A semismooth Newton method for obstacle-type quasivariational inequalities
Research Seminar on Mathematical Optimization / Non-smooth Variational Problems and Operator Equations  đ
Head
Michael HintermĂŒller
Number of talks
36
Comment
Currently past talks are not included because the homepage lists some talks without a clear date.
October 2024
September 2024
Verifying the equivalence or non-equivalence of quantum circuits with tensor networks
Abstract.
The development of quantum computers and algorithms is currently rapidly accelerating and will likely continue to do so in the next few years and even decades. As these systems continue to increase in size and complexity, there is an increasing need for methods to aid in their design. In particular, there is a need for debugging tools at each level of the quantum computing stack, specifically in the compilation and optimization of quantum circuits. In this work, we bridge the gap between quantum many-body physics and computer science by using tensor network techniques to verify the equivalence or non-equivalence of quantum circuits in order to detect errors that may occur during the many steps of this process.
June 2024
Model predictive control for generalized Nash Equilibrium problems
Abstract.
We study model predictive control (MPC) schemes for non-cooperative dynamic games. The dynamic games are modelled as jointly convex generalized Nash equilibrium problems (GNEP) governed by a jointly controlled linear time-discrete dynamics. We are particularly interested in the asymptotic stability of the resulting closed-loop dynamics. To this end, we introduce a family of auxiliary problems, α-Quasi-GNEPs, which approximate the original GNEPs. For MPC schemes based on α-Quasi-GNEPs, stability guarantees can be derived if stabilizing end-constraints are enforced. This analysis is based on showing that the underlying optimal-value function is a Lyapunov function for the closed-loop. Passing to a limit, we identify a suitable Lyapunov function for MPC schemes based on the original GNEPs.
May 2024
Quantum noise characterization with a tensor network quantum jump method
Abstract.
In this talk, we will discuss a novel approach to characterizing the noise in noisy quantum circuits through the Tensorized Quantum Jump Method (TJM). The well-known Quantum Jump Method (Monte Carlo wave function), which is used to approximate Lindbladian dynamics, can be transferred to a tensor network algorithm via Strang splitting of the Lindbladian and the help of a dynamical low-rank approximation through the Time-Dependent Variational Principle. Choosing the sparse Pauli strings of the Sparse Pauli-Lindblad Model (SPLM) as Lindblad operators makes this method a new approach to characterizing quantum noise in large systems by learning the corresponding noise parameters.
Robust Multilevel Training of Artificial Neural Networks
Abstract.
In this talk, we will introduce a multilevel optimizier for training of an artificial neural network. We are particularly interested in nerual networks to learn the hidden physical law or nonlinear mapping from the given data using algebraic multigrid strategies. And we would like to give some further insight into the potential of multilevel optimization methods in the end.
Rate independent evolutions: some basics, some progress
Abstract.
We discuss some elementary rate independent evolutions, in particular the stop and the play, and offer remarks on the historical development. We also elaborate on issues concerning related optimal control problems.
February 2024
Computing multiple solutions of topology optimization problems
Abstract.
Topology optimization finds the optimal material distribution of a fluid or solid in a domain, subject to PDE and volume constraints. The models often result in a PDE, volume and inequality constrained, nonconvex, infinite-dimensional optimization problem that may support many local minima. In practice, heuristics are used to obtain the global minimum, but these can fail even in the simplest of cases. In this talk, we will introduce the deflated barrier method, a second-order algorithm that solves such problems, has local superlinear convergence, and can systematically discover many of these local minima. We will present examples which include finding 42 solutions of the topology optimization of a fluid satisfying the Navier-Stokes equations and more recent work involving the three-dimensional topology optimization of a fluid in Stokes flow.Underpinning the algorithm is the deflation mechanism. Deflation prevents a Newton-like solver from converging to a solution that has already been discovered. Deflation is computationally cheap, it does not affect the conditioning of the discretized systems, it may be coupled with a finite difference, finite volume or finite element discretization, and it is easy to implement.
December 2023
Maximal parabolic regularity for the treatment of real world problems
Abstract.
This is a series of three lectures on non-smooth problems, the first dedicated to elliptic ones and the third to parabolic ones. We start by explainig what means 'nonsmooth' and describe effects which cancel classical regularity results. Then we introduce a general geometric setting which allows in consequence to prove elliptic regularity results sufficient for attacking two- and three dimensional real world problems.Having this at hand, elliptic operators on the scale of (negatively indexed) Sobolev spaces and Lp spaces are introduced and their regularity properties are investigated, among them Lâ estimates, Hölder estimates and W 1,q estimates for the solution. After these preparations we pass to parabolic equations in the third and last lecture. Here the notion of 'maximal parabolic regularity' for an elliptic operator A is the central one - being now for some decades the ultimative instrument also for the investigation in particular of nonlinar problems. Since into this notion the domain of A enters explicitly, it becomes clear that an exact knowledge of dom(A), prepared in the first two lectures, is highly desirable. So, after having introduced maximal parabolic regularity and explained some of its properties, we show that second order divergence operators satisfy this property even if the domain is highly non-smooth, the coefficient function is only bounded, measurable and elliptic and the boundary conditions are mixed
Super-resolved Lasso
Abstract.
Super-resolution of pointwise sources is of utmost importance in various areas of imaging sciences. Specific instances of this problem arise in single molecule fluorescence, spike sorting in neuroscience, astrophysical imaging, radar imaging, and nuclear resonance imaging. In all these applications, the Lasso method (also known as Basis Pursuit or l1-regularization) is the de facto baseline method for recovering sparse vectors from low-resolution measurements. This approach requires discretization of the domain, which leads to quantization artifacts and consequently, an overestimation of the number of sources. While grid-less methods, such as Prony-type methods or non-convex optimization over the source position, can mitigate this, the Lasso remains a strong baseline due to its versatility and simplicity. In this work, we introduce a simple extension of the Lasso, termed ``super-resolved Lasso" (SR-Lasso). Inspired by the Continuous Basis Pursuit (C-BP) method, our approach introduces an extra parameter to account for the shift of the sources between grid locations. Our method is more comprehensive than C-BP, accommodating both arbitrary real-valued or complex-valued sources. Furthermore, it can be solved similarly to the Lasso as it boils down to solving a group-Lasso problem. A notable advantage of SR-Lasso is its theoretical properties, akin to grid-less methods. Given a separation condition on the sources and a restriction on the shift magnitude outside the grid, SR-Lasso precisely estimates the correct number of sources.
November 2023
Regularity for non-smooth elliptic problems II
Abstract.
This is a series of three lectures on non-smooth problems, the first dedicated to elliptic ones and the third to parabolic ones. We start by explainig what means 'nonsmooth' and describe effects which cancel classical regularity results. Then we introduce a general geometric setting which allows in consequence to prove elliptic regularity results sufficient for attacking two- and three dimensional real world problems.Having this at hand, elliptic operators on the scale of (negatively indexed) Sobolev spaces and Lp spaces are introduced and their regularity properties are investigated, among them Lâ estimates, Hölder estimates and W 1,q estimates for the solution. After these preparations we pass to parabolic equations in the third and last lecture. Here the notion of 'maximal parabolic regularity' for an elliptic operator A is the central one - being now for some decades the ultimative instrument also for the investigation in particular of nonlinar problems. Since into this notion the domain of A enters explicitly, it becomes clear that an exact knowledge of dom(A), prepared in the first two lectures, is highly desirable. So, after having introduced maximal parabolic regularity and explained some of its properties, we show that second order divergence operators satisfy this property even if the domain is highly non-smooth, the coefficient function is only bounded, measurable and elliptic and the boundary conditions are mixed
October 2023
Analysis of a variational contact problem arising in thermoelasticity
Abstract.
We study a model of a thermoforming process involving a membrane and a mould as implicit obstacle problems. Mathematically, the model consists of parabolic and elliptic PDEs coupled to a (quasi-)variational inequality. We study the related stationary (elliptic) model. We look at the existence of weak solutions, and by exploring the fine properties of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions. Under certain contraction conditions, we also show a uniqueness result. This is based on a joint paper with Jose-Francisco Rodrigues (Lisbon, Portugal) and Carlos N. Rautenberg (Virginia, USA).
Regularity for non-smooth elliptic problems I
Abstract.
This is a series of three lectures on non-smooth problems, the first dedicated to elliptic ones and the third to parabolic ones. We start by explainig what means 'nonsmooth' and describe effects which cancel classical regularity results. Then we introduce a general geometric setting which allows in consequence to prove elliptic regularity results sufficient for attacking two- and three dimensional real world problems.Having this at hand, elliptic operators on the scale of (negatively indexed) Sobolev spaces and Lp spaces are introduced and their regularity properties are investigated, among them Lâ estimates, Hölder estimates and W 1,q estimates for the solution. After these preparations we pass to parabolic equations in the third and last lecture. Here the notion of 'maximal parabolic regularity' for an elliptic operator A is the central one - being now for some decades the ultimative instrument also for the investigation in particular of nonlinar problems. Since into this notion the domain of A enters explicitly, it becomes clear that an exact knowledge of dom(A), prepared in the first two lectures, is highly desirable. So, after having introduced maximal parabolic regularity and explained some of its properties, we show that second order divergence operators satisfy this property even if the domain is highly non-smooth, the coefficient function is only bounded, measurable and elliptic and the boundary conditions are mixed
Mazen
Ali
Fraunhofer-Institut ITWM, Kaiserslautern
Quantum Computing for Differential Equations and Surrogate Modeling
Abstract.
Quantum computing has transitioned from theoretical promise to practical reality, with multiple devices now accessible to the public. This technological evolution has catalyzed a multidisciplinary race to achieve the first 'quantum advantage,' drawing experts from fields as diverse as physics, computer science, finance, mathematics, and chemistry. However, despite the immense potential, the practical utility of current quantum computing implementations remains modest. Much of the research is concentrated on similar, easily-attainable goals, often accompanied by overstated claims and unwarranted optimism. Consequently, pivotal questions about the true nature of 'quantum advantage,' the roadmap to achieving it, and its fundamental relevance remain Our focus is on harnessing the capabilities of quantum computing for material simulations at the macroscopic scale. In this presentation, I will offer an overview of the current state of quantum computing, discuss methodologies for solving differential equations directly on quantum platforms, and explore the use of quantum machine learning to create surrogate models for complex systems.
August 2023
A proximal trust-region method for nonsmooth optimization with inexact function and gradient evaluations
Abstract.
We develop a novel trust-region method to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. This class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. Our method is unique in that it permits and systematically controls the use of inexact objective function and derivative evaluations inherent in large-scale system solves and compression techniques, e.g. randomized sketching. When using a quadratic Taylor model for the trust-region subproblem, our algorithm is an inexact, matrix-free proximal Newton-type method that permits indefinite Hessians. We prove global convergence of our method in Hilbert space and elaborate on potential nonsmooth subproblem solvers based on ideas taken from their smooth counter-parts. Under additional assumptions, we can also prove superlinear, or even quadratic convergence to local minima. We demonstrate its efficacy on examples from data science and PDE-constrained optimization.
July 2023
Uncertainty quantification for models involving hysteresis operators
Abstract.
Parameters within models involving hysteresis operators that are supposed to describe with real world objects like, e.g. magneto mechanical devices, have to be identified from measurements. Hence, they are subject to corresponding errors. The methods of Uncertainty Quantification (UQ) are applied to investigate the influence of these errors. As an example, results of forward UQ for a play operator with uncertain yield limit will be presented. Afterwards, the model for a magneto mechanical devices involving a generalized Prandtl-IshlinskiÄ operator considered in Sec. 5 in Davino-KrejÄĂ--Visone-2013, Fully coupled modeling of magneto-mechanical hysteresis through `thermodynamic' compatibility. Smart Mater. Struct. https://doi.org/10.1088/0964-1726/22/9/095009 will be considered. Starting from data used to generated a First-Order-Reversal-Curves (FORC)-diagram inverse UQ is performed by formulating appropriate Bayesian Inverse Problems (BIPs) and applying Bayes' Theorem. The density of the resulting posterior density is represented by samples resulting from MCMC-computations using UQLab, the âThe Framework for Uncertainty Quantificationâ, see https://www.uqlab.com/. Afterwards, forward UQ is performed and the results are compared to measurements. These are results of a joined work with Carmine Stefano Clemente and Daniele Davino of the UniversitĂ degli Studi del Sannio, Benevento, Italy and Ciro Visone of UniversitĂ di Napoli Federico II, Napoli, Italy, see also: K.-Davino-Visone-2020, On forward and inverse uncertainty quantification for models involving hysteresis operators, Math. Model. Nat. Phenom 15, https://doi.org/10.1051/mmnp/2020009 and Clemente-Davino-K.-Visone-2023, Forward and Inverse Uncertainty Quantification for a model for a magneto mechanical device involving a hysteresis operator, WIAS Preprint 3009
A semismooth Newton solver with automatic differentiation written in C++
Abstract.
In this talk we consider problems of the form F(x)=0 where F is a nonlinear Newton differentiable mapping between Solbolev spaces. It is well-known that a semismooth Newton method ensures local superlinear convergence towards a solution. The function spaces are discretized by suitable finite elements over a given grid. A major difficulty is the practical implementation of generalized Jacobians. To this end, we present automatic differentiation techniques to obtain discrete subgradients of F. The resulting sparse linear problems are solved by efficient linear solvers. The framework is easy to use and to implement: The user only needs to implement a local evaluation of F in the weak form for a given set of test functions. An example implementation is given for a thermoforming model from a recent paper. To verify the solver, the results of this model are reproduced.
June 2023
Deep Learning with variable time stepping
Abstract.
Feature propagation in Deep Neural Networks (DNNs) can be associated to nonlinear discrete dynamical systems. The novelty, in this talk, lies in letting the discretization parameter (time step-size) vary from layer to layer, which needs to be learned, in an optimization framework. The proposed framework can be applied to any of the existing networks such as ResNet, DenseNet or Fractional-DNN. This framework is shown to help overcome the vanishing and exploding gradient issues. Stability of some of the existing continuous DNNs such as Fractional-DNN is also studied. The proposed approach is applied to an ill-posed 3D-Maxwell's equation.
Physics-informed neural control of partial differential equations with applications to numerical homogenisation
Abstract.
In this talk we discuss a model for numerical homogenisation based on the combination of physics-informed neural networks and standard numerical approximation techniques. From a continuous viewpoint, the formulation corresponds to a non-standard PDE-constrained optimisation problem with a neural network objective. From a discrete viewpoint, the formulation represents a hybrid neural network numerical solver. We discuss physics-informed neural networks, the numerical homogenisation modelling framework and its aforementioned optimisation interpretation, as well as related discrete concepts and routes towards applications in materials science and fluid flows in porous media.
May 2023
Proximal Galerkin: Structure-preserving finite element analysis for free boundary problems, maximum principles, and optimal design
Abstract.
One of the longest-standing challenges in finite element analysis is to develop a stable, scalable, high-order Galerkin method that strictly enforces pointwise bound constraints. The latent variable proximal Galerkin finite element method is a nonlinear, structure-preserving method with these properties. This talk will introduce proximal Galerkin and describe its capability for treating free-boundary problems, enforcing discrete maximum principles, and designing scalable, mesh-independent algorithms for optimal design. The talk begins with a derivation of the latent variable proximal point (LVPP) method: an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive (Bayesian) barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Thereupon, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this talk, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between positivity-preserving discretizations and infinite-dimensional Lie groups; and (3) a gradient-based, structure-preserving algorithm for two-field density-based topology optimization. The overall latent variable proximal Galerkin combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis.
Degenerate hysteresis in partially saturated porous media
Abstract.
We propose a model for fluid diffusion in partially saturated porous media taking into account hysteresis effects in the pressure-saturation relation. The resulting mathematical problem leads to a diffusion equation with Robin boundary condition for the pressure in an N-dimensional domain with a Preisach hysteresis operator under the time derivative. The problem is doubly degenerate in the sense that the saturation range is bounded, and no a priori control of the time derivative of the pressure is available. A bootstrapping argument based on particular geometric properties of the hysteresis operator makes it possible to prove the existence and uniqueness of a strong solution to the problem for arbitrarily large data. This is a joint work with Chiara Gavioli from TU Wien.
Convergence analysis of the nonoverlapping Robin-Robin method for nonlinear elliptic equations
Abstract.
The nonoverlapping Robin-Robin method is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. In this talk we present a convergence analysis for the Robin-Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. The analysis relies on a new theory for nonlinear Steklov-Poincare operators based on the p-structure and the Lp-generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. This is joint work with Emil Engström (Lund University)
April 2023
Optimality conditions for problems with probabilistic state constraints
Abstract.
In this talk, we discuss optimization problems subject to probabilistic constraints. Our focus is on the setting in which the control variable belongs to a reflexive and separable Banach space, which is of interest, for instance, in physics-based models where the control acts on a system described by a partial differential equation (PDE). Incorporating uncertainty into such models has been of increasing interest, since in practice, one might only have access to empirical measurements or ranges of values for model parameters and inputs. We present different possibilities for incorporating uncertainty in state constraints and derive their optimality conditions. The conditions are applied to a simple example from PDE-constrained optimization under uncertainty. Perspectives for the numerical solution of these problems are discussed, as well as planned research directions.
March 2023
On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs
Abstract.
We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc. This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation unctions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero.
February 2023
The Hamilton-Jacobi Formulation of Optimal Path Planning for Autonomous Vehicles
Abstract.
We present a partial-differential-equation-based optimal path planning framework for simple self-driving cars. This formulation relies on optimal control theory, dynamic programming, and a Hamilton-Jacobi-Bellman equation, and thus provides an interpretable alternative to black-box machine learning algorithms. We design grid-based numerical methods used to resolve the solution to the Hamilton-Jacobi-Bellman equation and generate optimal trajectories. We then describe how efficient and scalable algorithms for solutions of high dimensional Hamilton-Jacobi equations can be used to solve similar problems in higher dimensions and in nearly real-time. We demonstrate all of our methods with several examples.
January 2023
Clemens
Sirotenko
Machine Learning for Quantitative MRI
Abstract.
The field of quantitative Magnetic Resonance Imaging aims at extracting physical tissue parameters from a sequence of highly under sampled MR images. One recently proposed approach attempts to solve this problem by estimating a set of unknown parameters in a system of ordinary differential equations. While classical approaches such as the Levenberg Marquardt algorithm or Landweber iteration yield good results under small noise levels, numerical experiments with low sampling rates and noise of large magnitude lead to unsatisfactory outcomes and unstable convergence behavior. Therefore, a spatial regularization approach based on coupled dictionary learning is proposed. From a mathematical viewpoint this ends up in a variety of non convex and non smooth optimization problems. Iterative schemes to solve these problems are discussed and convergence to equilibrium points is studied. Moreover numerical results and open questions are presented.
December 2022
Mike
Theiss
Deriving a constrained Mean-Field Game
Abstract.
Mean-Field Games (MFGs) have a wide area of applications, i.e. crowd motion, flocking models, or behavior of investors. In most of these applications, it makes sense to assume constraints to the control or the state. We will start with some basic properties of a specific linear quadratic N-player game with mean field interaction. Afterward, we let the number of players N go to infinity, for deriving a "constrained MFG". Therefore we have to analyze the mean-field interaction, which describes the behavior of the whole group modeled as a flow of probability measures. Interesting is also the connection of the MFG to the original N-player problem. In the end, we will discuss some ideas on how to solve such constrained MFGs.
November 2022
Analysis of stochastic gradient descent in continuous time
Abstract.
Optimisation problems with discrete and continuous data appear in statistical estimation, machine learning, functional data science, robust optimal control, and variational inference. The 'full' target function in such an optimisation problem is given by the integral over a family of parameterised target functions with respect to a discrete or continuous probability measure. Such problems can often be solved by stochastic optimisation methods: performing optimisation steps with respect to the parameterised target function with randomly switched parameter values. In this talk, we discuss a continuous-time variant of the stochastic gradient descent algorithm. This so-called stochastic gradient process couples a gradient flow minimising a parameterised target function and a continuous-time 'index' process which determines the parameter. We first briefly introduce the stochastic gradient processes for finite, discrete data which uses pure jump index processes. Then, we move on to continuous data. Here, we allow for very general index processes: reflected diffusions, pure jump processes, as well as other LĂ©vy processes on compact spaces. Thus, we study multiple sampling patterns for the continuous data space. We show that the stochastic gradient process can approximate the gradient flow minimising the full target function at any accuracy. Moreover, we give convexity assumptions under which the stochastic gradient process with constant learning rate is geometrically ergodic. In the same setting, we also obtain ergodicity and convergence to the minimiser of the full target function when the learning rate decreases over time sufficiently slowly.
July 2022
Amal
Alphonse
Some aspects of elliptic quasi-variational inequalities
Abstract.
Quasi-variational inequalities (QVIs) can be thought of as generalisations of variational inequalities where the constraint set in which the solution is sought depends on the unknown solution itself. In this talk, I'll discuss various aspects of elliptic quasi-variational inequalities of obstacle type including existence results, sensitivity analysis of the source-to-solution map as well as optimal control problems with QVI constraints and associated stationarity systems.
June 2022
Denis
Korolev
Model order reduction techniques for electrical machines
Abstract.
In this talk, I will discuss model order reduction methods for parameterized elliptic and parabolic partial differential equations and their application to the modelling of magnetic fields in electrical machines. If time permits, modern deep learning methods of model order reduction will be discussed.
Clemens
Sirotenko
Dictionary learning for quantitative MRI
Abstract.
A nonlinear inverse problem related to quantitative Magnetic Resonance Imaging (qMRI) is under consideration. In general, qMRI summarizes techniques that aim at extracting physical tissue parameters from a sequence of highly under sampled MR images. Recently, a mathematical setup was introduced that addresses this problem by estimating a set of unknown parameters in a system of ODEs called Bloch equations. While classical approaches such as the Levenberg Marquardt algorithm or Landweber iteration yield good results under small noise levels, numerical experiments show that low sampling rates and noise of large magnitude lead to unsatisfactory outcomes and unstable convergence behavior. Therefore, a spatial regularization approach based on (coupled) dictionary learning is proposed, which has already shown excellent results in the linear inverse problem of classical MRI. From a mathematical viewpoint this ends up in a variety of non convex and non smooth optimization problems. Iterative schemes to solve these problems are discussed and convergence to equilibrium points is studied. Moreover numerical results are presented and open questions such as regularization properties, parameter choice and acceleration strategies are discussed.
May 2022
Sarah
Essadi
From N-player games to mean-field games
Abstract.
We consider deterministic differential games with a large, but finite, population of symmetric interacting players. The interaction term is of mean-field type and exhibits heterogeneity both via the linear dynamics of the players and in their non-smooth cost functionals. We proceed on a first-step with only constraints on the control and with no additional state constraints. We characterise optimal solutions by deriving first-order optimality conditions. However, due to the non-smoothness of the objectives, set-valued mappings appear in the adjoint equation. To overcome this issue, we make use of a Huber-type regularisation. Furthermore, we aim at analysing the asymptotic behaviour of this system, for infinitely many players. This limiting analysis renders possible the construction of approximate Nash equilibria for the N-player games based on a solution of the corresponding mean-field game.
April 2022
Steven-Marian
Stengl
Combined Regularization and Discretization of Equilibrium Problems and Primal-Dual Gap Estimators
Abstract.
In this talk, we adress the treatment of finite element discretizations of a class of equilibrium problems involving moving constraints. Therefore, a Moreau-Yosida based regularization technique, controlled by a parameter, is discussed. A generalized Î-convergence concept is utilized to obtain a priori results. The same technique is applied to the discretization and the combination of both. In addition, a primal-dual gap technique is used for the derivation of error estimators and a strategy for balancing between a refinement of the mesh and an update of the regularization parameter is established. The theoretical findings are illustrated for the obstacle problem as well as numerical experiments are performed for two quasi-variational inequalities with application to thermoforming and biomedicine, respectively.
March 2022
Olivier
Huber
Topics in gas transport: Nash equilibrium and constrained exact boundary controllability
Abstract.
We present two results related to the transport of gas: the existence of a solution to a Generalized Nash Equilibrium Problem (GNEP) arising from the modeling of the gas market as an oligopoly, that is only the producers are players, and the consumers just react to the quantity of gas available. In a second part, the constrained exact boundary controllability of a semilinear hyperbolic PDE is investigated. The existence of an absolutely continuous solution and boundary control will be shown, under appropriate assumptions.
February 2022
Axel
Kröner
Optimal control of a semilinear heat equation subject to state and control constraints
Abstract.
In this talk 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 and the cost functional is of tracking type and contains a linear term in the control variable. We derive second-order necessary and sufficient conditions relying on the concept of alternative costates, quasi-radial critical directions, and the Goh transformation.
December 2021
André
Uschmajew
Optimization on low-rank manifolds
Abstract.
Low-rank matrix and tensor models are important in many applications for representing and embedding high-dimensional data or functions. They typically lead to non-convex optimization problems on sets of matrices or tensors of given rank. In this talk, we give a basic introduction to the geometry of such sets and how it can be used to derive and study optimization algorithms. Compared to direct optimization of the factors in the model, the geometric approach is more intrinsic and can lead to improved methods. For a class of quadratic cost functions on matrices we also discuss how the geometric viewpoint is useful for studying the non-convex optimization landscape under low-rank constraints.
November 2021
Jo Andrea
BrĂŒggemann
On the existence of solutions and solution methods for elliptic obstacle-type quasi-variational inequalities with volume constraints