develop a function space framework for a class of optimal control problems posed on infinite time horizons, which is otherwise not available.

The problem of finite-dimensional compensator design for the monodomain equations is investigated. Exponential stabilizability and detectability of the linearized infinite-dimensional control system is studied. It is shown that the system is not exactly null-controllable but still can be exponentially stabilized by finite-rank input and output operators provided the desired stability margin is small enough. Based on existing results on model order reduction of infinite-dimensional systems, a finite-dimensional compensator is obtained by LQG-balanced truncation. The compensator is shown to be locally stabilizing for the infinite-dimensional nonlinear control system. Examples motivated by cardiophysiology are used to illustrate these results in a numerical setup.