Publikationen

Monografien

  • M. Hintermüller, T. Keil, Chapter 3: Optimal Control of Geometric Partial Differential Equations, in: Geometric Partial Differential Equations: Part 2, A. Bonito, R.H. Nochetto, eds., 22 of Handbook of Numerical Analysis, Elsevier, 2021, pp. 213--270, (Chapter Published), DOI 10.1016/bs.hna.2020.10.003 .

Artikel in Referierten Journalen

  • C. Geiersbach, E. Loayza-Romero, K. Welker, Stochastic approximation for optimization in shape spaces, SIAM Journal on Optimization, 31 (2021), pp. 348--376, DOI 10.1137/20M1316111 .

  • C. Geiersbach, T. Scarinci, Stochastic proximal gradient methods for nonconvex problems in Hilbert Spaces., Computational Optimization and Applications. An International Journal, (2021), published online on 12.01.2021, DOI 10.1007/s10589-020-00259-y .

  • M. Hintermüller, S. Rösel, Duality results and regularization schemes for Prandtl--Reuss perfect plasticity, ESAIM. Control, Optimisation and Calculus of Variations, published online on 01.03.2021, 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.

  • C. Geiersbach, W. Wollner, A stochastic gradient method with mesh refinement for PDE-constrained optimization under uncertainty, SIAM Journal on Scientific Computing, 42 (2020), pp. A2750--A2772, DOI 10.1137/19M1263297 .

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

  • N.A. Dao, J.I. Díaz, Q.B.H. Nguyen, Pointwise gradient estimates in multi-dimensional slow diffusion equations with a singular quenching term, Advanced Nonlinear Studies, 20 (2020), pp. 373--384, DOI 10.1515/ans-2020-2076 .

  • 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 .

  • M. Brokate, Newton and Bouligand derivatives of the scalar play and stop operator, Mathematical Modelling of Natural Phenomena, 15 (2020), pp. 51/1--51/34, DOI 10.1051/mmnp/2020013 .

  • 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.

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

  • 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.

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

Preprints, Reports, Technical Reports

  • M. Brokate, P. Krejči, A variational inequality for the derivative of the scalar play operator, Preprint no. 2803, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2803 .
    Abstract, PDF (302 kByte)
    We show that the directional derivative of the scalar play operator is the unique solution of a certain variational inequality. Due to the nature of the discontinuities involved, the variational inequality has an integral form based on the Kurzweil-Stieltjes integral.

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

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

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

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

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

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

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

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

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

Vorträge, Poster

  • A. Alphonse, Directional differentiability and optimal control for elliptic quasi-variational inequalities, Workshop ``Challenges in Optimization with Complex PDE-Systems'' (Online Workshop), February 14 - 20, 2021, Mathematisches Forschungsinstitut Oberwolfach, February 17, 2021.

  • A. Alphonse, Directional differentiability and optimal control for elliptic quasi-variational inequalities (online talk), Meeting of the Scientific Advisory Board of WIAS, WIAS Berlin, March 12, 2021.

  • A. Alphonse, Some aspects of sensitivity analysis and optimal control for elliptic QVIs (coauthors: Michael Hintermüller and Carlos Rautenberg, online talk), 91th Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Session DFG-PP 1962 Non-smooth and Complementarity-based Distributed Parameter Systems, March 15 - 19, 2021, Universität Kassel, March 16, 2021.

  • A. Alphonse, Some aspects of sensitivity analysis and optimal control for elliptic QVIs (online talk), Annual Meeting of the DFG SPP 1962 (Virtual Conference), March 24 - 25, 2021, WIAS Berlin, March 25, 2021.

  • C. Geiersbach, Almost sure state constraints with an application to stochastic Nash equilibrium problems, SIAM Conference on Computational Science and Engineering -- CSE21 (Virtual Conference), Minisymposium MS 114 ``Risk-Averse PDE-Constrained Optimization'', March 1 - 5, 2021, Virtual Conference Host: National Security Agency (NSA), March 2, 2021.

  • C. Geiersbach, Optimality conditions and regularization for convex stochastic optimization with almost sure state constraints, Workshop ``Challenges in Optimization with Complex PDE-Systems'' (Online Workshop), February 14 - 20, 2021, Mathematisches Forschungsinstitut Oberwolfach, February 16, 2021.

  • C. Geiersbach, Stochastic approximation with applications to PDE-constrained optimization under uncertainty (online talk), WIAS Seminar ``Modern Methods in Applied Stochastics and Nonparametric Statistics'', March 9, 2021.

  • C. Geiersbach, Stochastic approximation with applications to PDE-constrained optimization under uncertainty -- Part two (online talk), WIAS Seminar ``Modern Methods in Applied Stochastics and Nonparametric Statistics'', April 20, 2021.

  • G. Dong, Physics-integrated models and data-driven regularization methods for qMRI (online talk), online meeting between Physikalisch-Technische Bundesanstalt (PTB) division 8 and WIAS FG 8 to exchange ideas in quantitative MRI, April 16, 2021.

  • M. Hintermüller, Non smooth and complementarity-based distributed parameter systems: Simulation and hierarchical optimization (online talk), 91th Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Session DFG-PP 1962 Non-smooth and Complementarity-based Distributed Parameter Systems, March 15 - 19, 2021, Universität Kassel, March 16, 2021.

  • M. Hintermüller, Optimization with learning-informed differential equation constraints and its applications, Online Conference ``Industrial and Applied Mathematics'', January 11 - 15, 2021, The Hong Kong University of Science and Technology, Institute for Advanced Study, January 13, 2021.

  • M. Hintermüller, Optimization with learning-informed differential equation constraints and its applications (online talk), Seminar CMAI, George Mason University, Center for Mathematics and Artificial Intelligence, Fairfax, USA, March 19, 2021.

  • M. Hintermüller, Optimization with learning-informed differential equation constraints and its applications (online talk), One World Optimization Seminar, Universität Wien, Fakultät für Mathematik, Austria, May 10, 2021.

  • K. Papafitsoros, Optimization with learning-informed differential equation constraints and its applications (online talk), University of Graz, Institute of Mathematics and Scientific Computing, January 21, 2021.

  • K. Papafitsoros, Optimization with learning-informed differential equation constraints and its applications (online talk), Seminar Modern Methods in Applied Stochastics and Nonparametric Statistics, WIAS Berlin, March 16, 2021.

  • S.-M. Stengl, Combined regularization and discretization of equilibrium problems and primal-dual gap estimators, Seminar Mathematical Optimization / Non-smooth Variational Problems and Operator Equations, WIAS Berlin, April 15, 2021.

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

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

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

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

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

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

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

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

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

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