Publications
Monographs

M. Hintermüller, K. Papafitsoros, Chapter 11: Generating Structured Nonsmooth Priors and Associated Primaldual 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. 437502, (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

A. Alphonse, M. Hintermüller, C.N. Rautenberg, Directional differentiability for elliptic quasivariational inequalities of obstacle type, Calculus of Variations and Partial Differential Equations, (2019), published online on 24.01.2019, DOI 10.1007/s0052601814730 .
Abstract
The directional differentiability of the solution map of obstacle type quasivariational inequal ities (QVIs) with respect to perturbations on the forcing term is studied. The classical result of Mignot is then extended to the quasivariational 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. 13901412, 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. 257283, 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 GeonSi microbridge are given. The highly favorable electronic properties of this design are demonstrated by steadystate simulations of the corresponding van Roosbroeck (driftdiffusion) system. 
H. Antil, C.N. Rautenberg, Sobolev spaces with nonMuckenhoupt weights, fractional elliptic operators, and applications, SIAM Journal on Mathematical Analysis, 51 (2019), pp. 24792503, DOI 10.1137/18M1224970 .
Abstract
We propose a new variational model in weighted Sobolev spaces with nonstandard 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 EulerLagrange 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. 114001/1114001/37, DOI 10.1088/13616420/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 L^{1} and L^{2} 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 L^{1} and L^{2} 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 physicsbased models, SIAM Journal on Imaging Sciences, 2 (2019), pp. 927971, 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. 187193], magnetic resonance fingerprinting (MRF) was introduced as a technique being capable of simultaneously recovering such quantitative parameters by using a twostep 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 MRFtype algorithms is limited by the ?discretization size? of the dictionary, a novel physicsbased method for qMRI is proposed. In contrast to the conventional twostep method, our model is dictionaryfree and is rather governed by a single nonlinear equation, which is studied analytically. This nonlinear equation is efficiently solved via robustified Newtontype 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.
Contributions to Collected Editions

A. Alphonse, M. Hintermüller, C.N. Rautenberg, Recent trends and views on elliptic quasivariational 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. 131.

M. Hintermüller, N. Strogies, On the consistency of RungeKutta methods up to order three applied to the optimal control of scalar conservation laws, in: Numerical Analysis and Optimization, M. AlBaali, L. Grandinetti, A. Purnama, eds., 235 of Springer Proceedings in Mathematics & Statistics, Springer Nature Switzerland AG, Cham, 2019, pp. 119154.
Abstract
Higherorder RungeKutta (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 RKcoefficients are derived that coincide with those characterizing strong stability preserving RungeKutta 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 thirdorder method. Finally, numerical tests for controlling Burgers equation validate the theoretical results.
Preprints, Reports, Technical Reports

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)
PDEconstrained (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 PDEconstrained GNEPs with uncertain data. The deterministic nonlinear problems are analyzed using the theory of generalized convexity for setvalued operators, and a variational approximation approach is proposed. The stochastic setting includes a detailed overview of the recently developed theory and algorithms for riskaverse PDEconstrained optimization problems. These new results open the way to a rigorous study of stochastic PDEconstrained 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 CahnHilliard NavierStokes 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 CahnHilliard NavierStokes 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 goaloriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (PODMOR) in order to replace highfidelity models by low order surrogates. In particular, we combine POD with spaceadapted 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 nonsmoothness. The latter might stem from energies containing nonsmooth constituents such as obstacletype 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 Yosidatype 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 dualweighted 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, electrowetting 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 HeleShaw flow with geometric boundary conditions involving a complementarity system due to contact line pinning, and the latter gives rise to a CahnHilliard NavierStokes model including a nonsmooth obstacle type potential leading to a variational inequality constraint. 
A. Alphonse, M. Hintermüller, C.N. Rautenberg, Existence, iteration procedures and directional differentiability for parabolic QVIs, Preprint no. 2592, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2592 .
Abstract, PDF (408 kByte)
We study parabolic quasivariational 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, M. Hintermüller, Y. Zhang, A class of secondorder 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 secondorder dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of secondorder damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped secondorder total variation flow, which is primarily motivated by the application of image denoising; the other is a damped secondorder mean curvature flow for level sets of scalar functions, which is related to a nonconvex 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 secondorder 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 firstorder flows are also documented. 
A. Alphonse, M. Hintermüller, C.N. Rautenberg, Stability of the solution set of quasivariational 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 obstacletype the stability of its solution set and associated optimal control problems are considered. These optimal control problems are nonstandard in the sense that they involve an objective with setvalued 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 wellposed.
Talks, Poster

J.A. Brüggemann, Elliptic obstacletype quasivariational 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 obstacletype quasivariational inequalities with volume constraints, Joint Research Seminar on Mathematical Optimization / Nonsmooth Variational Problems and Operator Equations, WIAS, November 21, 2019.

J.A. Brüggemann, Solution methods for a class of obstacletype quasi variational inequalities with volume constraints, ICCOPT 2019  Sixth International Conference on Continuous Optimization, Session ``QuasiVariational Inequalities and Generalized Nash Equilibrium Problems (Part II)'', August 5  8, 2019, Berlin, August 7, 2019.

A. Alphonse, Directional differentiability for elliptic quasivariational inequalities, Workshop ``Surface, Bulk, and Geometric Partial Differential Equations: Interfacial, stochastic, nonlocal and discrete structures'', January 20  26, 2019, Mathematisches Forschungsinstitut Oberwolfach, January 25, 2019.

A. Alphonse, Directional differentiability for elliptic quasivariational 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 CahnHilliardNavierStokes 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 ``QuasiVariational Inequalities and Generalized Nash Equilibrium Problems (Part I)'', August 5  8, 2019, Berlin, August 6, 2019.

C. Löbhard, Spacetime 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 discretisations of Brinkman optimal control problems, ICCOPT 2019  Sixth International Conference on Continuous Optimization, Session ``QuasiVariational Inequalities and Generalized Nash Equilibrium Problems (Part II)'', August 5  8, 2019, Berlin, August 7, 2019.

G. Dong, Direct reconstruction of biophysical parameters using dictionary learning and robust regularization, MATH+ Annual Meeting, 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 A113 5: ``Computationally Efficient Methods for Largescale 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 ME14 1: ``Recent Advances in PDEconstrained 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, FrenchGermanSwiss 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 ofmultiphase 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 ComplementarityBased 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: MultiModality/MultiSpectral Imaging and Structural Priors, July 8  12, 2019, Grenoble, France, August 8, 2019.

K. Papafitsoros, Generating structure nonsmooth 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 nonsmooth 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 physicsbased models, Synergistic Reconstruction Symposium CCP PETMR, 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 quasivariational inequalities with gradient and obstacle type constraints, Modern Maximal Monotone Operator Theory: From Nonsmooth Optimization to Differential Inclusions, January 28  February 1, 2019, Erwin Schrödinger International Institute for Mathematics and Physics (ESI), Wien, Austria, January 31, 2019.
Research Groups
 Partial Differential Equations
 Laser Dynamics
 Numerical Mathematics and Scientific Computing
 Nonlinear Optimization and Inverse Problems
 Interacting Random Systems
 Stochastic Algorithms and Nonparametric Statistics
 Thermodynamic Modeling and Analysis of Phase Transitions
 Nonsmooth Variational Problems and Operator Equations