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Publications

Monographs

  • M. Hintermüller, K. Papafitsoros, Chapter 11: Generating Structured Nonsmooth Priors and Associated Primal-dual Methods, in: Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, R. Kimmel, X.-Ch. Tai, eds., 20 of Handbook of Numerical Analysis, Elsevier, 2019, pp. 437--502, (Chapter Published), DOI 10.1016/bs.hna.2019.08.001 .

  • M. Hintermüller, M. Hinze, J. Sokołowski, S. Ulbrich, eds., Special issue to honour Guenter Leugering on his 65th birthday, 1 of Control & Cybernetics, Systems Research Institute, Polish Academy of Sciences, Warsaw, 2019, (Collection Published).

  • M. Hintermüller, J.F. Rodrigues, eds., Topics in Applied Analysis and Optimisation -- Partial Differential Equations, Stochastic and Numerical Analysis, CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, 396 pages, (Collection Published).

Articles in Refereed Journals

  • M.S. Aronna, J.F. Bonnans, A. Kröner, State-constrained control-affine parabolic problems I: First and second order necessary optimality conditions, Set-Valued and Variational Analysis. Theory and Applications. Springer, Dordrecht. English., (2020), published online on 17.10.2020, DOI 10.1007/s11228-020-00560-2 .
    Abstract
    In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions.

  • N.A. Dao, J.I. Díaz, Q.B.H. Nguyen, Pointwise gradient estimates in multi-dimensional slow diffusion equations with a singular quenching term, Advanced Nonlinear Studies, published online on 19.03.2020, DOI 10.1515/ans-2020-2076 .

  • A. Alphonse, M. Hintermüller, C.N. Rautenberg, Existence, iteration procedures and directional differentiability for parabolic QVIs, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 95/1--95/53, DOI 10.1007/s00526-020-01732-6 .
    Abstract
    We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.

  • G. Dong, H. Guo, Parametric polynomial preserving recovery on manifolds, SIAM Journal on Scientific Computing, 42 (2020), pp. A1885--A1912, DOI 10.1137/18M1191336 .

  • J. Polzehl, K. Papafitsoros, K. Tabelow, Patch-wise adaptive weights smoothing in R, Journal of Statistical Software, 95 (2020), pp. 1--27, DOI 10.18637/jss.v095.i06 .
    Abstract
    Image reconstruction from noisy data has a long history of methodological development and is based on a variety of ideas. In this paper we introduce a new method called patch-wise adaptive smoothing, that extends the Propagation-Separation approach by using comparisons of local patches of image intensities to define local adaptive weighting schemes for an improved balance of reduced variability and bias in the reconstruction result. We present the implementation of the new method in an R package aws and demonstrate its properties on a number of examples in comparison with other state-of-the art image reconstruction methods.

  • A. Alphonse, M. Hintermüller, C.N. Rautenberg, Directional differentiability for elliptic quasi-variational inequalities of obstacle type, Calculus of Variations and Partial Differential Equations, 58 (2019), pp. 39/1--39/47, DOI 10.1007/s00526-018-1473-0 .
    Abstract
    The directional differentiability of the solution map of obstacle type quasi-variational inequal- ities (QVIs) with respect to perturbations on the forcing term is studied. The classical result of Mignot is then extended to the quasi-variational case under assumptions that allow multiple solu- tions of the QVI. The proof involves selection procedures for the solution set and represents the directional derivative as the limit of a monotonic sequence of directional derivatives associated to specific variational inequalities. Additionally, estimates on the coincidence set and several sim- plifications under higher regularity are studied. The theory is illustrated by a detailed study of an application to thermoforming comprising of modelling, analysis and some numerical experiments.

  • M.S. Aronna, J.F. Bonnans, A. Kröner, Optimal control of PDEs in a complex space setting: Application to the Schrödinger equation, SIAM Journal on Control and Optimization, 57 (2019), pp. 1390--1412, DOI 10.1137/17M1117653 .

  • L. Adam, M. Hintermüller, D. Peschka, Th.M. Surowiec, Optimization of a multiphysics problem in semiconductor laser design, SIAM Journal on Applied Mathematics, 79 (2019), pp. 257--283, DOI 10.1137/18M1179183 .
    Abstract
    A multimaterial topology optimization framework is suggested for the simultaneous optimization of mechanical and optical properties to be used in the development of optoelectronic devices. Based on the physical aspects of the underlying device, a nonlinear multiphysics model for the elastic and optical properties is proposed. Rigorous proofs are provided for the sensitivity of the fundamental mode of the device with respect to the changes in the underlying topology. After proving existence and optimality results, numerical experiments leading to an optimal material distribution for maximizing the strain in a Ge-on-Si microbridge are given. The highly favorable electronic properties of this design are demonstrated by steady-state simulations of the corresponding van Roosbroeck (drift-diffusion) system.

  • H. Antil, C.N. Rautenberg, Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications, SIAM Journal on Mathematical Analysis, 51 (2019), pp. 2479--2503, DOI 10.1137/18M1224970 .
    Abstract
    We propose a new variational model in weighted Sobolev spaces with non-standard weights and applications to image processing. We show that these weights are, in general, not of Muckenhoupt type and therefore the classical analysis tools may not apply. For special cases of the weights, the resulting variational problem is known to be equivalent to the fractional Poisson problem. The trace space for the weighted Sobolev space is identified to be embedded in a weighted L2 space. We propose a finite element scheme to solve the Euler-Lagrange equations, and for the image denoising application we propose an algorithm to identify the unknown weights. The approach is illustrated on several test problems and it yields better results when compared to the existing total variation techniques.

  • L. Calatroni, K. Papafitsoros, Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt & pepper noise removal, Inverse Problems and Imaging, 35 (2019), pp. 114001/1--114001/37, DOI 10.1088/1361-6420/ab291a .
    Abstract
    We analyse a variational regularisation problem for mixed noise removal that was recently proposed in [14]. The data discrepancy term of the model combines L1 and L2 terms in an infimal convolution fashion and it is appropriate for the joint removal of Gaussian and Salt & Pepper noise. In this work we perform a finer analysis of the model which emphasises on the balancing effect of the two parameters appearing in the discrepancy term. Namely, we study the asymptotic behaviour of the model for large and small values of these parameters and we compare it to the corresponding variational models with L1 and L2 data fidelity. Furthermore, we compute exact solutions for simple data functions taking the total variation as regulariser. Using these theoretical results, we then analytically study a bilevel optimisation strategy for automatically selecting the parameters of the model by means of a training set. Finally, we report some numerical results on the selection of the optimal noise model via such strategy which confirm the validity of our analysis and the use of popular data models in the case of "blind” model selection.

  • G. Dong, M. Hintermüller, K. Papafitsoros, Quantitative magnetic resonance imaging: From fingerprinting to integrated physics-based models, SIAM Journal on Imaging Sciences, 2 (2019), pp. 927--971, DOI 10.1137/18M1222211 .
    Abstract
    Quantitative magnetic resonance imaging (qMRI) is concerned with estimating (in physical units) values of magnetic and tissue parameters, e.g., relaxation times $T_1$, $T_2$, or proton density $rho$. Recently, in [Ma et al., Nature, 495 (2013), pp. 187--193], magnetic resonance fingerprinting (MRF) was introduced as a technique being capable of simultaneously recovering such quantitative parameters by using a two-step procedure: (i) given a probe, a series of magnetization maps are computed and then (ii) matched to (quantitative) parameters with the help of a precomputed dictionary which is related to the Bloch manifold. In this paper, we first put MRF and its variants into perspective with optimization and inverse problems to gain mathematical insights concerning identifiability of parameters under noise and interpretation in terms of optimizers. Motivated by the fact that the Bloch manifold is nonconvex and that the accuracy of the MRF-type algorithms is limited by the ?discretization size? of the dictionary, a novel physics-based method for qMRI is proposed. In contrast to the conventional two-step method, our model is dictionary-free and is rather governed by a single nonlinear equation, which is studied analytically. This nonlinear equation is efficiently solved via robustified Newton-type methods. The effectiveness of the new method for noisy and undersampled data is shown both analytically and via extensive numerical examples, for which improvement over MRF and its variants is also documented.

  • S. Hajian, M. Hintermüller, C. Schillings, N. Strogies, A Bayesian approach to parameter identification in gas networks, Control and Cybernetics, 48 (2019), pp. 377--402.
    Abstract
    The inverse problem of identifying the friction coefficient in an isothermal semilinear Euler system is considered. Adopting a Bayesian approach, the goal is to identify the distribution of the quantity of interest based on a finite number of noisy measurements of the pressure at the boundaries of the domain. First well-posedness of the underlying non-linear PDE system is shown using semigroup theory, and then Lipschitz continuity of the solution operator with respect to the friction coefficient is established. Based on the Lipschitz property, well-posedness of the resulting Bayesian inverse problem for the identification of the friction coefficient is inferred. Numerical tests for scalar and distributed parameters are performed to validate the theoretical results.

  • M. Hintermüller, N. Strogies, Identification of the friction function in a semilinear system for gas transport through a network, Optimization Methods & Software, 35 (2020), pp. 576--617 (published online on 10.12.2019), DOI 10.1080/10556788.2019.1692206 .

Contributions to Collected Editions

  • A. Alphonse, M. Hintermüller, C.N. Rautenberg, Recent trends and views on elliptic quasi-variational inequalities, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 1--31.

  • M. Hintermüller, N. Strogies, On the consistency of Runge--Kutta methods up to order three applied to the optimal control of scalar conservation laws, in: Numerical Analysis and Optimization, M. Al-Baali, L. Grandinetti, A. Purnama, eds., 235 of Springer Proceedings in Mathematics & Statistics, Springer Nature Switzerland AG, Cham, 2019, pp. 119--154.
    Abstract
    Higher-order Runge-Kutta (RK) time discretization methods for the optimal control of scalar conservation laws are analyzed and numerically tested. The hyperbolic nature of the state system introduces specific requirements on discretization schemes such that the discrete adjoint states associated with the control problem converge as well. Moreover, conditions on the RK-coefficients are derived that coincide with those characterizing strong stability preserving Runge-Kutta methods. As a consequence, the optimal order for the adjoint state is limited, e.g., to two even in the case where the conservation law is discretized by a third-order method. Finally, numerical tests for controlling Burgers equation validate the theoretical results.

Preprints, Reports, Technical Reports

  • M.S. Aronna, J.F. Bonnans, A. Kröner, State-constrained control-affine parabolic problems II: Second order sufficient optimality conditions, Preprint no. 2778, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2778 .
    Abstract, PDF (319 kByte)
    In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order sufficient conditions relying on the Goh transform.

  • M. Hintermüller, S.-M. Stengl, On the convexity of optimal control problems involving non-linear PDEs or VIs and applications to Nash games, Preprint no. 2759, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2759 .
    Abstract, PDF (381 kByte)
    Generalized Nash equilibrium problems in function spaces involving PDEs are considered. One of the central issues arising in this context is the question of existence, which requires the topological characterization of the set of minimizers for each player of the associated Nash game. In this paper, we propose conditions on the operator and the functional that guarantee the reduced formulation to be a convex minimization problem. Subsequently, we generalize results of convex analysis to derive optimality systems also for non-smooth operators. Our theoretical findings are illustrated by examples.

  • A. Alphonse, M. Hintermüller, C.N. Rautenberg, On the differentiability of the minimal and maximal solution maps of elliptic quasi-variational inequalities, Preprint no. 2758, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2758 .
    Abstract, PDF (259 kByte)
    In this short note, we prove that the minimal and maximal solution maps associated to elliptic quasi-variational inequalities of obstacle type are directionally differentiable with respect to the forcing term and for directions that are signed. On the way, we show that the minimal and maximal solutions can be seen as monotone limits of solutions of certain variational inequalities and that the aforementioned directional derivatives can also be characterised as the monotone limits of sequences of directional derivatives associated to variational inequalities.

  • A. Alphonse, M. Hintermüller, C.N. Rautenberg, Optimal control and directional differentiability for elliptic quasi-variational inequalities, Preprint no. 2756, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2756 .
    Abstract, PDF (381 kByte)
    We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs. We give three existence theorems based on an order approach, an iteration scheme and a sequential regularisation through partial differential equations. We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general unsigned data, thereby extending the results of our previous work which provided a first differentiability result for QVIs in infinite dimensions. Optimal control problems with QVI constraints are also considered and we derive various forms of stationarity conditions for control problems, thus supplying among the first such results in this area.

  • C. Geiersbach, W. Wollner, Optimality conditions for convex stochastic optimization problems in Banach spaces with almost sure state constraint, Preprint no. 2755, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2755 .
    Abstract, PDF (321 kByte)
    We analyze a convex stochastic optimization problem where the state is assumed to belong to the Bochner space of essentially bounded random variables with images in a reflexive and separable Banach space. For this problem, we obtain optimality conditions that are, with an appropriate model, necessary and sufficient. Additionally, the Lagrange multipliers associated with optimality conditions are integrable vector-valued functions and not only measures. A model problem is given demonstrating the application to PDE-constrained optimization under uncertainty.

  • G. Dong, M. Hintermüller, K. Papafitsoros, Optimization with learning-informed differential equation constraints and its applications, Preprint no. 2754, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2754 .
    Abstract, PDF (1761 kByte)
    Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.

  • A. Alphonse, C.N. Rautenberg, J.F. Rodrigues, Analysis of a quasi-variational contact problem arising in thermoelasticity, Preprint no. 2747, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2747 .
    Abstract, PDF (1168 kByte)
    We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions.

  • M. Hintermüller, S.-M. Stengl, Th.M. Surowiec, Uncertainty quantification in image segmentation using the Ambrosio--Tortorelli approximation of the Mumford--Shah energy, Preprint no. 2703, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2703 .
    Abstract, PDF (930 kByte)
    The quantification of uncertainties in image segmentation based on the Mumford-Shah model is studied. The aim is to address the error propagation of noise and other error types in the original image to the restoration result and especially the reconstructed edges (sharp image contrasts). Analytically, we rely on the Ambrosio-Tortorelli approximation and discuss the existence of measurable selections of its solutions as well as sampling-based methods and the limitations of other popular methods. Numerical examples illustrate the theoretical findings.

  • M. Hintermüller, K. Papafitsoros, C.N. Rautenberg, H. Sun, Dualization and automatic distributed parameter selection of total generalized variation via bilevel optimization, Preprint no. 2689, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2689 .
    Abstract, PDF (25 MByte)
    Total Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, two Newton type methods for the solution of the lower level problem, i.e. the image reconstruction problem, and two bilevel TGV algorithms are introduced, respectively. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.

  • D. Gahururu, M. Hintermüller, S.-M. Stengl, Th.M. Surowiec, Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms, and risk aversion, Preprint no. 2654, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2654 .
    Abstract, PDF (337 kByte)
    PDE-constrained (generalized) Nash equilibrium problems (GNEPs) are considered in a deterministic setting as well as under uncertainty. This includes a study of deterministic GNEPs with nonlinear and/or multivalued operator equations as forward problems and PDE-constrained GNEPs with uncertain data. The deterministic nonlinear problems are analyzed using the theory of generalized convexity for set-valued operators, and a variational approximation approach is proposed. The stochastic setting includes a detailed overview of the recently developed theory and algorithms for risk-averse PDE-constrained optimization problems. These new results open the way to a rigorous study of stochastic PDE-constrained GNEPs.

  • M. Hintermüller, K. Papafitsoros, C.N. Rautenberg, Variable step mollifiers and applications, Preprint no. 2628, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2628 .
    Abstract, PDF (562 kByte)
    We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as well as corresponding approximation results. The results are then applied to extend recently developed theory concerning the density of convex intersections.

  • C. Grässle, M. Hintermüller, M. Hinze, T. Keil, Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities, Preprint no. 2617, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2617 .
    Abstract, PDF (11 MByte)
    We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.

  • M. Hintermüller, T. Keil, Optimal control of geometric partial differential equations, Preprint no. 2612, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2612 .
    Abstract, PDF (15 MByte)
    Optimal control problems for geometric (evolutionary) partial differential inclusions are considered. The focus is on problems which, in addition to the nonlinearity due to geometric evolution, contain optimization theoretic challenges because of non-smoothness. The latter might stem from energies containing non-smooth constituents such as obstacle-type potentials or terms modeling, e.g., pinning phenomena in microfluidics. Several techniques to remedy the resulting constraint degeneracy when deriving stationarity conditions are presented. A particular focus is on Yosida-type mollifications approximating the original degenerate problem by a sequence of nondegenerate nonconvex optimal control problems. This technique is also the starting point for the development of numerical solution schemes. In this context, also dual-weighted residual based error estimates are addressed to facilitate an adaptive mesh refinement. Concerning the underlying state model, sharp and diffuse interface formulations are discussed. While the former always allows for accurately tracing interfacial motion, the latter model may be dictated by the underlying physical phenomenon, where near the interface mixed phases may exist, but it may also be used as an approximate model for (sharp) interface motion. In view of the latter, (sharp interface) limits of diffuse interface models are addressed. For the sake of presentation, this exposition confines itself to phase field type diffuse interface models and, moreover, develops the optimal control of either of the two interface models along model applications. More precisely, electro-wetting on dielectric is used in the sharp interface context, and the control of multiphase fluids involving spinodal decomposition highlights the phase field technique. Mathematically, the former leads to a Hele-Shaw flow with geometric boundary conditions involving a complementarity system due to contact line pinning, and the latter gives rise to a Cahn-Hilliard Navier-Stokes model including a non-smooth obstacle type potential leading to a variational inequality constraint.

  • G. Dong, M. Hintermüller, Y. Zhang, A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science, Preprint no. 2591, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2591 .
    Abstract, PDF (1181 kByte)
    In this paper, we study damped second-order dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped second-order total variation flow, which is primarily motivated by the application of image denoising; the other is a damped second-order mean curvature flow for level sets of scalar functions, which is related to a non-convex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces which are described by level sets of scalar functions, and show the existence and uniqueness of the solution for a regularized version of the equation. The latter is used in our algorithmic development. A general algorithm for numerical discretization of the two nonlinear PDEs is proposed and analyzed. Its efficiency is demonstrated by various numerical examples, where simulations on the behavior of solutions of the new equations and comparisons with first-order flows are also documented.

  • A. Alphonse, M. Hintermüller, C.N. Rautenberg, Stability of the solution set of quasi-variational inequalities and optimal control, Preprint no. 2582, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2582 .
    Abstract, PDF (321 kByte)
    For a class of quasivariational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements to the solution set of the QVI with respect to monotonic perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed.

Talks, Poster

  • S.-M. Stengl, Uncertainty quantification of the Ambrosio--Tortorelli approximation in image segmentation, Workshop on PDE Constrained Optimization under Uncertainty and Mean Field Games, January 28 - 30, 2020, WIAS, Berlin, January 30, 2020.

  • G. Dong, Integrated physics-based method, learning-informed model and hyperbolic PDEs for imaging, Efficient Algorithms in Data Science, Learning and Computational Physics, Sanya, China, January 12 - 16, 2020.

  • M. Hintermüller, Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction, The Fifth International Conference on Numerical Analysis and Optimization, January 6 - 9, 2020, Sultan Qaboos University, Oman, January 6, 2020.

  • M. Hintermüller, Magnetic resonance fingerprinting of integrated physics models, Efficient Algorithms in Data Science, Learning and Computational Physics, January 12 - 16, 2020, Sanya, China, January 15, 2020.

  • A. Kröner, Optimal control of a semilinear heat equation subject to state and control constraints, Joint Research Seminar on Nonsmooth Variational Problems and Operator Equations / Mathematical Optimization, WIAS, February 27, 2020.

  • H. Nguyen, A shape optimization problem for stationary Navier--Stokes flows in three-dimensional tubes, Model Order Reduction Summer School 2020 (MORSS 2020), September 7 - 10, 2020, École polytechnique fédérale de Lausanne (EPFL), Lausanne, Switzerland, September 7, 2020.

  • K. Papafitsoros, Automatic distributed regularization parameter selection in Total Generalized Variation image reconstruction via bilevel optimization, Seminar, Southern University of Science and Technology, Shenzhen, China, January 17, 2020.

  • K. Papafitsoros, Automatic distributed regularization parameter selection in Total Generalized Variation image reconstruction via bilevel optimization, Seminar, Shenzhen MSU-BIT University, Department of Mathematics, Shenzhen, China, January 16, 2020.

  • K. Papafitsoros, Automatic distributed regularization parameter selection in imaging via bilevel optimization, Workshop on PDE Constrained Optimization under Uncertainty and Mean Field Games, January 28 - 30, 2020, WIAS, Berlin, January 30, 2020.

  • K. Papafitsoros, Spatially dependent parameter selection in TGV based image restoration via bilevel optimization, Efficient Algorithms in Data Science, Learning and Computational Physics, Sanya, China, January 12 - 16, 2020.

  • J.A. Brüggemann, Elliptic obstacle-type quasi-variational inequalities (QVIs) with volume constraints motivated by a contact problem in biomedicine, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Berlin, August 5 - 8, 2019.

  • J.A. Brüggemann, On the existence of solutions and solution methods for elliptic obstacle-type quasi-variational inequalities with volume constraints, Joint Research Seminar on Mathematical Optimization / Non-smooth Variational Problems and Operator Equations, WIAS, November 21, 2019.

  • J.A. Brüggemann, Solution methods for a class of obstacle-type quasi variational inequalities with volume constraints, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Quasi-Variational Inequalities and Generalized Nash Equi-librium Problems (Part II)'', August 5 - 8, 2019, Berlin, August 7, 2019.

  • A. Alphonse, Directional differentiability for elliptic quasi-variational inequalities, Workshop ``Surface, Bulk, and Geometric Partial Differential Equations: Interfacial, Stochastic, Non-local and Discrete Structures'', January 20 - 26, 2019, Mathematisches Forschungsinstitut Oberwolfach, January 25, 2019.

  • A. Alphonse, Directional differentiability for elliptic quasi-variational inequalities, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Best Paper Session, August 5 - 8, 2019, Berlin, August 5, 2019.

  • T. Keil, Optimal control of a coupled Cahn--Hilliard--Navier--Stokes system with variable fluid densities, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Optimal Control of Phase Field Models'', August 5 - 8, 2019, Berlin, August 5, 2019.

  • S.-M. Stengl, M. Hintermüller, On the convexity of optimal control problems involving nonlinear PDEs or VIs, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Berlin, August 5 - 8, 2019.

  • S.-M. Stengl, On the convexity of optimal control problems involving nonlinear PDEs or VIs, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Quasi-Variational Inequalities and Generalized Nash Equilibrium Problems (Part I)'', August 5 - 8, 2019, Berlin, August 6, 2019.

  • C. Löbhard, Space-time discretization for parabolic optimal control problems with state constraints, ICCOPT 2019 -- Sixth International Conference on Continuos Optimization, Session ``Optimal Control and Dynamical Systems (Part VI)'', August 5 - 8, 2019, Berlin, August 7, 2019.

  • R. Sandilya, Error bounds for discontinuous finite volume discretisa-tions of Brinkman optimal control problems, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, Session ``Quasi-Variational Inequalities and Generalized Nash Equi-librium Problems (Part II)'', August 5 - 8, 2019, Berlin, August 7, 2019.

  • G. Dong, Direct reconstruction of biophysical parameters using dictionary learning and robust regularization, 1st MATH+ Day, Berlin, December 13, 2019.

  • M. Hintermüller, (Pre)Dualization, dense embeddings of convex sets, and applications in image processing, Thematic Programme ``Modern Maximal Monotone Operator Theory: From Nonsmooth Optimization to Differential Inclusions'', Workshop ``Numerical Algorithms in Nonsmooth Optimization'', February 25 - March 1, 2019, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria, February 28, 2019.

  • M. Hintermüller, A function space framework for structural total variation regularization with applications in inverse problems, 71st Workshop: Advances in Nonsmooth Analysis and Optimization (NAO2019), June 25 - 30, 2019, International School of Mathematics ``Guido Stampacchia'', Erice, Italy, June 26, 2019.

  • M. Hintermüller, A function space framework for structural total variation regularization with applications in inverse problems, Thematic Programme ``Modern Maximal Monotone Operator Theory: From Nonsmooth Optimization to Differential Inclusions'', Workshop ``Nonsmooth and Variational Analysis'', January 28 - February 1, 2019, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria, February 1, 2019.

  • M. Hintermüller, A physically oriented method for quantitative magnetic resonance imaging, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Session MS A1-1-3 5 ``Computationally Efficient Methods for Large-scale Inverse Problems in Imaging Applications'', July 15 - 19, 2019, Valencia, Spain, July 17, 2019.

  • M. Hintermüller, Applications in image processing, Workshop on Efficient Operator Splitting Techniques for Complex System and Large Scale Data Analysis, January 15 - 18, 2019, Sanya, China, January 14, 2019.

  • M. Hintermüller, Generalized Nash equilibrium problems with PDEs connected to spot markets with (gas) transport, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Session MS ME-1-4 1 ``Recent Advances in PDE-constrained Optimization'', July 15 - 19, 2019, Valencia, Spain, July 15, 2019.

  • M. Hintermüller, Generalized Nash equilibrium problems with application to spot markets with gas transport, Workshop ``Electricity Systems of the Future: Incentives, Regulation and Analysis for Efficient Investment'', March 18 - 22, 2019, Isaac Newton Institute, Cambridge, UK, March 21, 2019.

  • M. Hintermüller, Generalized Nash games with PDEs and applications in energy markets, French-German-Swiss Conference on Optimization (FGS'2019), September 17 - 20, 2019, Nice, France, September 20, 2019.

  • M. Hintermüller, Lecture Series: Optimal control of nonsmooth structures, Thematic Programme ``Modern Maximal Monotone Operator Theory: From Nonsmooth Optimization to Differential Inclusions'', February 4 - 7, 2019, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria.

  • M. Hintermüller, Math4NFDI -- A consortium for mathematics, National Research Data Infrastructure Conference 2019, May 13 - 14, 2019, Bonn.

  • M. Hintermüller, Optimal control of multiphase fluids and droplets, Colloquium of the Mathematical Institute, University of Oxford, UK, June 7, 2019.

  • M. Hintermüller, Optimal control of multiphase fluids and droplets, Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland, December 3, 2019.

  • M. Hintermüller, Optimal control problems involving nonsmooth structures, Autumn School 2019 ``Optimal Control and Optimization with PDEs'' (ALOP), October 7 - 10, 2019, Universität Trier.

  • M. Hintermüller, Structural total variation regularization with applications in inverse problems, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), DFG Priority Programme 1962 ``Non Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization'', February 18 - 22, 2019, Technische Universität Wien, Austria, February 19, 2019.

  • K. Papafitsoros, A function space framework for structural total variation regularization with applications in inverse problems, Applied Inverse Problems Conference, Minisymposium ``Multi-Modality/Multi-Spectral Imaging and Structural Priors'', July 8 - 12, 2019, Grenoble, France, August 8, 2019.

  • K. Papafitsoros, Generating structure non-smooth priors for image reconstruction, Young Researchers in Imaging Seminars, March 20 - 27, 2019, Henri Poincaré Institute, Paris, France, March 27, 2019.

  • K. Papafitsoros, Generating structure non-smooth priors for image reconstruction, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, August 5 - 8, 2019, Berlin, August 6, 2019.

  • K. Papafitsoros, Quantitative MRI: From fingerprinting to integrated physics-based models, Synergistic Reconstruction Symposium, November 3 - 6, 2019, Chester, UK, November 4, 2019.

  • C.N. Rautenberg, A nonlocal variational model in image processing associated to the spatially variable fractional Laplacian, ICCOPT 2019 -- Sixth International Conference on Continuous Optimization, August 5 - 8, 2019, Berlin, August 6, 2019.

  • C.N. Rautenberg, Parabolic quasi-variational inequalities with gradient and obstacle type constraints, Thematic Programme ``Modern Maximal Monotone Operator Theory: From Nonsmooth Optimization to Differential Inclusions'', January 28 - February 1, 2019, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria, January 31, 2019.

Research Groups