Prof. Michael Hintermüller

Publications since 2016

Articles in Refereed Journals

M. Bongarti, M. Hintermüller, Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 89 (2024), pp. 36/1--36/48, DOI 10.1007/s00245-023-10088-0 .
Abstract
The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush--Kuhn--Tucker (KKT) stationarity system with an almost surely non--singular Lagrange multiplier is derived.
G. Dong, M. Hintermüller, K. Papafitsoros, A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives, SIAM Journal on Optimization, 34 (2024), pp. 2314--2349, DOI 10.1137/22M1534420 .
Abstract
We propose and analyze a numerical algorithm for solving a class of optimal control problems for learning-informed semilinear partial differential equations. The latter is a class of PDEs with constituents that are in principle unknown and are approximated by nonsmooth ReLU neural networks. We first show that a direct smoothing of the ReLU network with the aim to make use of classical numerical solvers can have certain disadvantages, namely potentially introducing multiple solutions for the corresponding state equation. This motivates us to devise a numerical algorithm that treats directly the nonsmooth optimal control problem, by employing a descent algorithm inspired by a bundle-free method. Several numerical examples are provided and the efficiency of the algorithm is shown.
M. Hintermüller, S.-M. Stengl, A generalized $Gamma$-convergence concept for a type of equilibrium problems, Journal of Nonlinear Science, 34 (2024), pp. 83/1--83/28, DOI 10.1007/s00332-024-10059-x .
Abstract
A novel generalization of Γ-convergence applicable to a class of equilibrium problems is studied. After the introduction of the latter, a variety of its applications is discussed. The existence of equilibria with emphasis on Nash equilibrium problems is investigated. Subsequently, our Γ-convergence notion for equilibrium problems, generalizing the existing one from optimization, is introduced and discussed. The work ends with its application to a class of penalized generalized Nash equilibrium problems and quasi-variational inequalities.
M. Hintermüller, Th.M. Surowiec, M. Theiss, On a differential generalized Nash equilibrium problem with mean field interaction, SIAM Journal on Optimization, 34 (2024), pp. 2821--2855, DOI 10.1137/22M1489952 .
M. Gugat, J. Habermann, M. Hintermüller, O. Huber, Constrained exact boundary controllability of a semilinear model for pipeline gas flow, European Journal of Applied Mathematics, 34 (2023), pp. 532--553, DOI 10.1017/S0956792522000389 .
Abstract
While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.
M. Hintermüller, T. Keil, Strong stationarity conditions for the optimal control of a Cahn--Hilliard--Navier--Stokes system, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 89 (2024), pp. 12/1--12/28 (published online on 05.12.2023), DOI 10.1007/s00245-023-10063-9 .
Abstract
This paper is concerned with the distributed optimal control of a time-discrete Cahn-Hilliard-Navier-Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a variational inequality of fourth order and the Navier-Stokes equation. The existence of solutions to the primal system and of optimal controls is established. The Lipschitz continuity of the constraint mapping is derived and used to characterize the directional derivative of the constraint mapping via a system of variational inequalities and partial differential equations. Finally, strong stationarity conditions are presented following an approach from Mignot and Puel.
M. Hintermüller, A. Kröner, Differentiability properties for boundary control of fluid-structure interactions of linear elasticity with Navier--Stokes equations with mixed-boundary conditions in a channel, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 87 (2023), pp. 15/1--15/38, DOI 10.1007/s00245-022-09938-0 .
Abstract
In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from [Lasiecka, Szulc, and Zochoswki, Nonl. Anal.: Real World Appl., 44, 2018]. An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generaziling the results of [Wollner and Wick, J. Math. Fluid Mech., 21, 2019] to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.
D.G. Gahururu, M. Hintermüller, Th.M. Surowiec, Risk-neutral PDE-constrained generalized Nash equilibrium problems, Mathematical Programming. A Publication of the Mathematical Programming Society, 198 (2023), pp. 1287--1337 (published online on 29.03.2022), DOI 10.1007/s10107-022-01800-z .
G. Dong, M. Hintermüller, K. Papafitsoros, Optimization with learning-informed differential equation constraints and its applications, ESAIM. Control, Optimisation and Calculus of Variations, 28 (2022), pp. 3/1--3/44, DOI 10.1051/cocv/2021100 .
Abstract
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.
C. Geiersbach, M. Hintermüller, Optimality conditions and Moreau--Yosida regularization for almost sure state constraints, ESAIM. Control, Optimisation and Calculus of Variations, 28 (2022), pp. 80/1--80/36, DOI 10.1051/cocv/2022070 .
Abstract
We analyze a potentially risk-averse convex stochastic optimization problem, where the control is deterministic and the state is a Banach-valued essentially bounded random variable. We obtain strong forms of necessary and sufficient optimality conditions for problems subject to equality and conical constraints. We propose a Moreau--Yosida regularization for the conical constraint and show consistency of the optimality conditions for the regularized problem as the regularization parameter is taken to infinity.
A. Alphonse, M. Hintermüller, C.N. Rautenberg, On the differentiability of the minimal and maximal solution maps of elliptic quasi-variational inequalities, Journal of Mathematical Analysis and Applications, 507 (2022), pp. 125732/1--125732/19, DOI 10.1016/j.jmaa.2021.125732 .
Abstract
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, Set-Valued and Variational Analysis. Theory and Applications. Springer, Dordrecht. English., 30 (2022), pp. 873--922, DOI 10.1007/s11228-021-00624-x .
Abstract
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.
M. Hintermüller, K. Papafitsoros, C.N. Rautenberg, H. Sun, Dualization and automatic distributed parameter selection of total generalized variation via bilevel optimization, Numerical Functional Analysis and Optimization. An International Journal, 43 (2022), pp. 887--932, DOI 10.1080/01630563.2022.2069812 .
Abstract
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.
G. Dong, M. Hintermüller, Y. Zhang, A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging, SIAM Journal on Imaging Sciences, 14 (2021), pp. 645--688, DOI 10.1137/20M1366277 .
Abstract
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, On the differentiability of the minimal and maximal solution maps of elliptic quasi-variational inequalities, Journal of Mathematical Analysis and Applications, published online on 27.10.2021, DOI 10.1016/j.jmaa.2021.125732 .
Abstract
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.
M. Hintermüller, S.-M. Stengl, Th.M. Surowiec, Uncertainty quantification in image segmentation using the Ambrosio--Tortorelli approximation of the Mumford--Shah energy, Journal of Mathematical Imaging and Vision, 63 (2021), pp. 1095--1117, DOI 10.1007/s10851-021-01034-2 .
Abstract
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, S. Rösel, Duality results and regularization schemes for Prandtl--Reuss perfect plasticity, ESAIM. Control, Optimisation and Calculus of Variations, 27 (2021), pp. S1/1--S1/32, DOI 10.1051/cocv/2018004 .
Abstract
We consider the time-discretized problem of the quasi-static evolution problem in perfect plasticity posed in a non-reflexive Banach space and we derive an equivalent version in a reflexive Banach space. A primal-dual stabilization scheme is shown to be consistent with the initial problem. As a consequence, not only stresses, but also displacement and strains are shown to converge to a solution of the original problem in a suitable topology. This scheme gives rise to a well-defined Fenchel dual problem which is a modification of the usual stress problem in perfect plasticity. The dual problem has a simpler structure and turns out to be well-suited for numerical purposes. For the corresponding subproblems an efficient algorithmic approach in the infinite-dimensional setting based on the semismooth Newton method is proposed.
L. Banz, M. Hintermüller, A. Schröder, A posteriori error control for distributed elliptic optimal control problems with control constraints discretized by $hp$-finite elements, Computers & Mathematics with Applications. An International Journal, 80 (2020), pp. 2433--2450, DOI 10.1016/j.camwa.2020.08.007 .
C. Rautenberg, M. Hintermüller, A. Alphonse, Stability of the solution set of quasi-variational inequalities and optimal control, SIAM Journal on Control and Optimization, 58 (2020), pp. 3508--3532, DOI 10.1137/19M1250327 .
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.
M. Hintermüller, K. Papafitsoros, C.N. Rautenberg, Variable step mollifiers and applications, Integral Equations and Operator Theory, 92 (2020), pp. 53/1--53/34, DOI 10.1007/s00020-020-02608-2 .
Abstract
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.
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.
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.
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 .
L. Adam, M. Hintermüller, Th.M. Surowiec, A PDE-constrained optimization approach for topology optimization of strained photonic devices, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; Springer-Verlag, Heidelberg. English, French, German, Italian, English abstracts., 19 (2018), pp. 521--557, DOI 10.1007/s11081-018-9394-5 .
Abstract
Recent studies have demonstrated the potential of using tensile-strained, doped Germanium as a means of developing an integrated light source for (amongst other things) future microprocessors. In this work, a multi-material phase-field approach to determine the optimal material configuration within a so-called Germanium-on-Silicon microbridge is considered. Here, an “optimal" configuration is one in which the strain in a predetermined minimal optical cavity within the Germanium is maximized according to an appropriately chosen objective functional. Due to manufacturing requirements, the emphasis here is on the cross-section of the device; i.e. a socalled aperture design. Here, the optimization is modeled as a non-linear optimization problem with partial differential equation (PDE) and manufacturing constraints. The resulting problem is analyzed and solved numerically. The theory portion includes a proof of existence of an optimal topology, differential sensitivity analysis of the displacement with respect to the topology, and the derivation of first and second-order optimality conditions. For the numerical experiments, an array of first and second-order solution algorithms in function-space are adapted to the current setting, tested, and compared. The numerical examples yield designs for which a significant increase in strain (as compared to an intuitive empirical design) is observed.
L. Adam, M. Hintermüller, Th.M. Surowiec, A semismooth Newton method with analytical path-following for the $H^1$-projection onto the Gibbs simplex, IMA Journal of Numerical Analysis, 39 (2019), pp. 1276--1295 (published online on 07.06.2018), DOI 10.1093/imanum/dry034 .
Abstract
An efficient, function-space-based second-order method for the $H^1$-projection onto the Gibbs-simplex is presented. The method makes use of the theory of semismooth Newton methods in function spaces as well as Moreau-Yosida regularization and techniques from parametric optimization. A path-following technique is considered for the regularization parameter updates. A rigorous first and second-order sensitivity analysis of the value function for the regularized problem is provided to justify the update scheme. The viability of the algorithm is then demonstrated for two applications found in the literature: binary image inpainting and labeled data classification. In both cases, the algorithm exhibits mesh-independent behavior.
M. Hintermüller, M. Hinze, Ch. Kahle, T. Keil, A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn--Hilliard--Navier--Stokes system, Optimization and Engineering. International Multidisciplinary Journal to Promote Optimization Theory & Applications in Engineering Sciences, 19 (2018), pp. 629--662, DOI 10.1007/s11081-018-9393-6 .
Abstract
This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a time-discrete Cahn--Hilliard--Navier--Stokes system with variable densities. The free energy density associated to the Cahn--Hilliard system incorporates the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier--Stokes equation. A dual-weighed residual approach for goal-oriented adaptive finite elements is presented which is based on the concept of C-stationarity. The overall error representation depends on primal residual weighted by approximate dual quantities and vice versa as well as various complementary mismatch errors. Details on the numerical realization of the adaptive concept and a report on numerical tests are given.
M. Hintermüller, M. Holler, K. Papafitsoros, A function space framework for structural total variation regularization with applications in inverse problems, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 34 (2018), pp. 064002/1--064002/39, DOI 10.1088/1361-6420/aab586 .
Abstract
In this work, we introduce a function space setting for a wide class of structural/weighted total variation (TV) regularization methods motivated by their applications in inverse problems. In particular, we consider a regularizer that is the appropriate lower semi-continuous envelope (relaxation) of a suitable total variation type functional initially defined for sufficiently smooth functions. We study examples where this relaxation can be expressed explicitly, and we also provide refinements for weighted total variation for a wide range of weights. Since an integral characterization of the relaxation in function space is, in general, not always available, we show that, for a rather general linear inverse problems setting, instead of the classical Tikhonov regularization problem, one can equivalently solve a saddle-point problem where no a priori knowledge of an explicit formulation of the structural TV functional is needed. In particular, motivated by concrete applications, we deduce corresponding results for linear inverse problems with norm and Poisson log-likelihood data discrepancy terms. Finally, we provide proof-of-concept numerical examples where we solve the saddle-point problem for weighted TV denoising as well as for MR guided PET image reconstruction.
M. Hintermüller, C.N. Rautenberg, N. Strogies, Dissipative and non-dissipative evolutionary quasi-variational inequalities with gradient constraints, Set-Valued and Variational Analysis. Theory and Applications. Springer, Dordrecht. English., 27 (2019), pp. 433--468 (published online on 14.07.2018), DOI 10.1007/s11228-018-0489-0 .
Abstract
Evolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A semi-discretization in time is employed for the study of the problems and the derivation of a numerical solution scheme, respectively. Convergence of the discretization procedure is proven and properties of the original infinite dimensional problem, such as existence, extra regularity and non-decrease in time, are derived. The proposed numerical solver reduces to a finite number of gradient-constrained convex optimization problems which can be solved rather efficiently. The paper ends with a report on numerical tests obtained by a variable splitting algorithm involving different nonlinearities and types of constraints.
H. Antil, M. Hintermüller, R.H. Nochetto, Th.M. Surowiec, D. Wegner, Finite horizon model predictive control of electrowetting on dielectric with pinning, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 19 (2017), pp. 1--30, DOI 10.4171/IFB/375 .
S. Hajian, M. Hintermüller, S. Ulbrich, Total variation diminishing schemes in optimal control of scalar conservation laws, IMA Journal of Numerical Analysis, 39 (2019), pp. 105--140 (published online on 14.12.2017), DOI 10.1093/imanum/drx073 .
Abstract
In this paper, optimal control problems subject to a nonlinear scalar conservation law are studied. Such optimal control problems are challenging both at the continuous and at the discrete level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore discretization of the underlying optimal control problem should be designed with care. Here the discretize-then-optimize approach is employed where first the full discretization of the objective function as well as the underlying PDE is considered. Then, the derivative of the reduced objective is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta (TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods, also called strong stability preserving (SSP), are originally designed to preserve total variation of the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is too strong. Also approximation properties that the discrete adjoint inherits from the discretization of the state equation are studied. Moreover order conditions are derived. In addition, optimal choices with respect to CFL constant are discussed and numerical experiments are presented.
M. Hintermüller, T. Keil, D. Wegner, Optimal control of a semidiscrete Cahn--Hilliard--Navier--Stokes system with non-matched fluid densities, SIAM Journal on Control and Optimization, 55 (2017), pp. 1954--1989.
M. Hintermüller, K. Papafitsoros, C.N. Rautenberg, Analytical aspects of spatially adapted total variation regularisation, Journal of Mathematical Analysis and Applications, 454 (2017), pp. 891--935, DOI 10.1016/j.jmaa.2017.05.025 .
Abstract
In this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem, motivated by its application in signal recovery tasks. We study in depth the relationship between the weight function and the creation of new discontinuities in the solution. A partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed. We prove that the weighted total variation minimisation problem is well-posed even in the case of vanishing weight function, despite the lack of coercivity. This is based on the fact that the total variation of the solution is bounded by the total variation of the data, a result that it also shown here. Finally the relationship to the corresponding weighted fidelity problem is explored, showing that the two problems can produce completely different solutions even for very simple data functions.
M. Hintermüller, C.N. Rautenberg, M. Mohammadi, M. Kanitsar, Optimal sensor placement: A robust approach, SIAM Journal on Control and Optimization, 55 (2017), pp. 3609--3639.
Abstract
We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. The paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem and finalizes with a range of numerical tests.
M. Hintermüller, C.N. Rautenberg, S. Rösel, Density of convex intersections and applications, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 473 (2017), pp. 20160919/1--20160919/28, DOI 10.1098/rspa.2016.0919 .
Abstract
In this paper we address density properties of intersections of convex sets in several function spaces. Using the concept of Gamma-convergence, it is shown in a general framework, how these density issues naturally arise from the regularization, discretization or dualization of constrained optimization problems and from perturbed variational inequalities. A variety of density results (and counterexamples) for pointwise constraints in Sobolev spaces are presented and the corresponding regularity requirements on the upper bound are identified. The results are further discussed in the context of finite element discretizations of sets associated to convex constraints. Finally, two applications are provided, which include elasto-plasticity and image restoration problems.
M. Hintermüller, C.N. Rautenberg, T. Wu, A. Langer, Optimal selection of the regularization function in a generalized total variation model. Part II: Algorithm, its analysis and numerical tests, Journal of Mathematical Imaging and Vision, 59 (2017), pp. 515--533.
Abstract
Based on the generalized total variation model and its analysis pursued in part I (WIAS Preprint no. 2235), in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distributed regularization weight. Further, its numerical realization is discussed and results obtained for image denoising and deblurring as well as Fourier and wavelet inpainting are reported on.
M. Hintermüller, C.N. Rautenberg, On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 74 (2017), pp. 1--35.
Abstract
A class of abstract nonlinear evolution quasi-variational inequality (QVI) problems in function space is considered. The abstract framework developed in this paper includes constraint sets of obstacle and gradient type. The paper address the existence, uniqueness and approximation of solutions when the constraint set mapping is of a special form. Uniqueness is addressed through contractive behavior of a nonlinear mapping whose fixed points are solutions to the QVI. An axiomatic semi-discrete approximation scheme is developed, which is proven to be convergent and which is numerically implemented. The paper ends by a report on numerical tests for several nonlinear constraints of gradient-type.
M. Hintermüller, C.N. Rautenberg, Optimal selection of the regularization function in a weighted total variation model. Part I: Modeling and theory, Journal of Mathematical Imaging and Vision, 59 (2017), pp. 498--514.
Abstract
Based on the generalized total variation model and its analysis pursued in part I (WIAS Preprint no. 2235), in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distributed regularization weight. Further, its numerical realization is discussed and results obtained for image denoising and deblurring as well as Fourier and wavelet inpainting are reported on.
K. Sturm, M. Hintermüller, D. Hömberg, Distortion compensation as a shape optimisation problem for a sharp interface model, Computational Optimization and Applications. An International Journal, 64 (2016), pp. 557--588.
Abstract
We study a mechanical equilibrium problem for a material consisting of two components with different densities, which allows to change the outer shape by changing the interface between the subdomains. We formulate the shape design problem of compensating unwanted workpiece changes by controlling the interface, employ regularity results for transmission problems for a rigorous derivation of optimality conditions based on the speed method, and conclude with some numerical results based on a spline approximation of the interface.
M. Hintermüller, S. Rösel, A duality-based path-following semismooth Newton method for elasto-plastic contact problems, Journal of Computational and Applied Mathematics, 292 (2016), pp. 150--173.
M. Hintermüller, Th. Surowiec, A bundle-free implicit programming approach for a class of elliptic MPECs in function space, Mathematical Programming Series A, 160 (2016), pp. 271--305.

Contributions to Collected Editions

R. Danabalan, M. Hintermüller, Th. Koprucki, K. Tabelow, MaRDI: Building research data infrastructures for mathematics and the mathematical sciences, in: 1st Conference on Research Data Infrastructure (CoRDI) -- Connecting Communities, Y. Sure-Vetter, C. Goble, eds., 1 of Proceedings of the Conference on Research Data Infrastructure, TIB Open Publishing, Hannover, 2023, pp. 69/1--69/4, DOI 10.52825/cordi.v1i.397 .
Abstract
MaRDI is building a research data infrastructure for mathematics and beyond based on semantic technologies (metadata, ontologies, knowledge graphs) and data repositories. Focusing on the algorithms, models and workflows, the MaRDI infrastructure will connect with other disciplines and NFDI consortia on data processing methods, solving real world problems and support mathematicians on research datamanagement
A. Alphonse, M. Hintermüller, C.N. Rautenberg, Stability and sensitivity analysis for quasi-variational inequalities, in: Non-Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization, M. Hintermüller, R. Herzog, Ch. Kanzow, M. Ulbrich, S. Ulbrich, eds., 172 of International Series of Numerical Mathematics, Birkhäuser, Springer Nature Switzerland AG, Cham, 2022, pp. 183--210.
D. Gahururu, M. Hintermüller, S.-M. Stengl, Th.M. Surowiec, Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms and risk aversion, in: Non-Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization, M. Hintermüller, R. Herzog, Ch. Kanzow, M. Ulbrich, S. Ulbrich, eds., 172 of International Series of Numerical Mathematics, Birkhäuser, Springer Nature Switzerland AG, Cham, 2022, pp. 145--181.
C. Grässle, M. Hintermüller, M. Hinze, T. Keil, Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities, in: Non-Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization, M. Hintermüller, R. Herzog, Ch. Kanzow, M. Ulbrich, S. Ulbrich, eds., 172 of International Series of Numerical Mathematics, Birkhäuser, Springer Nature Switzerland AG, Cham, 2022, pp. 211--240.
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.
M. Hintermüller, T. Keil, Some recent developments in optimal control of multiphase flows, in: Shape Optimization, Homogenization and Optimal Control. DFG-AIMS Workshop held at the AIMS Center Senegal, March 13--16, 2017, V. Schulz, D. Seck, eds., 169 of International Series of Numerical Mathematics, Birkhäuser, Springer Nature Switzerland AG, Cham, 2018, pp. 113--142, DOI 10.1007/978-3-319-90469-6_7 .
M. Hintermüller, A. Langer, C.N. Rautenberg, T. Wu, Adaptive regularization for image reconstruction from subsampled data, in: Imaging, Vision and Learning Based on Optimization and PDEs IVLOPDE, Bergen, Norway, August 29 -- September 2, 2016, X.-Ch. Tai, E. Bae, M. Lysaker, eds., Mathematics and Visualization, Springer International Publishing, Berlin, 2018, pp. 3--26, DOI 10.1007/978-3-319-91274-5 .
Abstract
Choices of regularization parameters are central to variational methods for image restoration. In this paper, a spatially adaptive (or distributed) regularization scheme is developed based on localized residuals, which properly balances the regularization weight between regions containing image details and homogeneous regions. Surrogate iterative methods are employed to handle given subsampled data in transformed domains, such as Fourier or wavelet data. In this respect, this work extends the spatially variant regularization technique previously established in [15], which depends on the fact that the given data are degraded images only. Numerical experiments for the reconstruction from partial Fourier data and for wavelet inpainting prove the efficiency of the newly proposed approach.

Preprints, Reports, Technical Reports

A. Alphonse, M. Hintermüller, A. Kister, Ch.H. Lun, C. Sirotenko, A neural network approach to learning solutions of a class of elliptic variational inequalities, Preprint no. 3152, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3152 .
Abstract, PDF (21 MByte)
We develop a weak adversarial approach to solving obstacle problems using neural networks. By employing (generalised) regularised gap functions and their properties we rewrite the obstacle problem (which is an elliptic variational inequality) as a minmax problem, providing a natural formulation amenable to learning. Our approach, in contrast to much of the literature, does not require the elliptic operator to be symmetric. We provide an error analysis for suitable discretisations of the continuous problem, estimating in particular the approximation and statistical errors. Parametrising the solution and test function as neural networks, we apply a modified gradient descent ascent algorithm to treat the problem and conclude the paper with various examples and experiments. Our solution algorithm is in particular able to easily handle obstacle problems that feature biactivity (or lack of strict complementarity), a situation that poses difficulty for traditional numerical methods.
G. Dong, M. Hintermüller, C. Sirotenko, Dictionary learning based regularization in quantitative MRI: A nested alternating optimization framework, Preprint no. 3135, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3135 .
Abstract, PDF (5706 kByte)
In this article we propose a novel regularization method for a class of nonlinear inverse problems that is inspired by an application in quantitative magnetic resonance imaging (MRI). It is a special instance of a general dynamical image reconstruction problem with an underlying time discrete physical model. Our regularization strategy is based on dictionary learning, a method that has been proven to be effective in classical MRI. To address the resulting non-convex and non-smooth optimization problem, we alternate between updating the physical parameters of interest via a Levenberg-Marquardt approach and performing several iterations of a dictionary learning algorithm. This process falls under the category of nested alternating optimization schemes. We develop a general such algorithmic framework, integrated with the Levenberg-Marquardt method, of which the convergence theory is not directly available from the literature. Global sub-linear and local strong linear convergence in infinite dimensions under certain regularity conditions for the sub-differentials are investigated based on the Kurdyka?Lojasiewicz inequality. Eventually, numerical experiments demonstrate the practical potential and unresolved challenges of the method.
A. Alphonse, C. Christof, M. Hintermüller, I. Papadopoulos, A globalized inexact semismooth Newton method for nonsmooth fixed--point equations involving variational inequalities, Preprint no. 3132, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3132 .
Abstract, PDF (23 MByte)
We develop a semismooth Newton framework for the numerical solution of fixed-point equations that are posed in Banach spaces. The framework is motivated by applications in the field of obstacle-type quasi-variational inequalities and implicit obstacle problems. It is discussed in a general functional analytic setting and allows for inexact function evaluations and Newton steps. Moreover, if a certain contraction assumption holds, we show that it is possible to globalize the algorithm by means of the Banach fixed-point theorem and to ensure q-superlinear convergence to the problem solution for arbitrary starting values. By means of a localization technique, our Newton method can also be used to determine solutions of fixed-point equations that are only locally contractive and not uniquely solvable. We apply our algorithm to a quasi-variational inequality which arises in thermoforming and which not only involves the obstacle problem as a source of nonsmoothness but also a semilinear PDE containing a nondifferentiable Nemytskii operator. Our analysis is accompanied by numerical experiments that illustrate the mesh-independence and q -superlinear convergence of the developed solution algorithm.
G. Dong, M. Flaschel, M. Hintermüller, K. Papafitsoros, C. Sirotenko, K. Tabelow, Data--driven methods for quantitative imaging, Preprint no. 3105, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3105 .
Abstract, PDF (7590 kByte)
In the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative Magnetic Resonance Imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill-posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically-oriented overview on how data-driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps.
A. Alphonse, M. Hintermüller, C.N. Rautenberg, G. Wachsmuth, Minimal and maximal solution maps of elliptic QVIs: Penalisation, Lipschitz stability, differentiability and optimal control, Preprint no. 3093, WIAS, Berlin, 2024.
Abstract, PDF (501 kByte)
Quasi-variational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We prove that the solution maps are locally Lipschitz continuous and directionally differentiable and show existence of optimal controls for problems that incorporate these maps as the control-to-state operator. We also consider a Moreau?Yosida-type penalisation for the QVI wherein we show that it is possible to approximate the minimal and maximal solutions by sequences of minimal and maximal solutions (respectively) of certain PDEs, which have a simpler structure and offer a convenient characterisation in particular for computation. For solution mappings of these penalised problems, we prove a number of properties including Lipschitz and differential stability. Making use of the penalised equations, we derive (in the limit) C-stationarity conditions for the control problem, in addition to the Bouligand stationarity we get from the differentiability result.
M. Hintermüller, D. Korolev, A hybrid physics-informed neural network based multiscale solver as a partial differential equation constrained optimization problem, Preprint no. 3052, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3052 .
Abstract, PDF (1045 kByte)
In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating multiscale PDEs. From a continuous perspective, our formulation corresponds to a non-standard PDE-constrained optimization problem with a PINN-type objective. From a discrete standpoint, the formulation represents a hybrid numerical solver that utilizes both neural networks and finite elements. We propose a function space framework for the problem and develop an algorithm for its numerical solution, combining an adjoint-based technique from optimal control with automatic differentiation. The multiscale solver is applied to a heat transfer problem with oscillating coefficients, where the neural network approximates a fine-scale problem, and a coarse-scale problem constrains the learning process. We show that incorporating coarse-scale information into the neural network training process through our modelling framework acts as a preconditioner for the low-frequency component of the fine-scale PDE, resulting in improved convergence properties and accuracy of the PINN method. The relevance of the hybrid solver to numerical homogenization is discussed.
A. Alphonse, C. Geiersbach, M. Hintermüller, Th.M. Surowiec, Risk-averse optimal control of random elliptic VIs, Preprint no. 2962, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2962 .
Abstract, PDF (1541 kByte)
We consider a risk-averse optimal control problem governed by an elliptic variational inequality (VI) subject to random inputs. By deriving KKT-type optimality conditions for a penalised and smoothed problem and studying convergence of the stationary points with respect to the penalisation parameter, we obtain two forms of stationarity conditions. The lack of regularity with respect to the uncertain parameters and complexities induced by the presence of the risk measure give rise to new challenges unique to the stochastic setting. We also propose a path-following stochastic approximation algorithm using variance reduction techniques and demonstrate the algorithm on a modified benchmark problem.
G. Dong, M. Hintermüller, K. Papafitsoros, K. Völkner, First-order conditions for the optimal control of learning-informed nonsmooth PDEs, Preprint no. 2940, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2940 .
Abstract, PDF (408 kByte)
In this paper we study the optimal control of a class of semilinear elliptic partial differential equations which have nonlinear constituents that are only accessible by data and are approximated by nonsmooth ReLU neural networks. The optimal control problem is studied in detail. In particular, the existence and uniqueness of the state equation are shown, and continuity as well as directional differentiability properties of the corresponding control-to-state map are established. Based on approximation capabilities of the pertinent networks, we address fundamental questions regarding approximating properties of the learning-informed control-to-state map and the solution of the corresponding optimal control problem. Finally, several stationarity conditions are derived based on different notions of generalized differentiability.
M. Hintermüller, S.-M. Stengl, On the convexity of optimal control problems involving non-linear PDEs or VIs and applications to Nash games (changed title: Vector-valued convexity of solution operators with application to optimal control problems), Preprint no. 2759, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2759 .
Abstract, PDF (338 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.

External Preprints

P. Dvurechensky, C. Geiersbach, M. Hintermüller, A. Kannan, S. Kater, G. Zöttl, A Cournot-Nash model for a coupled hydrogen and electricity market, Preprint no. arxiv:2410.20534, Cornell University, 2024, DOI 10.48550/arXiv.2410.20534 .
Abstract
We present a novel model of a coupled hydrogen and electricity market on the intraday time scale, where hydrogen gas is used as a storage device for the electric grid. Electricity is produced by renewable energy sources or by extracting hydro- gen from a pipeline that is shared by non-cooperative agents. The resulting model is a generalized Nash equilibrium problem. Under certain mild assumptions, we prove that an equilibrium exists. Perspectives for future work are presented.
V. Grimm, M. Hintermüller, O. Huber, L. Schewe, M. Schmidt, G. Zöttl, A PDE-constrained generalized Nash equilibrium approach for modeling gas markets with transport, Preprint no. 458, Dokumentserver des Sonderforschungsbereichs Transregio 154, urlhttps://opus4.kobv.de/opus4-trr154/home, 2021.
M. Hintermüller, N. Strogies, On the identification of the friction coefficient in a semilinear system for gas transport through a network, Preprint, DFG SFB Transregio 154 ``Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks'', 2017.

Publications before 2016

M. Hintermüller, T. Valkonen, T. Wu, Limiting aspects of non-convex ^phi$ models [pdf], SIAM J. Imaging Sciences 8(4), pp. 2581-2621, 2015 [bib]
M. Hintermüller, T. Wu, Bilevel Optimization for Calibrating Point Spread Functions in Blind Deconvolution, Inverse Problems in Imaging 9(4), pp. 1139-1169, 2015 [bib]
M. Hintermüller, T. Surowiec, A. Kämmler, Generalized Nash Equilibrium Problems in Banach Spaces: Theory, Nikaido-Isoda-Based Path-Following Methods, and Applications [pdf], SIAM J. Optimization 25(3), pp. 1826-1856, 2015 [bib]
M. Hintermüller, C. Löbhard, H.M. Tber, An l1-penalty scheme for the optimal control of elliptic variational inequalities, in: Numerical Analysis and Optimization Volume 134 of the series Springer Proceedings in Mathematics & Statistics, pp. 151-190, 2015 [bib]
M. Hintermüller, A. Laurain, I. Yousept, Shape Sensitivities for an Inverse Problem in Magnetic Induction Tomography Based on the Eddy Current Model [pdf], Inverse Problems 31(6), 2015 [bib]
M. Hintermüller, C.N. Rautenberg, On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces [pdf], J. Math. Anal. Appl. 426(1) pp. 585-593, 2015 [bib]
M. Hintermüller, J. Rasch, Several path-following methods for a class of gradient constrained variational inequalities [pdf], Computers and Mathematics with Applications 69(10),2015, pp. 1045-1067 [bib]
C. Brett, C. Elliott, M. Hintermüller, C. Löbhard, Mesh Adaptivity in Optimal Control of Elliptic Variational Inequalities with Point-Tracking of the State [pdf], Interfaces and Free Boundaries 17(1), pp. 21-53, 2015 [bib]
A. Gaevskaja, M. Hintermüller, R.H.W. Hoppe, C. Löbhard, Adaptive finite elements for optimally controlled elliptic variational inequalities of obstacle type, in: Optimization with PDE Constraints, pp. 95-105, 2014, Ronald Hoppe (Ed.) [bib]
M. Hintermüller, T. Wu, Robust Principal Component Pursuit via Alternating Minimization on Matrix Manifolds [pdf], (the final publication is available at Springer via http://dx.doi.org/10.1007/s10851-014-0527-y), Journal of Mathematical Imaging and Vision 51(3), pp. 361-377, 2015 [bib]
M. Hintermüller, A. Langer, Non-Overlapping Domain Decomposition Methods For Dual Total Variation Based Image Denoising, Journal of Scientific Computing 62(2), pp. 456-481, 2015 [bib]
M. Hintermüller, C.N. Rautenberg, J. Hahn, Functional-analytic and numerical issues in splitting methods for total variation-based image reconstruction, Inverse Problems 30(5), pp. 055014, 2014 [bib]
M. Hintermüller, A. Langer, Surrogate Functional Based Subspace Correction Methods for Image Processing, in Domain Decomposition Methods in Science and Engineering XXI Volume 98 of the series Lecture Notes in Computational Science and Engineering, pp. 829-837, 2014 [bib]
M. Hintermüller, M.M. Rincon-Camacho, An adaptive finite element method in $L2$-TV-based image denoising, Inverse Problems and Imaging, 8(3), 2014, pp. 685-711 [bib]
M. Hintermüller, B.S. Mordukhovich, T. Surowiec, Several Approaches for the Derivation of Stationarity Conditions for Elliptic MPECs with Upper-Level Control Constraints, Mathematical Programming 146(1-2), pp. 555-582 [bib]
M. Hintermüller, D. Wegner, Optimal control of a semi-discrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control and Optimization 52(1), pp. 747-772, 2014 [bib]
M. Hintermüller, A. Schiela, W. Wollner, The length of the primal-dual path in Moreau-Yosida-based path-following for state-constrained optimal control, SIAM J. Optimization 24(1), 108-126, 2014 [bib]
M. Hintermüller, R.H.W. Hoppe, C. Löbhard, A dual-weighted residual approach to goal-oriented adaptivity for optimal control of elliptic variational inequalities, ESAIM: Control, Optimisation and Calculus of Variations 20(2), 2014, pp. 524-546 [bib]
M. Hintermüller, T. Wu, A superlinearly convergent R-regularized Newton scheme for variational models with concave sparsity-promoting priors, Computational Optimization and Applications 57(1), pp. 1-25, 2014 [bib]
M. Hintermüller, A. Langer, Subspace Correction Methods for a Class of Nonsmooth and Nonadditive Convex Variational Problems with Mixed $L^1/L^2$ Data-Fidelity in Image Processing, SIAM J. Imaging Sci., 6(4), 2134-2173, 2013 [bib]
M. Hintermüller, C.N. Rautenberg, Parabolic quasi-variational inequalities with gradient-type constraints, SIAM J. on Optimization 23(4), pp. 2090-2123, 2013 [bib]
M. Hintermüller, T. Wu, A Smoothing Descent Method for Nonconvex TV^q-Models, Lecture Notes in Computer Science 8293 (Efficient Algorithms for Global Optimization Methods in Computer Vision), pp. 119-133, 2014 [bib]
M. Hintermüller, T. Wu, Nonconvex TV^q-Models in Image Restoration: Analysis and a Trust-Region Regularization Based Superlinearly Convergent Solver, SIAM Journal on Imaging Science 6, 2013, pp. 1385-1415 [bib]
M. Hintermüller, T. Surowiec, A PDE-constrained generalized Nash equilibrium problem with pointwise control and state constraints, Pacific Journal on Optimization 9(2), pp. 251-273, 2013 [bib]
M. Hintermüller, D. Marahrens, P.A. Markowich, C. Sparber, Optimal bilinear control of Gross-Pitaevskii equations, SIAM J. Control and Optimization 51(3), pp. 2509-2543, 2013 [bib]
M. Freiberger, M. Hintermüller, A. Laurain, H. Scharfetter, Topological sensitivity analysis in fluorescence optical tomography, Inverse Problems 29(2), 2013 [bib]
M. Hintermüller, M. Hinze, C. Kahle, An adaptive finite element Moreau-Yosida-based solver for a coupled Cahn-Hilliard/Navier-Stokes system, Journal of Computational Physics, 2012 [bib]
M. Hintermüller, C.N. Rautenberg, A Sequential Minimization Technique for Elliptic Quasi-Variational Inequalities with Gradient Constraints, SIAM Journal on Optimization 22(4), 2012, pp. 1224-1257 [bib]
K. Bredies, Y. Dong, M. Hintermüller, Spatially dependent regularization parameter selection in total generalized variation models for image restoration, International Journal of Computer Mathematics 90(1), 2013, pp. 109-123 [bib]
M. Hintermüller, D. Wegner, Distributed optimal control of the Cahn-Hilliard system including the case of a double-obstacle homogeneous free energy density [pdf], SIAM J. Control Optim. 50, 2012, pp. 388-418 [bib]
M. Hintermüller, C.Y. Kao, A. Laurain, Principal Eigenvalue Minimization for an Elliptic Problem with Indefinite Weight and Robin Boundary Conditions, Applied Mathematics & Optimization 65(1), 111-146, 2012, DOI 10.1007/s00245-011-9153-x [bib]
M. Hintermüller, J.C. de los Reyes, A Duality-Based Semismooth Newton Framework for Solving Variational Inequalities of the Second Kind, Interfaces and Free Boundaries 13, 2011, pp. 437-462 [bib]
M. Hintermüller, T. Surowiec, First Order Optimality Conditions for Elliptic Mathematical Programs with Equilibrium Constraints via Variational Analysis, SIAM J. Optim. 21(4), pp. 1561-1593, 2012 [bib]
C. Elliott, M. Hintermüller, G. Leugering, J. Sokolowski, Advances in Shape and Topology Optimization: Theory, Numerics and New Application Areas., Optimization Methods and Software 26, Special Journal Issue 4-5, 2011, pp. 511-894
M. Hintermüller, M. Hinze, R.H.W. Hoppe, Weak-Duality Based Adaptive Finite Element Methods for PDE-Constrained Optimization with Pointwise Gradient State-Constraints, J. Comp. Math. 30, 2012, pp. 101-123 [bib]
C. Clason, M. Hintermüller, S.L. Keeling, F. Knoll, A. Laurain, G. Von Winckel, An image space approach to Cartesian based parallel MR imaging with total variation regularization, Medical Image Analysis 16(1), pp. 189-200, 2012 [bib]
M. Hintermüller, V. Kovtunenko, K. Kunisch, Obstacle Problems with Cohesion: A Hemi-Variational Inequality Approach and its Efficient Numerical Solution [pdf], SIAM Journal on Optimization 21 (2), 2011, pp. 491-516 [bib]
S.L. Keeling, M. Hintermüller, F. Knoll, D. Kraft, A. Laurain, A Total Variation Based Approach to Correcting Surface Coil Magnetic Resonance Images, Applied Mathematics and Computation 218(2), 2011, pp. 219-232 [bib]
K. Chen, Y. Dong, M. Hintermüller, A Nonlinear Multigrid Solver with Line Gauss-Seidel-Semismooth-Newton-Smoother for the Fenchel-Pre-Dual in Total Variation based Image Restoration, Inverse Problems and Imaging 5(2), pp. 323 - 339, 2011 [bib]
M. Hintermüller, A. Laurain, A.A. Novotny, Second-Order Topological Expansion for Electrical Impedance Tomography, Advances in Computational Mathematics 36(2), pp. 235-265, 2012, DOI: 10.1007/s10444-011-9205-4 [bib]
M. Hintermüller, M. Hinze, M.H. Tber, An Adaptive Finite-Element Moreau-Yosida-Based Solver for a Non-smooth Cahn-Hilliard Problem, Optimization Methods and Software 26 (4-5), Special Issue: Advances in Shape and Topology Optimization: Theory, Numerics and New Applications Areas, 2011, pp. 777-811 [bib]
M. Hintermüller, A. Laurain, Optimal Shape Design Subject to Elliptic Variational Inequalities, SIAM J. Control Optim. 49 (3), 2011, pp. 1015 - 1047 [bib]
M. Hintermüller, Y. Dong, M.M. Rincon-Camacho, Automated Regularization Parameter Selection in Multi-Scale Total Variation Models for Image Restoration, Journal of Mathematical Imaging and Vision 40 (1), 2011, pp. 82-104 [bib]
M. Hintermüller, V.A. Kovtunenko, From Shape Variation to Topology Changes in Constrained Minimization: A Velocity Method-Based Concept, Optimization Methods and Software 26 (4-5), Special Issue: Advances in Shape and Topology Optimization: Theory, Numerics and New Applications Areas, 2011, pp. 513-532 [bib]
F. Knoll, Y. Dong, M. Hintermüller, C. Langkammer, R. Stollberger, Total Variation Denoising with Spatially Dependent Regularization [pdf], ISMRM 18th Annual Scientific Meeting and Exhibition 2010 Proceedings, p. 5088 [bib]
Y. Dong, M. Hintermüller, M.M. Rincon-Camacho, A Multi-Scale Vectorial L-tau-TV Framework for Color Image Restoration, International Journal of Computer Vision 92 (3), 2011, pp. 296 - 307, DOI: 10.1007/s11263-010-0359-1 [bib]
M. Hintermüller, R.H.W. Hoppe, Goal-Oriented Adaptivity in Pointwise State Constrained Optimal Control of Partial Differential Equations, SIAM J. Control Optim. 48, p. 5468-5487 (2010) [bib]
M. Hintermüller, M.M. Rincon-Camacho, Expected Absolute Value Estimators for a Spatially Adapted Regularization Parameter Choice Rule in L1-TV-Based Image Restoration, Inverse Problems, Vol. 26, No. 8 (2010) [bib]
M. Hintermüller, R.H.W. Hoppe, Goal-oriented mesh adaptivity for mixed control-state constrained elliptic optimal control problems, Computational Methods in Applied Sciences 15, 2010, pp. 97-111, DOI: 10.1007/978-90-481-3239-3_8 [bib]
M. Hintermüller, M.H. Tber, An inverse problem in American options as a mathematical program with equilibrium constraints: C-stationarity and an active-set-Newton solver., SIAM J. on Control and Optimization 48, 2010 [bib]
R. H. Chan, Y. Dong, M. Hintermüller, An efficient two-phase L1-TV method for restoring blurred images with impulse noise, IEEE Transactions on Image Processing 19 (4), 2010 [bib]
M. Hintermüller, I. Yousept, A sensitivity-based extrapolation technique for the numerical solution of state-constrained optimal control problems, ESAIM Control, Optimization and Calculus of Variations 16 (3), 2010, pp. 503-522 [bib]
M. Hintermüller, A. Laurain, A shape and topology optimization technique for solving a class of linear complementarity problems in function space, Computational Optimization and Applications 46 (3), 2010, pp. 535-569 [bib]
F. Knoll, Y. Dong, M. Hintermüller, R. Stollberger, Automatic spatially dependent parameter selection for TV denoising of MR images with non-uniform noise distribution, Biomed Tech 55 (Suppl. 1), 2010, pp. 198 - 202 [bib]
M. Hintermüller, S.L. Keeling, Image registration and segmentation based on energy minimization, Handbook of Optimization in Medicine. Series: Springer Optimization and Its Applications , Vol. 26 Pardalos, P.M.; Romeijn, H.E.(Eds.) 2009, XI I I, 442 p. 129 illus., Hardcover ISBN: 978-0-387-09769-5 [bib]
M. Hintermüller, M. Hinze, Moreau-Yosida Regularization in State Constrained Elliptic Control Problems: Error Estimates and Parameter Adjustment, SIAM J. on Numerical Analysis, 47 (3), 2009, pp. 1666-1683 [bib]
M. Hintermüller, I. Kopacka, A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs, Computational Optimization and Applications 50 (1), 2011, pp. 111-145 [bib]
Y. Dong, M. Hintermüller, M. Neri, An Efficient Primal-Dual Method for L1-TV Image Restoration., SIAM J. Imaging Science, Vol. 2, Issue 4, pp. 1168-1189 (2009) [bib]
M. Hintermüller, K. Kunisch, PDE-Constrained Optimization Subject to Pointwise Constraints on the Control, the State and its Derivative, SIAM J. Optim., Vol. 20, Issue 3, pp. 1133-1156 (2009) [bib]
M. Hintermüller, V. Kovtunenko, K. Kunisch, A Papkovich-Neuber-based approach to cracks with contact in 3D, IMA J. Appl. Math. 74 (2009), pp. 325-343 [bib]
M. Hintermüller, I. Kopacka, Mathematical Programs with Complementarity Constraints in Function Space: C- and Strong Stationarity and a Path-Following Algorithm, SIAM J. Optim. Volume 20, Issue 2, pp. 868-902 (2009) [bib]
Y. Dong, M. Hintermüller, Multi-Scale Vectorial Total Variation with Automated Regularization Parameter Selection for Color Image Restoration, Springer Lecture Notes in Computer Science, no. 5567 (2009), pp.271-281 [bib]
M. Hintermüller, A.Laurain, Multiphase Image Segmentation and Modulation Recovery Based on Shape and Topological Sensitivity, Journal of Mathematical Imaging and Vision, 35 (2009) 1, pp. 1-22 [bib]
M. Hintermüller, I. Kopacka, S. Volkwein, Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints, ESAIM: COCV 15 3 (2009), 626-652, DOI: 10.1051/cocv:2008042 [bib]
M. Hintermüller, R.H.W. Hoppe, Adaptive finite element methods for control constrained distributed and boundary optimal control problems. Numerical PDE Constrained Optimization., Series: Lecture Notes in Computational Science and Engineering , Vol. 72 Heinkenschloss, Matthias; Vicente, Luis Nunes; Fernandes, Luis Merca (Eds.) 2009, Softcover ISBN: 978-3-540-77328-3 [bib]
M. Hintermüller, K. Kunisch, Stationary optimal control problems with pointwise state constraints. Numerical PDE Constrained Optimization., Series: Lecture Notes in Computational Science and Engineering , Vol. 72 Heinkenschloss, Matthias; Vicente, Luis Nunes; Fernandes, Luis Merca (Eds.) 2009, Softcover ISBN: 978-3-540-77328-3 [bib]
M. Hintermüller, A. Laurain, Electrical Impedance Tomography: From Topology to Shape., Control and Cybernetics, Vol. 37, No. 4 (2008) [bib]
M. Hintermüller, An active-set equality constrained Newton solver with feasibility restoration for inverse coefficient problems in elliptic variational inequalities, Inverse Problems 24 (2008), no. 3, 034017 [bib]
M. Hintermüller, R.H.W. Hoppe, Goal-oriented adaptivity in control constrained optimal control of partial differential equations, SIAM Journal on Control and Optimization, 47 (2008), pp. 1721-1743 [bib]
M. Hintermüller, F. Tröltzsch, I. Yousept, Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems, Numerische Mathematik, 108 (2008), no. 4, pp. 571-603 [bib]
M. Hintermüller, R.H.W. Hoppe, Y. Iliash, M. Kieweg, An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints, ESAIM: Control, Optimisation and Calculus of Variations (COCV), 14 3 (2008), pp. 540-560 [bib]
M. Hintermüller, A. Laurain, Where to create a hole?, European Consortium for Mathematics in Industry, ECMI Newsletter 41, 2007 [bib]
M. Hintermüller, S. Volkwein, F. Diwoky, Fast solution techniques in constrained optimal boundary control of the semilinear heat equation, Internat. Ser. Numer. Math., 155, Birkhäuser, Basel, 2007, pp. 119-147 [bib]
M. Hintermüller, Mesh-independence and fast local convergence of a primal-dual active-set method for mixed control-state constrained elliptic control problems, ANZIAM Journal, 49 (2007), no. 1, pp. 1-38 [bib]
M. Hintermüller, V. Kovtunenko, K. Kunisch, Constrained optimization for interface cracks in composite materials subject to non-penetration conditions, Engineering Mathematics, 59 (2007), no. 3, pp. 301-321 [bib]
M. Hintermüller, A combined shape Newton and topology optimization technique in real time image segmentation, Real-Time PDE-Constrained Optimization, Comput. Sci. Eng., 3, SIAM, Philadelphia, PA, 2007, pp. 253 - 274 [bib]
M. Hintermüller, K. Kunisch, Feasible and non-interior path-following in constrained minimization with low multiplier regularity, SIAM J. Control and Optimization 45 (2006) 4, pp. 1198-1221 [bib]
M. Hintermüller, K. Kunisch, Path-following methods for a class of constrained minimization problems in function space, SIAM J. Optimization, 17 (2006) 1, pp. 159-187 [bib]
M. Hintermüller, V. Kovtunenko, K. Kunisch, An optimization approach for the delamination of a composite material with non-penetration, In Free and Moving Boundaries: Analysis, Simulation and Control, eds. R. Glowinski and J.-P. Zolesio, Lecture Notes in Pure and Applied Mathematics, no. 252, Taylor & Francis CRC Press, London, 2006. [bib]
M. Hintermüller, G. Stadler, A primal-dual algorithm for TV-based inf-convolution-type image restoration, SIAM J. Scientific Computing, 28 (2006) 1, pp. 1-23 [bib]
M. Hintermüller, M. Hinze, A SQP-semismooth Newton-type algorithm applied to control of the instationary Navier-Stokes system sub ject to control constraints, SIAM J. Optimization, 16 (2006) 4, pp. 1177-2000 [bib]
M. Hintermüller, R. Griesse, M. Hinze, Differential stability of optimal control problems for the Navier Stokes equations, Numerical Functional Analysis and Optimization, 26 (2005) 7-8, pp. 829-850 [bib]
M. Hintermüller, M. Burger, Projected gradient flows for BV/Level set relaxation, Proc. Appl. Math. Mech., 5 (2005), pp. 11-14 [bib]
M. Hintermüller, Fast level-set based algorithms using shape and topological sensitivity information, Control and Cybernetics, 34 (2005) 1, pp. 305-324 [bib]
M. Hintermüller, V. Kovtunenko, K. Kunisch, Generalized Newton methods for crack problems with non-penetration condition, Numerical Methods for Partial Differential Equations, 21 (2005) 3, pp. 586-610 [bib]
M. Hintermüller, L.N. Vicente, Space mapping for optimal control of partial differential equations, SIAM J. Optimization, 15 (2005), pp. 1002-1025 [bib]
M. Hintermüller, K. Kunisch, Totally bounded variation regularization as bilaterally constrained optimization problem, SIAM J. Applied Mathematics, 64 (2004), pp.1311-1333 [bib]
M. Hintermüller, V. Kovtunenko, K. Kunisch, Semismooth Newton methods for a class of unilaterally constrained variational problems, Advances in Mathematical Sciences and Applications, 14 (2004) 2, pp. 513-535 [bib]
M. Hintermüller, M. Ulbrich, A mesh independence result for semismooth Newton methods, Mathematical Programming, 101 (2004) 1, pp. 151-184 [bib]
M. Hintermüller, K. Kunisch, Y. Spasov, S. Volkwein, Dynamical system based optimal control of incompressible fluids, International Journal for Numerical Methods in Fluids, 46 (2004), pp. 345-359 [bib]
M. Hintermüller, V. Kovtunenko, K. Kunisch, The primal-dual active set method for a crack problem with non-penetration, IMA J. on Applied Mathematics, 69 (2004), pp. 1-26 [bib]
M. Hintermüller, W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional, Journal of Mathematical Imaging and Vision, 20 (2004), pp. 19-42 [bib]
M. Hintermüller, W. Ring, A level set approach for the solution of a state-constrained optimal control problem, Numerische Mathematik 98 (2004) 1, pp. 135-166 [bib]
M. Hintermüller, G. Stadler, A semi-smooth Newton method for constrained linear-quadratic control problems, ZAMM, 83 (2003)4, pp. 219-237 [bib]
M. Hintermüller, W. Ring, Numerical aspects for a level set based algorithm for state constrained optimal control problems, CAMES-Computer Assisted Mechanics and Engineering Sciences, 10 (2003)2, pp. 149-161 [bib]
M. Hintermüller, W. Ring, A second order shape optimization approach for image segmentation, SIAM J. on Applied Mathematics, 64 (2003)2, pp. 442-467 [bib]
M. Hintermüller, W. Hinterberger, K. Kunisch, M. von Oehsen, O. Scherzer, Tube methods for BV regularization, Journal of Mathematical Imaging and Vision, 19 (2003)3, pp.219-235 [bib]
M. Hintermüller, K. Ito, K. Kunisch, The primal-dual active set strategy as a semi-smooth Newton method, SIAM J. Optimization, 13 (2003) 3, pp. 865-888 [bib]
M. Hintermüller, A primal-dual active set algorithm for bilaterally control constrained optimal control problems, Quarterly of Applied Mathematics, LXI 1 (2003), pp. 131-161 [bib]
M. Hintermüller, M. Hinze, Globalization of SQP-methods in control of the instationary Navier-Stokes equations, M2AN - Mathematical Modelling and Numerical Analysis, 36 (2002), pp. 725-746 [bib]
M. Hintermüller, Solving nonlinear programming problems with noisy function values and gradients, JOTA, 114 (2002)1, pp. 133-169 [bib]
M. Hintermüller, K. Kunisch, Inverse problems for elastohydrodynamic models, ZAMM, 81 (2001) Suppl.1, pp. S17-S20 [bib]
M. Hintermüller, On a globalized augmented Lagrangian SQP-algorithm for nonlinear optimal control problems with box constraints, in: Fast solution methods for discretized optimization problems. K.-H. Hoffmann, R.H.W. Hoppe, V. Schulz (eds.), International Series of Numerical Mathematics 138, Birkhäuser publishers, Basel, 2001, pp. 139-153 [bib]
M. Hintermüller, P. Bachhiesl, H. Hutten, F. Kappel, H. Scharfetter, Efficient computation of optimal controls for the exchange process during the dialysis therapy, Computational Optimization and Applications, 18 (2001),pp.161-175 [bib]
M. Hintermüller, A proximal bundle method based on approximate subgradients, Computational Optimization and Applications, 20 (2001), pp. 245-266. [bib]
M. Hintermüller, Inverse coefficient problems for variational inequalities: optimality conditions and numerical realization, Mathematical Modelling and Numerical Analysis (M2AN), 35 (2001), pp. 129-152 [bib]
M. Hintermüller, K. Stüwe, Topology and isotherms revisited: the influence of laterally migrating drainage divides, Earth and Planetary Science Letters, 184 (2000), pp. 287-303 [bib]
M. Hintermüller, M. Bergounioux, M. Haddou, K. Kunisch, A comparison of a Moreau-Yosida based active set strategy and interior point methods for constrained optimal control problems, SIAM Journal on Optimization, 11 (2000), pp. 495-521. [bib]
M. Hintermüller, An algorithm for solving nonlinear programs with noisy inequality constraints, Nonlinear Optimization and Related Topics, Kluwer, 1999, pp. 143-168. [bib]
M. Hintermüller, Algorithms for Solving Nonlinear Programming Problems with Noisy Data, University publisher R. Trauner, Series C - Technology and Science, No. 23, 1998, 224 pp. [bib]