Nonsmooth Variational Problems and Operator Equations

Publications

Monographs

• M. Hintermüller, R. Herzog, Ch. Kanzow, M. Ulbrich, S. Ulbrich, eds., Non-Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization, 172 of International Series of Numerical Mathematics, Birkhäuser, Springer Nature Switzerland AG, Cham, 2022, viii + 519 pages, (Collection Published), DOI 10.1007/978-3-030-79393-7 .

Articles in Refereed Journals

• A. Alphonse, D. Caetano, A. Djurdjevac, Ch.M. Elliot, Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs, Journal of Differential Equations, 353 (2023), pp. 268-338, DOI 10.1016/j.jde.2022.12.032 .
  Abstract
  We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev?Bochner spaces. An Aubin?Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev?Bochner spaces. We conclude with the proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising the evolutionary p-Laplace equation on a moving domain or surface) and identify some additional problems that can be formulated with the setting developed in this work.

• M. Flaschel, H. Yu, N. Reiter, J. Hinrichsen, S. Budday, P. Steinmann, S. Kumar, L. De Lorenzis, Automated discovery of interpretable hyperelastic material models for human brain tissue with EUCLID, Journal of the Mechanics and Physics of Solids, 180 (2023), pp. 105404/1--105404/23, DOI 10.1016/j.jmps.2023.105404 .

• M. Brokate, C. Christof, Strong stationarity conditions for optimal control problems governed by a rate-independent evolution variational inequality, SIAM Journal on Control and Optimization, 61 (2023), pp. 2222--2250, DOI 10.1137/22M1494403 .

• 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.

• E. Marino, M. Flaschel, S. Kumar, L. De Lorenzis, Automated identification of linear viscoelastic constitutive laws with EUCLID, Mechanics of Materials, 181 (2023), pp. 104643/1--104643/12, DOI 10.1016/j.mechmat.2023.104643 .

• C. Geiersbach, T. Scarinci, A stochastic gradient method for a class of nonlinear PDE-constrained optimal control problems under uncertainty, Journal of Differential Equations, 364 (2023), pp. 635-666, DOI 10.1016/j.jde.2023.04.034 .

• Q. Wang, D. Dachunyang, Y. Zhang, Real-variable characterizations and their applications of matrix-weighted Triebel-Lizorkin spaces, Journal of Mathematical Analysis and Applications, 529 (2023), pp. 127629/1--127629/37, DOI 10.1016/j.jmaa.2023.127629 .

• 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.

• M. Bongarti, I. Lasiecka, J.H. Rodrigues, Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity, Discrete and Continuous Dynamical Systems -- Series S, 15 (2022), pp. 1355--1376, DOI 10.3934/dcdss.2022020 .

• M. Bongarti, I. Lasiecka, Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary, Discrete and Continuous Dynamical Systems -- Series S, 15 (2022), pp. 1957--1985, DOI 10.3934/dcdss.2022107 .

• M. Bongarti, L.D. Galvan, L. Hatcher, M.R. Lindstrom, Ch. Parkinson, Ch. Wang, A.L. Bertozzi , Alternative SIAR models for infectious diseases and applications in the study of non-compliance, Mathematical Models & Methods in Applied Sciences, 32 (2022), pp. 1987--2015, DOI 10.1142/S0218202522500464 .
  Abstract
  In this paper, we use modified versions of the SIAR model for epidemics to propose two ways of understanding and quantifying the effect of non-compliance to non-pharmaceutical intervention measures on the spread of an infectious disease. The SIAR model distinguishes between symptomatic infected (I) and asymptomatic infected (A) populations. One modification, which is simpler, assumes a known proportion of the population does not comply with government mandates such as quarantining and social-distancing. In a more sophisticated approach, the modified model treats non-compliant behavior as a social contagion. We theoretically explore different scenarios such as the occurrence of multiple waves of infections. Local and asymptotic analyses for both models are also provided.

• J.C. De Los Reyes, K. Herrera, Parameter space study of optimal scale-dependent weights in TV image denoising, Applicable Analysis. An International Journal, published online on 03.02.2022, DOI 10.1080/00036811.2022.2033231 .

• 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 .

• M. Brokate, M. Ulbrich, Newton differentiability of convex functions in normed spaces and of a class of operators, SIAM Journal on Optimization, 32 (2022), pp. 1265--1287, DOI 10.1137/21M1449531 .

• 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, C.N. Rautenberg, J.F. Rodrigues, Analysis of a quasi-variational contact problem arising in thermoelasticity, Nonlinear Analysis. An International Mathematical Journal, 217 (2022), pp. 112728/1--112728/40, DOI 10.1016/j.na.2021.112728 .
  Abstract
  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.

• 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.

Contributions to Collected Editions

• R. Danabalan, M. Hintermüller, Th. Koprucki, K. Tabelow, MaRDI: Building research data infrastructures for mathematics and the mathematical sciences, in: Vol. 1 (2023): 1st Conference on Research Data Infrastructure (CoRDI) - Connecting Communities, Y. Sure-Vetter, C. Goble, eds., Proceedings of the Conference on Research Data Infrastructure, TIB Open Publishing, Hannover, pp. published online on 07.09.2023 (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.

• J.C. De Los Reyes, D. Villacís, Bilevel optimization methods in imaging, in: Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging, K. Chen, C.-B. Schönlieb, X.-Ch. Tai, L. Younces, eds., Springer, Cham, pp. published online on 17.02.2022, DOI 10.1007/978-3-030-03009-4_66-1 .

• 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.

• S.H.K. Narayanan, Th. Propson, M. Bongarti, J. Hückelheim, P. Hovland, Reducing memory requirements of quantum optimal control, in: ICCS 2022: Computational Science -- ICCS 2022, D. Groen, C. DE Mulatier, M. Paszynski, V.V. Krzhizhanovskaya, J.J. Dongarra, P.M.A. Sloot, eds., 13353 of Lecture Notes in Computer Science, Springer, Cham, 2022, pp. 129--142, DOI 10.1007/978-3-031-08760-8_11 .

• C. Geiersbach, E. Loayza-Romero, K. Welker, PDE-constrained shape optimization: Towards product shape spaces and stochastic models, in: Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging, K. Chen, C.-B. Schönlieb, X.-Ch. Tai, L. Younces, eds., Springer, Cham, pp. published online on 07.04.2022, DOI 10.1007/978-3-030-03009-4_120-1 .
  Abstract
  Shape optimization models with one or more shapes are considered in this chapter. Of particular interest for applications are problems in which a so-called shape functional is constrained by a partial differential equation (PDE) describing the underlying physics. A connection can be made between a classical view of shape optimization and the differential geometric structure of shape spaces. To handle problems where a shape functional depends on multiple shapes, a theoretical framework is presented, whereby the optimization variable can be represented as a vector of shapes belonging to a product shape space. The multi-shape gradient and multi-shape derivative are defined, which allows for a rigorous justification of a steepest descent method with Armijo backtracking. As long as the shapes as subsets of a hold-all domain do not intersect, solving a single deformation equation is enough to provide descent directions with respect to each shape. Additionally, a framework for handling uncertainties arising from inputs or parameters in the PDE is presented. To handle potentially high-dimensional stochastic spaces, a stochastic gradient method is proposed. A model problem is constructed, demonstrating how uncertainty can be introduced into the problem and the objective can be transformed by use of the expectation. Finally, numerical experiments in the deterministic and stochastic case are devised, which demonstrate the effectiveness of the presented algorithms.

Preprints, Reports, Technical Reports

• 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.

• C. Geiersbach, T. Suchan, K. Welker, Optimization of piecewise smooth shapes under uncertainty using the example of Navier--Stokes flow, Preprint no. 3037, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3037 .
  Abstract, PDF (1911 kByte)
  We investigate a complex system involving multiple shapes to be optimized in a domain, taking into account geometric constraints on the shapes and uncertainty appearing in the physics. We connect the differential geometry of product shape manifolds with multi-shape calculus, which provides a novel framework for the handling of piecewise smooth shapes. This multi-shape calculus is applied to a shape optimization problem where shapes serve as obstacles in a system governed by steady state incompressible Navier--Stokes flow. Numerical experiments use our recently developed stochastic augmented Lagrangian method and we investigate the choice of algorithmic parameters using the example of this application.

• C. Geiersbach, R. Henrion, Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form, Preprint no. 3021, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3021 .
  Abstract, PDF (355 kByte)
  In this paper, we discuss optimality conditions for optimization problems subject to random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution.

• M. Bongarti, M. Hintermüller, Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network, Preprint no. 3016, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3016 .
  Abstract, PDF (457 kByte)
  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.

• C. Geiersbach, T. Suchan, K. Welker, Stochastic augmented Lagrangian method in shape spaces, Preprint no. 3010, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3010 .
  Abstract, PDF (679 kByte)
  In this paper, we present a stochastic Augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints. We investigate the convergence of the method, which is based on a stochastic approximation approach with random stopping combined with an iterative procedure for updating Lagrange multipliers. The algorithm is applied to a multi-shape optimization problem with geometric constraints and demonstrated numerically.

• A. Alphonse, D. Caetano, A. Djurdjevac, Ch.M. Elliott, Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs, Preprint no. 2994, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.2994 .
  Abstract, PDF (527 kByte)
  We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev-Bochner spaces. An Aubin-Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev-Bochner spaces. We conclude with the proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising the evolutionary p-Laplace equation on a moving domain or surface) and identify some additional problems that can be formulated with the setting developed in this work.

• G. Dong, M. Hintermüller, K. Papafitsoros, A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives, Preprint no. 2964, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2964 .
  Abstract, PDF (748 kByte)
  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.

• 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, T. Keil, Strong stationarity conditions for the optimal control of a Cahn--Hilliard--Navier--Stokes system, Preprint no. 2924, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2924 .
  Abstract, PDF (290 kByte)
  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.

Talks, Poster

• S. Essadi, A deterministic nonsmooth mean field game with control and state constraints, 9th International Conference on Modeling, Simulation and Applied Optimization (ICMSAO'23), April 26 - 28, 2023, American University of Sharjah, UAE, Marrakesh, Morocco, April 27, 2023.

• S. Essadi, On nonsmooth mean field games with control and state constraints, SIAM Conference on Optimization (OP23), MS90 ``On Addressing Nonsmoothness, Hierarchy, and Uncertainty in Optimization and Games'', May 31 - June 3, 2023, Seattle, USA, June 1, 2023.

• Q. Wang, Robust multilevel training of artificial neural networks, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

• M. Bongarti, Network boundary control of the semilinear isothermal Euler equation modeling gas transport on a network of pipelines, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), May 30 - June 2, 2023, Technische Universität Dresden, June 2, 2023.

• D. Korolev, ML4SIM: Mathematical Architecture, ML4SIM Consortium Meeting, WIAS, Berlin, November 15, 2023.

• D. Korolev, Machine Learning for Simulation Intelligence in Composite Process Design, Leibniz MMS Days 2023, Potsdam, April 17 - 19, 2023.

• D. Korolev, Physics-informed neural control of partial differential equations with applications to numerical homogenization, Kaiserslautern Applied and Industrial Mathematics Days - KLAIM 2023, September 25 - 27, 2023, Fraunhofer ITWM, Kaiserslautern, September 26, 2023.

• M. Brokate, Derivatives and optimal control of a scalar sweeping process, Variational Analysis and Optimization Seminar, University of Michigan, Ann Arbor, USA, March 31, 2023.

• M. Brokate, Derivatives and optimal control of a sweeping process, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), May 30 - June 2, 2023, Technische Universität Dresden, June 2, 2023.

• M. Brokate, Strong stationarity conditions for an optimal control problem involving a rate-independent variational inequality, International Conference on Optimization: SIGOPT 2023, March 14 - 16, 2023, BTU Cottbus - Senftenberg, March 15, 2023.

• C. Geiersbach, Optimality Conditions in Control Problems with Probabilistic State Constraints, International Conference Stochastic Programming 2023, July 24 - 28, 2023, University of California, Davis, USA, July 25, 2023.

• C. Geiersbach, Optimality conditions for problems with probabilistic state constraints, Joint Research Seminar on Nonsmooth Variational Problems and Operator Equations / Mathematical Optimization, WIAS, Berlin, April 25, 2023.

• C. Geiersbach, Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form, Frontiers of Stochastic Optimization and its Applications in Industry, May 10 - 12, 2023, WIAS, Berlin, May 11, 2023.

• C. Geiersbach, Optimization with random state constraints in probabilistic or almost-sure form, Thematic Einstein Semester Mathematical Optimization for Machine Learning, Summer Semester 2023, September 13 - 15, 2023, Zuse Instutite Berlin (ZIB), Berlin, September 15, 2023.

• C. Geiersbach, Optimization with random uniform state constraints, Optimal Control Theory and Related Fields, December 4 - 7, 2023, Universidad Tecnica Federico Santa Maria, Valparaiso, Chile, December 4, 2023.

• A. Alphonse, Analysis of a quasi-variational contact problem arising in thermoelasticity, European Conference on Computational Optimization (EUCCO), Heidelberg University, September 25, 2023.

• P. Dvurechensky, C. Geiersbach, M. Hintermüller, A. Kannan, S. Kater, Equilibria for distributed multi-modal energy systems under uncertainty, MATH+ Day, Humboldt-Universität zu Berlin, October 20, 2023.

• M. Hintermüller, A descent algorithm for the optimal control of ReLU neural network Informed PDEs based on approximate directional derivatives, FoCM 2023 -- Foundations of Computational Mathematics, Session II.2: ``Continuous Optimization'', June 12 - 21, 2023, Sorbonne University, Paris, France, June 15, 2023.

• M. Hintermüller, A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives, SIAM Conference on Computational Science and Engineering (CSE23), MS390 ``Algorithms for Applications in Nonconvex, Nonsmooth Optimization'', February 26 - March 3, 2023, Amsterdam, Netherlands, March 3, 2023.

• M. Hintermüller, A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives, SIAM Conference on Optimization (OP23), MS 35 ``PDE-Constrained Optimization with Nonsmooth Structures or under Uncertainty'', May 31 - June 3, 2023, Seattle, USA, May 31, 2023.

• M. Hintermüller, Learning-informed and PINN-based multi scale PDE models in optimization, Conference on Deep Learning for Computational Physics, July 4 - 6, 2023, UCL, London, UK, July 6, 2023.

• M. Hintermüller, Optimal control of (quasi)variational inequalities: Stationarity, risk-aversion, and numerical solution, Workshop on Optimization, Equilibrium and Complementarity, August 16 - 19, 2023, The Hong Kong Polytechnic University, Department of Applied Mathematic, August 19, 2023.

• M. Hintermüller, Optimal control of multiphase fluids and droplets (online talk), Workshop ``Control Methods in Hyperbolic Partial Differential Equations'' (Hybrid Event), November 5 - 10, 2023, Mathematisches Forschungsinstitut Oberwolfach, November 7, 2023.

• M. Hintermüller, Short Course: Mathematics of PDE Constrained Optimization, Recent Trends in Optimization and Control: Short Course and Workshop, September 18 - 22, 2023, University of Pretoria, Future Africa Campus, South Africa, September 19, 2023.

• M. Hintermüller, PDE-constrained optimization with non-smooth learning-informed structures, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00711 ``Recent Advances in Optimal Control and Optimization'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 21, 2023.

• C. Sirotenko, Dictionary learning for an inverse problem in quantitative MRI, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00687 ``Recent advances in deep learning--based inverse and imaging problems'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 22, 2023.

• C. Sirotenko, Machine Learning for Quantitative MRI, Joint Research Seminar on Nonsmooth Variational Problems and Operator Equations / Mathematical Optimization, WIAS, January 26, 2023.

• S. Essadi, Constrained deterministic non-smooth mean field games, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), DFG Priority Program 1962 ``Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization'', August 15 - 19, 2022, Rheinisch-Westfälische Technische Hochschule Aachen, August 16, 2022.

• S. Essadi, Constrained mean field games: Analysis and algorithms, SPP 1962 Annual Meeting 2022, October 24 - 26, 2022, Novotel Berlin Mitte, October 25, 2022.

• A. Alphonse, Directional differentiability and optimal control for quasi-variational inequalities (online talk), ``Partial Differential Equations and their Applications'' Seminar, University of Warwick, Mathematics Institute, UK, January 25, 2022.

• A. Alphonse, Risk-averse optimal control of elliptic random variational inequalities, SPP 1962 Annual Meeting 2022, October 24 - 26, 2022, Novotel Berlin Mitte, October 25, 2022.

• M. Bongarti, Boundary stabilization of nonlinear dynamics of acoustic waves under the JMGT equation, Oberseminar Partielle Differentialgleichungen, Universität Konstanz, November 17, 2022.

• M. Bongarti, Boundary stabilization of nonlinear dynamics of acoustics waves under the JMGT equation (online talk), Early Career Math Colloquium, University of Arizona, Tucson, USA, October 12, 2022.

• M. Bongarti, Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary, Waves Conference 2022, July 24 - 29, 2022, ENSTA Institut Polytechnique de Paris, France, July 25, 2022.

• M. Bongarti, Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary, IFIP TC7 System Modeling and Optimization, July 4 - 8, 2022, University of Technology, Warsaw, Poland, July 4, 2022.

• M. Bongarti, Nonlinear gas transport on a network of pipelines, IFIP TC7 System Modeling and Optimization, July 4 - 8, 2022, University of Technology, Warsaw, Poland, July 4, 2022.

• J.C. De Los Reyes, Bilevel learning for inverse problems, Seminar SFB 1060, Universität Bonn, Fachbereich Mathematik, April 14, 2022.

• J. Leake, Lorentzian polynomials on cones and the Heron--Rota--Welsh conjecture, Workshop ``The Laguerre-Pólya Class and Combinatorics'', March 13 - 19, 2022, Mathematisches Forschungsinstitut Oberwolfach, March 18, 2022.

• J. Leake, Lorentzian polynomials on cones and the Heron--Rota--Welsh conjecture, Technische Universität Braunschweig, Carl-Friedrich-Gauß-Fakultät, June 16, 2022.

• M. Theiss, Constrained MFG: Analysis and algorithms, SPP 1962 Annual Meeting 2022, October 24 - 26, 2022, Novotel Berlin Mitte, October 25, 2022.

• M. Brokate, Derivatives of hysteresis operators, MURPHYS 2022 -- Interdisciplinary Conference on Multiple Scale Systems, Systems with Hysteresis, May 29 - June 3, 2022, Ostravice, Czech Republic, May 30, 2022.

• M. Brokate, Newton derivatives of convex functionals, Conference on Multiple Scale Systems, Silesian University, Opava, Czech Republic, January 16, 2022.

• M. Brokate, Rate independent evolutions, Charles University, Department of Numerical Mathematics, Prague, Czech Republic, March 10, 2022.

• M. Brokate, Rate independent evolutions: Derivatives and control, Universität Kiel, Department of Mathematics, April 29, 2022.

• M. Brokate, Strong stationarity for an optimal control problem for a rate independent evolution, Conference on Differential Equations and Their Applications (EQUADIFF 15), Minisymposium NAA-03: ``Evolution Differential Equations with Application to Physics and Biology'', July 11 - 15, 2022, Masaryk University, Brno, Czech Republic, July 12, 2022.

• H. Kremer, J.-J. Zhu, K. Muandet, B. Schölkopf, Functional generalized empirical likelihood estimation for conditional moment restrictions, ICML 2022: 39th International Conference on Machine Learning (Online Event), Baltimore, USA, July 18 - 23, 2022.

• C. Geiersbach, Game-theoretical modeling for green hydrogen markets, Future WiNS: New Energies for a Sustainable World, December 7 - 9, 2022, Humboldt-Universität zu Berlin, December 9, 2022.

• C. Geiersbach, Optimality conditions and regularization for OUU with almost sure state constraints (online talk), SIAM Conference on Uncertainty Quantification (Hybrid Event), Minisymposium 24 ``PDE-Constrained Optimization Under Uncertainty'', April 12 - 15, 2022, Atlanta, Georgia, USA, April 12, 2022.

• C. Geiersbach, Optimality conditions and regularization for stochastic optimization with almost sure state constraints (online talk), 2022 SIAM Conference on Imaging Science (IS22) (Online Event), Minisymposium ``Stochastic Iterative Methods for Inverse Problems'', March 21 - 25, 2022, March 25, 2022.

• C. Geiersbach, Problems and challenges in stochastic optimization (online talk), WIAS Days, March 2, 2022.

• C. Geiersbach, Shape optimization under uncertainty: Challenges and algorithms, Helmut Schmidt Universität Hamburg, Mathematik im Bauingenieurwesen, April 26, 2022.

• C. Geiersbach, State constraints in stochastic optimization, PGMO DAYS 2022, Session 15F: ``New Developments in Optimal Control Theory, Part II'', November 28 - 30, 2022, Gaspard Monge Program for Optimization, Operations Research and their Interaction with Data Science, EDF Lab Paris-Saclay, Palaiseau, France, November 30, 2022.

• C. Geiersbach, Optimality conditions and regularization for stochastic optimization with almost sure state constraints, 15th Viennese Conference on Optimal Control and Dynamic Games, July 12 - 15, 2022, TU Wien, Austria, July 14, 2022.

• C. Geiersbach, Optimality conditions and regularization for stochastic optimization with almost sure state constraints, International Conference on Continuous Optimization -- ICCOPT/MOPTA 2022, Cluster ``PDE-Constrained Optimization'', July 23 - 28, 2022, Lehigh University, Bethlehem, Pennsylvania, USA, July 26, 2022.

• C. Geiersbach, Optimization with almost sure state constraints, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), Session 19 ``Optimization of Differential Equations'', August 15 - 19, 2022, Rheinisch-Westfälische Technische Hochschule Aachen, August 16, 2022.

• M. Hintermüller, A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives (online talk), Workshop 2: Structured Optimization Models in High-Dimensional Data Analysis, December 12 - 16, 2022, National University of Singapore, Institute for Mathematical Sciences, December 15, 2022.

• M. Hintermüller, Optimization subject to learning informed PDEs, International Conference on Continuous Optimization -- ICCOPT/MOPTA 2022, Cluster ``PDE-Constrained Optimization'', July 23 - 28, 2022, Lehigh University, Bethlehem, Pennsylvania, USA, July 27, 2022.

• M. Hintermüller, Optimization with learning-informed differential equation constraints (online talk), Workshop on Control Problems (Online Event), October 17 - 20, 2022, Technische Universität Dortmund, October 17, 2022.

• M. Hintermüller, Optimization with learning-informed differential equations, Robustness and Resilience in Stochastic Optimization and Statistical Learning: Mathematical Foundations, May 20 - 24, 2022, Ettore Majorana Foundation and Centre for Scientific Culture, Erice, Italy, May 24, 2022.

• M. Hintermüller, PDE-constrained optimization with learning-informed structures (online talk), Optimization in Oslo (OiO) Seminar, Simula Research Laboratory, Norway, December 7, 2022.

• K. Papafitsoros, Automatic distributed parameter selection of regularization functionals via bilevel optimization (online talk), SIAM Conference on Imaging Science (IS22) (Online Event), Minisymposium ``Statistics and Structure for Parameter and Image Restoration'', March 21 - 25, 2022, March 22, 2022.

• K. Papafitsoros, Total variation methods in image reconstruction, Institute Colloquium, Foundation for Research and Technology Hellas (IACM-FORTH), Institute of Applied and Computational Mathematics, Heraklion, Greece, May 3, 2022.

• K. Papafitsoros, Optimization with learning-informed nonsmooth differential equation constraints, Second Congress of Greek Mathematicians SCGM-2022, Session Numerical Analysis & Scientific Computing, July 4 - 8, 2022, National Technical University of Athens, July 6, 2022.

• C. Sirotenko, Dictionary learning for an in inverse problem in quantitative MRI, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), Session 21 ``Mathematical Signal and Image Processing'', August 15 - 19, 2022, Rheinisch-Westfälische Technische Hochschule Aachen, August 16, 2022.

• C. Sirotenko, Dictionary learning for an inverse problem in quantitative MRI (online talk), SIAM Conference on Imaging Science (IS22) (Online Event), Minisymposium ``Recent Advances of Inverse Problems in Imaging'', March 21 - 25, 2022, March 25, 2022.

External Preprints

• M. Flaschel, H. Yu, N. Reiter, J. Hinrichsen, S. Budday, P. Steinmann, S. Kumar, L. De Lorenzis, Automated discovery of interpretable hyperelastic material models for human brain tissue with EUCLID, Preprint no. arXiv:2305.16362, Cornell University, 2023, DOI 10.48550/arXiv.2305.16362 .

• J. Boddapati, M. Flaschel, S. Kumar, L. De Lorenzis, Ch. Daralo, Single-test evaluation of directional elastic properties of anisotropic structured materials, Preprint no. arXiv:2304.09112, Cornell University, 2023, DOI 10.48550/arXiv.2304.09112 .

• A. Kofler, F. Altekrüger, F.A. Ba, Ch. Kolbitsch, E. Papoutsellis, D. Schote, C. Sirotenko, F.F. Zimmermann, K. Papafitsoros, Learning regularization parameter-maps for variational image reconstruction using deep neural networks and algorithm unrolling, Preprint no. arXiv:2301.05888, Cornell University, 2023, DOI 10.48550/arXiv.2301.05888 .

• A. Kofler, F. Altekrüger, F.A. Ba, Ch. Kolbitsch, E. Papoutsellis, D. Schote, C. Sirotenko, F.F. Zimmermann, K. Papafitsoros, Unrolled three-operator splitting for parameter-map learning in low dose X-ray CT reconstruction, Preprint no. arXiv:2304.08350, Cornell University, 2023, DOI 0.48550/arXiv.2304.08350 .

• M. Brokate, C. Christof, Strong stationarity conditions for optimal control problems governed by a rate-independent evolution variational inequality, Preprint no. arXiv:2205.01196, Cornell University, 2022, DOI 10.48550/arXiv.2205.01196 .