Articles in Refereed Journals

  • A. Alphonse, Ch.M. Elliott, Well-posedness of a fractional porous medium equation on an evolving surface, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 137 (2016) pp. 3--42.

  • 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.
    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.

Preprints, Reports, Technical Reports

  • M. Hintermüller, A. Langer, C.N. Rautenberg, T. Wu, Adaptive regularization for image reconstruction from subsampled data, Preprint no. 2379, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2379 .
    Abstract, PDF (1600 kByte)
    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.

  • L. Adam, M. Hintermüller, Th.M. Surowiec, A PDE-constrained optimization approach for topology optimization of strained photonic devices, Preprint no. 2377, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2377 .
    Abstract, PDF (936 kByte)
    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.

  • M. Hintermüller, S. Rösel, Duality results and regularization schemes for Prandtl--Reuss perfect plasticity, Preprint no. 2376, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2376 .
    Abstract, PDF (353 kByte)
    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.

  • A. Alphonse, Ch.M. Elliott, J. Terra, A coupled ligand-receptor bulk-surface system on a moving domain: Well posedness, regularity and convergence to equilibrium, Preprint no. 2357, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2357 .
    Abstract, PDF (536 kByte)
    We prove existence, uniqueness, and regularity for a reaction-diffusion system of coupled bulk-surface equations on a moving domain modelling receptor-ligand dynamics in cells. The nonlinear coupling between the three unknowns is through the Robin boundary condition for the bulk quantity and the right hand sides of the two surface equations. Our results are new even in the non-moving setting, and in this case we also show exponential convergence to a steady state. The primary complications in the analysis are indeed the nonlinear coupling and the Robin boundary condition. For the well posedness and essential boundedness of solutions we use several De Giorgi-type arguments, and we also develop some useful estimates to allow us to apply a Steklov averaging technique for time-dependent operators to prove that solutions are strong. Some of these auxiliary results presented in this paper are of independent interest by themselves.

  • L. Adam, M. Hintermüller, Th.M. Surowiec, A semismooth Newton method with analytical path-following for the $H^1$-projection onto the Gibbs simplex, Preprint no. 2340, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2340 .
    Abstract, PDF (1345 kByte)
    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, C.N. Rautenberg, S. Rösel, Density of convex intersections and applications, Preprint no. 2333, WIAS, Berlin, 2016.
    Abstract, PDF (361 kByte)
    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, 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, Preprint no. 2311, WIAS, Berlin, 2016.
    Abstract, PDF (640 kByte)
    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, K. Papafitsoros, C.N. Rautenberg, Analytical aspects of spatially adapted total variation regularisation, Preprint no. 2293, WIAS, Berlin, 2016.
    Abstract, PDF (877 kByte)
    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, Preprint no. 2287, WIAS, Berlin, 2016.
    Abstract, PDF (4835 kByte)
    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.

  • H. Egger, Th. Kugler, N. Strogies, Parameter identification in a semilinear hyperbolic system, Preprint no. 2278, WIAS, Berlin, 2016.
    Abstract, PDF (424 kByte)
    We consider the identification of a nonlinear friction law in a one-dimensional damped wave equation from additional boundary measurements. Well-posedness of the governing semilinear hyperbolic system is established via semigroup theory and contraction arguments. We then investigate the inverse problem of recovering the unknown nonlinear damping law from additional boundary measurements of the pressure drop along the pipe. This coefficient inverse problem is shown to be ill-posed and a variational regularization method is considered for its stable solution. We prove existence of minimizers for the Tikhonov functional and discuss the convergence of the regularized solutions under an approximate source condition. The meaning of this condition and some arguments for its validity are discussed in detail and numerical results are presented for illustration of the theoretical findings.

  • M. Hintermüller, C.N. Rautenberg, On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities, Preprint no. 2237, WIAS, Berlin, 2016.
    Abstract, PDF (10 MByte)
    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, T. Wu, A. Langer, Optimal selection of the regularization function in a generalized total variation model. Part II: Algorithm, its analysis and numerical tests, Preprint no. 2236, WIAS, Berlin, 2016.
    Abstract, PDF (6570 kByte)
    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, Optimal selection of the regularization function in a generalized total variation model. Part I: Modelling and theory, Preprint no. 2235, WIAS, Berlin, 2016.
    Abstract, PDF (417 kByte)
    A generalized total variation model with a spatially varying regularization weight is considered. Existence of a solution is shown, and the associated Fenchel-predual problem is derived. For automatically selecting the regularization function, a bilevel optimization framework is proposed. In this context, the lower-level problem, which is parameterized by the regularization weight, is the Fenchel predual of the generalized total variation model and the upper-level objective penalizes violations of a variance corridor. The latter object relies on a localization of the image residual as well as on lower and upper bounds inspired by the statistics of the extremes.

Talks, Poster

  • M. Hintermüller, Bilevel optimization and applications in imaging, Workshop ``Emerging Developments in Interfaces and Free Boundaries'', January 22 - 28, 2017, Mathematisches Forschungsinstitut Oberwolfach.

  • M. Hintermüller, Bilevel optimization and applications in imaging, Mathematisches Kolloquium, Universität Wien, Austria, January 18, 2017.

  • M. Hintermüller, Optimal control of nonsmooth phase-field models, DFG-AIMS Workshop on ``Shape Optimization, Homogenization and Control'', March 13 - 16, 2017, Mbour, Senegal, March 14, 2017.

  • M. Hintermüller, Non-smooth structures in PDE-constrained optimization, Mathematisches Kolloquium, Universität Duisburg-Essen, Fakultät für Mathematik, Essen, January 11, 2017.

  • M. Hintermüller, Recent trends in PDE-constrained optimization with non-smooth structures, Fourth Conference on Numerical Analysis and Optimization (NAOIV-2017), January 2 - 5, 2017, Sultan Qaboos University, Muscat, Oman, January 4, 2017.

  • A. Alphonse, Existence for a fractional porous medium equation on an evolving surface, Joint Research Seminar on Nonsmooth Variational Problems and Operator Equations / Mathematical Optimization, WIAS, Berlin, December 13, 2016.

  • T. Wu, Bilevel optimization and applications in imaging sciences, August 24 - 25, 2016, Shanghai Jiao Tong University, Institute of Natural Sciences, China.

  • K. Papafitsoros, Analytical aspects of spatially adapted total variation type regularisation, Seminar Nichtglatte Variationsprobleme und Operatorgleichungen, May 12, 2016.

  • M. Hintermüller, S. Hajian, N. Strogies, Subproject B02 -- Parameter id., sensor localization and quantification of uncertainties in switched PDE systems, Annual Meeting of the Collaborative Research Center/Transregio (TRR) 154 ``Mathematical Modeling, Simulation and Optimization Using the Example of Gas Networks'', Technische Universität Berlin, October 4 - 5, 2016.

  • M. Hintermüller, S. Hajian, N. Strogies, Subproject B02 -- Parameter id., sensor localization and quantification of uncertainties in switched PDE systems, Conference ``Mathematics of Gas Transport'', Konrad-Zuse-Zentrum für Informationstechnik Berlin, October 6 - 7, 2016.

  • M. Hintermüller, K. Papafitsoros, C. Rautenberg, A fine scale analysis of spatially adapted total variation regularisation, Imaging, Vision and Learning based on Optimization and PDEs, Bergen, Norway, August 29 - September 1, 2016.

  • M. Hintermüller, Adaptive finite elements in total variation based image denoising, SIAM Conference on Imaging Science, Minisymposium ``Leveraging Ideas from Imaging Science in PDE-constrained Optimization'', May 23 - 26, 2016, Albuquerque, USA, May 24, 2016.

  • M. Hintermüller, Bilevel optimization and applications in imaging, Imaging, Vision and Learning based on Optimization and PDEs, August 29 - September 1, 2016, Bergen, Norway, August 30, 2016.

  • M. Hintermüller, Bilevel optimization for a generalized total-variation model, SIAM Conference on Imaging Science, Minisymposium ``Non-Convex Regularization Methods in Image Restoration'', May 23 - 26, 2016, Albuquerque, USA, May 26, 2016.

  • M. Hintermüller, Nonsmooth structures in PDE constrained optimization, 66th Workshop ``Advances in Convex Analysis and Optimization'', July 5 - 10, 2016, International Centre for Scientific Culture ``E. Majorana'', School of Mathematics ``G. Stampacchia'', Erice, Italy, July 9, 2016.

  • M. Hintermüller, Optimal control of multiphase fluids and droplets, WIAS-PGMO Workshop on Nonsmooth and Stochastic Optimization with Applications to Energy Management, May 10 - 12, 2016, WIAS Berlin, May 11, 2016.

  • M. Hintermüller, Optimal control of multiphase fluids and droplets, The Fifth International Conference on Continuous Optimization, Session: ``Recent Developments in PDE-constrained Optimization I'', August 6 - 11, 2016, Tokyo, Japan, August 10, 2016.

  • M. Hintermüller, Optimal control of multiphase fluids and droplets, Salzburg Mathematics Colloquium, Universität Salzburg, Fachbereich Mathematik, Austria, June 9, 2016.

  • M. Hintermüller, Optimal selection of the regularisation function in a localised TV model, SIAM Conference on Imaging Science, Minisymposium ``Analysis and Parameterisation of Derivative Based Regularisation'', May 23 - 26, 2016, Albuquerque, USA, May 24, 2016.

  • M. Hintermüller, Recent trends in optimal control problems with nonsmooth structures, Computational Methods for Control of Infinite-dimensional Systems, March 14 - 18, 2016, Institute for Mathematics and its Applications, Minneapolis, USA, March 14, 2016.

  • M. Hintermüller, Shape and topological sensitivities in mathematical image processing, BMS Summer School ``Mathematical and Numerical Methods in Image Processing'', July 25 - August 5, 2016, Berlin Mathematical School, Technische Universität Berlin, Humboldt-Universität zu Berlin, Berlin, August 4, 2016.

  • M. Hintermüller, Towards sharp stationarity conditions for classes of optimal control problems for variational inequalities of the second kind, International INdAM Conference ``Optimal Control for Evolutionary PDEs and Related Topics (OCERTO 2016)'', June 20 - 24, 2016, Cortona, Italy, June 20, 2016.