Research Seminar on Mathematical Optimization / Non-smooth Variational Problems and Operator Equations

Optimization

Tue, 14.07.26 at 10:00

WIAS ESH and online

Alexey Izmailov Brown University, Providence

A Proximal Discontinuous Galerkin Method for Total Bounded Variation Based Image Denoising

Optimization

Thu, 30.04.26 at 10:00

WIAS ESH and online

Yixuan Wang California Institute of Technology

Data-Driven Discovery and Verification of Singularities in Nonlinear Partial Differential Equations

Abstract.

Motivated by the Clay Prize problem on the blowup of Navier-Stokes equations (NSE), I present numerical approaches that facilitate deeper insights into singularity formation, demonstrating that machine learning methods, especially Physics-Informed Neural Networks and Neural Operators, significantly enhance the capability to identify and characterize potential blowups with high precision. Motivated by an exact symbolic search for blowups, we pioneered the Kolmogorov-Arnold Network (KAN) architecture, a novel machine learning paradigm whose interpretability and excellent scaling properties are achieved through learnable nonlinearities, gaining broad recognition in the AI for science community. Building on these insights, I introduced a robust analytical framework to establish blowups with clear stability, with rates automatically inferred, and without explicit spectral information of the linearized operator. This formulation naturally suggests a robust numerical algorithm for tracking singularity formation and beyond, potentially applicable to singularities with multiple scales and complicated systems like NSE. Finally, I will talk about our resolution of a longstanding open problem of nonuniqueness of weak solutions to NSE, which is crucial in the understanding of turbulence, leveraging again the synergy between high-fidelity numerical solutions and rigorous computer-assisted proofs.

Optimization

Mon, 23.03.26 at 14:30

WIAS ESH and online

Xiao Meng Hong Kong Baptist University

Well-posedness and optimal control of a PDE-ODE spatial-network model on metric graphs and sub-domains

Abstract.

Mathematical modeling of dynamics on metric graphs arises in various contexts, from epidemic dynamics to chemical transport in fractured media. Diffusion processes and interactions within complex topological structures pose significant analytical challenges. In practical scenarios, effective intervention strategies are also critical. In this work, we analyze a coupled PDE-ODE system defined on a hybrid structure. The model is formulated as a nonlinear system with junction conditions that capture diffusion in subdomains, along edges, and at vertices. The well-posedness of the system, including the existence, uniqueness, and regularity of solutions, is established via Galerkin approximations and energy estimates. Building on these coupled dynamics, we extend our approach to optimal control. In this framework, a quadratic cost functional penalizes deviations of the state variables from desired targets while accounting for the energetic cost of control actions. Our analysis demonstrates that the state-to-control mapping is Fréchet differentiable, and we derive the corresponding linearized system and adjoint system, along with the first-order optimality conditions.

Optimization

Fri, 27.02.26 at 15:00

WIAS ESH and online

Kei Fong Lam Hong Kong Baptist University

Structural optimization with convex non-differentiable anisotropy: Application to 3D printing

Abstract.

3D printing is an umbrella term for a set of technologies that manufacture highly intricate and complex designs that are not feasible with traditional die-casting or injection molding methods. But despite their popularization in recent years, several limitations prevent further integration of 3D printing into existing production lines. One recurring issue relates to overhangs, which are regions of the constructed object that when placed in a certain orientation extend outwards without any underlying support. Some of these overhangs can deform under their own weight and, if not supported from below, present a risk in damaging the printed object. Conventional wisdom from practitioners says that overhangs whose outer normal makes an angle greater than 135 degrees with the upwards vertical direction should be supported from below with scaffolding. These are then removed after a successful print, but increase the material and processing costs. Another remedy is to modify the design to be self-supporting as much as possible without compromising its intended functionality. In this talk we consider the latter within a structural topology optimization framework. Extending previous studies with a linear elasticity model, we realize an overhang angle constraint with the help of a convex anisotropic perimeter functional, and study the corresponding optimal control problem. Earlier studies with non-convex functionals lead to instabilities termed the "dripping effect", which can be suppressed under our proposed framework. Numerical examples are provided to demonstrate how we discourage designs that develop overhangs not respecting the angle constraint. It turns out that for our approach we have to work with non-differentiable functionals, and thus we turn to subdifferential calculus to derive the first order optimality conditions. If time permits we will touch on a related aspect of 4D printing that builds on these technologies to create designs capable of changing their shape and functionalities via external stimulus. This is a joint work with Harald Garcke (Regensburg), Robert Nurnberg (Trento) and Andrea Signori (Pavia).

Optimization

Thu, 12.02.26 at 15:30

WIAS R406 and online

Sarah Essadi WIAS

Constrained Mean-Field Games under Uncertainty

Abstract.

We consider mean-field games (MFGs) under uncertainty arising from constrained optimal control problems governed by stochastic ordinary differential equations (SODEs). Starting from a finite N-player game of risk-neutral agents whose individual dynamics are described by linear time-invariant ODEs with random parameters, we establish well-posedness of the SODE and prove existence and uniqueness results of optimal solutions. A characterization of optimal controls is obtained via first-order necessary optimality conditions. Passing to the limit as N goes to infinity, the mean-field interaction term is approximated by a probability measure, leading to a limiting MFG formulation. We show the existence of a mean-field equilibrium and prove that it induces an ε-Nash equilibrium for the finite N-game, as N tends to infinity. This provides a rigorous justification of the mean-field approximation for a large population of agents under uncertainty.

Optimization

Thu, 29.01.26 at 10:00

WIAS R405 and Onl...

Clemens Sirotenko WIAS

Learning Elliptic Variational Inequalities via Weak Min-Max Formulations

Abstract.

In this talk, we will present a novel weak adversarial framework for solving obstacle problems using neural networks. By reformulating the problem via (generalised) regularised gap functions into a natural min-max structure, we create a setting well suited to learning based approaches. We will outline the error analysis, highlighting both discretisation and statistical errors. Then, we will explain how parametrising the solution and test functions as neural networks allows us to solve the resulting min-max problem using a modified gradient descent-ascent method. Numerical experiments illustrate the robustness of the method. We will conclude by discussing current limitations, open problems, and future challenges of the approach.

Optimization

Tue, 02.12.25 at 10:00

Online via Zoom

Shida Wang Universität des Saarlandes

Convergence rates of regularized quasi-Newton methods without strong convexity

Abstract.

In this talk, we discuss the convergence rates of the cubic regularized proximal quasi-Newton method for solving non-smooth additive composite problems that satisfy the Kurdyka-Łojasiewicz (KL) property with respect to some desingularizing function, instead of relying on strong convexity. In particular, when the objective function is smooth and satisfies the Polyak-Łojasiewicz (PL) inequality, the algorithm attains a global superlinear convergence. Additionally, we introduce two practical and computationally efficient variants based on limited-memory quasi-Newton techniques.

Optimization

Wed, 02.07.25 at 11:00

WIAS R411 HVP5-7 ...

Eleonora Ficola Universität Hamburg

Existence and duality theory for linear-growth variational problems with measures

Abstract.

We consider functionals F with linear growth in the gradient variable coupled with a non-linear integral term respect to a (possibly signed) Radon measure on bounded domains in Rn. After achieving a generalized parametric lower-semicontinuity result, we then provide necessary and sufficient conditions for existence of BV-minimizers of F, discussing typical examples as well as limit cases. In parallel, we determine the corresponding dual maximization problem set in the class of divergence-measure vector fields and we reformulate the optimality relations in terms of a refined version of Anzellotti's pairing between measures and functions. By introducing a suitable notion of solutions to the Euler-Lagrange equation associated to F, we then demonstrate that our BV theory is complete and it provides natural extension to the Sobolev model. The seminar is based on joint work with Thomas Schmidt (Universität Hamburg).

Optimization

Tue, 20.05.25 at 14:00

WIAS R411 HVP5-7 ...

Max Fröhlich WIAS

Quantum circuit simulation with a localized dynamic time-dependent variational principle

Abstract.

We introduce a novel tensor network simulation method for quantum circuits that addresses key limitations inherent in the widely used time-evolving block decimation algorithm (TEBD). ... Benchmarking against conventional TEBD simulations demonstrates that our local TDVP simulation scheme achieves improved numerical stability, lower bond dimensions with at least the fidelity of TEBD, paving the way for more reliable large-scale quantum circuit simulations.

Optimization

Tue, 11.03.25 at 14:00

WIAS HVP5-7 R411 ...

Marcelo Bongarti WIAS

Multi-objective optimization with linear hyperbolic PDE constraints: generalized Nash equilibrium problems and gas market applications

Abstract.

The concept of Nash equilibrium is fundamental to a wide range of applications, spanning fields from particle mechanics to micro and macroeconomics. ... Finally, we present some recent results on the existence and characterization of equilibria, emphasizing optimality conditions as a framework for understanding such solutions.

Optimization

Tue, 17.12.24

WIAS HVP5-7 R411 ...

Marcelo Bongarti

Multi-objective optimization with linear hyperbolic PDE constraints: generalized Nash equilibrium problems and gas market applications

Abstract.

The concept of Nash equilibrium is fundamental to a wide range of applications, spanning fields from particle mechanics to micro and macroeconomics. However, much of the existing literature focuses on finite-dimensional settings. In this seminar, we draw on energy markets coupled with transport dynamics to motivate the study of multi-objective optimization problems with hyperbolic PDE constraints. We will explore the core ideas and challenges posed by generalized Nash equilibrium problems, particularly those related to dimensionality and regularity. Finally, we present some recent results on the existence and characterization of equilibria, emphasizing optimality conditions as a framework for understanding such solutions.

Optimization

Tue, 26.11.24 at 14:00

WIAS HVP5-7 R411 ...

Denis Korolev WIAS

Hybrid physics-informed neural network based dual-scale solver and its applications to learning-informed upscaling

Abstract.

Inspired by the Liquid Composite Molding process for fiber-reinforced composites and its associated multiscale fluid flow structure, we present a novel framework for optimizing physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs), with applications to dual-scale PDE systems. ... In this talk, we present the application setting, mathematical model, and highlights of its analysis, as well as outline perspectives on developing optimization algorithms for the hybrid framework in the infinite-dimensional setting.

Optimization

Thu, 14.11.24 at 10:00

WIAS HVP5-7 R411 ...

Amal Alphonse WIAS

Free boundary problems as limits of a bulk-surface model for receptor-ligand interactions on evolving domains

Abstract.

We derive various free boundary problems as reaction or singular limits of a coupled bulk-surface reaction-diffusion system on an evolving domain. ... In this talk, I will discuss the modelling, sketch the analysis, show some numerical simulations and finish with some open questions. Based on a joint work with Charlie Elliott, Chandrasekhar Venkataraman and Diogo Caetano.

Optimization

Tue, 22.10.24 at 14:00

WIAS HVP5-7 R411 ...

Ioannis Papadopoulos WIAS

A semismooth Newton method for obstacle-type quasivariational inequalities

Abstract.

Quasivariational inequalities (QVIs) are ubiquitous but, in particular, arise in PDE-constrained optimization in cases where the constraint set depends on the solution itself. ... We will see that the solver enjoys favourable properties such as local superlinear convergence and mesh independence.

Optimization

Tue, 10.09.24 at 10:30

WIAS HVP11 R 3.13...

Aaron Sander Technische Universität München

Verifying the equivalence or non-equivalence of quantum circuits with tensor networks

Abstract.

The development of quantum computers and algorithms is currently rapidly accelerating ... 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.

Optimization

Wed, 26.06.24 at 14:00

WIAS HVP5-7 R411 ...

Antonia Topalovic Humboldt-Universität zu Berlin

Model predictive control for generalized Nash Equilibrium problems

Abstract.

We study model predictive control (MPC) schemes for non-cooperative dynamic games. ... Passing to a limit, we identify a suitable Lyapunov function for MPC schemes based on the original GNEPs.

Optimization

Wed, 29.05.24 at 10:00

WIAS HVP5-7 R411 ...

Max Fröhlich WIAS

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). ... makes this method a new approach to characterizing quantum noise in large systems by learning the corresponding noise parameters.

Optimization

Mon, 27.05.24 at 14:00

WIAS ESH and online

Qi Wang WIAS

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 neural 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.

Optimization

Tue, 07.05.24 at 14:30

WIAS HVP5-7 R411 ...

Martin Brokate Technische Universität München

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.

Optimization

Wed, 21.02.24 at 14:00

WIAS HVP5-7 R411 ...

Ioannis Papadopoulos WIAS

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. ... Underpinning the algorithm is the deflation mechanism. Deflation prevents a Newton-like solver from converging to a solution that has already been discovered.

Optimization

Tue, 19.12.23 at 14:00

WIAS ESH and online

Joachim Rehberg WIAS

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 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

Optimization

Mon, 18.12.23 at 14:00

Online talk and W...

Clarice Poon Mathematics Institute, University of Warwick

Super-resolved Lasso

Abstract.

Super-resolution of pointwise sources is of utmost importance in various areas of imaging sciences. ... 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.

Optimization

Mon, 20.11.23 at 14:00

WIAS R406 and online

Joachim Rehberg WIAS

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 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

Optimization

Tue, 24.10.23 at 10:00

WIAS ESH and online

Amal Alphonse WIAS

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. ... 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).

Optimization

Tue, 17.10.23 at 14:00

WIAS ESH and online

Joachim Rehberg WIAS

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 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

Optimization

Thu, 12.10.23 at 10:15

WIAS R406 and online

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... 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.

Optimization

Tue, 29.08.23 at 14:00

WIAS ESH and online

Robert Baraldi Sandia National Laboratories, USA

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. ... We demonstrate its efficacy on examples from data science and PDE-constrained optimization.

Optimization

Thu, 20.07.23 at 14:00

WIAS R406 and online

Olaf Klein WIAS

Uncertainty quantification for models involving hysteresis operators

Abstract.

Parameters within models involving hysteresis operators ... These are results of a joined work with Carmine Stefano Clemente and Daniele Davino of the Università degli Studi del Sannio, Benevento, Italy and Ciro Visone of Università di Napoli Federico II, Napoli, Italy.

Optimization

Mon, 10.07.23 at 14:00

Online

Patrick Jaap WIAS

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 Sobolev spaces. ... An example implementation is given for a thermoforming model from a recent paper. To verify the solver, the results of this model are reproduced.

Optimization

Tue, 20.06.23 at 14:00

WIAS ESH and online

Evelyn Herberg Heidelberg University, Germany

Deep Learning with variable time stepping

Abstract.

Feature propagation in Deep Neural Networks (DNNs) can be associated to nonlinear discrete dynamical systems. ... The proposed approach is applied to an ill-posed 3D-Maxwell's equation.

Optimization

Tue, 06.06.23 at 14:00

WIAS HVP5-7 and o...

Denis Korolev WIAS

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. ... We discuss physics-informed neural networks, the numerical homogenisation modelling framework and related concepts.

Optimization

Thu, 25.05.23 at 14:00

WIAS ESH and online

Brendan Keith Brown University, USA

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 overall latent variable proximal Galerkin combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry.

Optimization

Wed, 24.05.23 at 15:15

WIAS ESH (joint w...

Pavel Krejči Technical University, Prague, Czechoslovakia

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. ... This is a joint work with Chiara Gavioli from TU Wien.

Optimization

Thu, 04.05.23 at 14:00

WIAS ESH and online

Eskil Hansen Lund University, Sweden

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. ... This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains.

Optimization

Tue, 25.04.23 at 14:00

WIAS HVP5-7 R411 ...

Caroline Geiersbach WIAS

Optimality conditions for problems with probabilistic state constraints

Abstract.

In this talk, we discuss optimization problems subject to probabilistic constraints. ... Perspectives for the numerical solution of these problems are discussed, as well as planned research directions.

Optimization

Wed, 22.03.23 at 10:00

WIAS R406 and online

Constantin Christof Technical University of Munich, Germany

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. ... 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 functions.

Optimization

Thu, 16.02.23 at 14:00

Online talk and W...

Christian Parkinson University of Arizona, USA

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. ... We demonstrate all of our methods with several examples.

Optimization

Thu, 26.01.23 at 10:00

WIAS HVP5-7 R411 ...

Clemens Sirotenko WIAS

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. ... Moreover numerical results and open questions are presented.

Optimization

Mon, 19.12.22 at 15:00

WIAS R406 and online

Mike Theiss WIAS

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 the end, we will discuss some ideas on how to solve such constrained MFGs.

Optimization

Thu, 10.11.22 at 11:00

WIAS HVP5-7 R411

Jonas Latz Heriot-Watt University, Edinburgh, Scotland

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. ... 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.

Optimization

Mon, 11.07.22 at 15:00

WIAS HVP5-7 R411

Amal Alphonse WIAS

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. ... and associated stationarity systems.

Optimization

Wed, 22.06.22 at 14:15

WIAS HVP5-7 R411

Denis Korolev WIAS

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.

Optimization

Wed, 08.06.22 at 14:00

WIAS HVP5-7 R411

Clemens Sirotenko WIAS

Dictionary learning for quantitative MRI

Abstract.

A nonlinear inverse problem related to quantitative Magnetic Resonance Imaging (qMRI) is under consideration. ... Numerical experiments are performed for two quasi-variational inequalities with application to thermoforming and biomedicine, respectively.

Optimization

Tue, 31.05.22 at 14:00

WIAS HVP5-7 R411

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.

Optimization

Mon, 16.05.22 at 14:00

WIAS HVP5-7 R411

Marcelo Bongarti WIAS

Nonlinear Transport in Gas Networks

Abstract.

In this talk, we discuss the nonlinear transport of gas in a network of pipelines. The evolution of the gas distribution on a given pipe is modeled by a suitable isothermal Euler semilinear system in one space dimension. On the network, solutions satisfying the so-called Kirchoff flux continuity conditions at the nodes are shown to exist within the vicinity of an equilibrium state. 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.

Optimization

Fri, 15.04.22

Online

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 Γ-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.

Optimization

Sat, 12.03.22 at 13:00

WIAS-R 406

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.

Optimization

Sun, 27.02.22 at 11:00

WIAS ESH

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.

Optimization

Thu, 16.12.21 at 14:00

WIAS ESH

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.

Optimization

Sun, 21.11.21

WIAS ESH

Jo Andrea Brüggemann

On the existence of solutions and solution methods for elliptic obstacle-type quasi-variational inequalities with volume constraints

Abstract.

In this talk, an elliptic obstacle-type quasi-variational inequality (QVI) with volume constraints is studied. This type of QVI is motivated by the reformulation of a compliant obstacle problem, where two elastic membranes are subject to external forces while enclosing a constant volume. The existence of solutions to this QVI is established building on fixed-point arguments and partly on the concept of Mosco-convergence. Since Mosco-convergence of the considered feasible sets usually requires complete continuity or compactness properties of the obstacle map, a two-fold approach is explored towards generalising the available existence results for the considered QVI. Based on the analytical findings, the solution of the QVI is approached by solving a sequence of variational inequalities (VIs). Each of these VIs is tackled in function space via a path-following semismooth Newton method. An a posteriori error estimator is derived towards enhancement of the algorithm's numerical performance by using adaptive finite element methods.

Optimization

Wed, 15.04.20 at 14:00

Online

Steven-Marian Stengl WIAS

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 Γ-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.

Optimization

Thu, 12.03.20 at 13:00

WIAS R406

Olivier Huber HU Berlin, Germany

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.

Optimization

Thu, 27.02.20 at 11:00

WIAS ESH

Axel Kröner WIAS

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. ... We derive second-order necessary and sufficient conditions relying on the concept of alternative costates, quasi-radial critical directions, and the Goh transformation.

Optimization

Mon, 16.12.19 at 14:00

WIAS ESH

André Uschmajew MPI Leipzig, Germany

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. ... 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.

Optimization

Thu, 21.11.19 at 11:00

WIAS ESH

Jo Andrea Brüggemann WIAS

On the existence of solutions and solution methods for elliptic obstacle-type quasi-variational inequalities with volume constraints

Abstract.

In this talk, an elliptic obstacle-type quasi-variational inequality (QVI) with volume constraints is studied. ... An a posteriori error estimator is derived towards enhancement of the algorithm's numerical performance by using adaptive finite element methods.

Optimization

Fri, 01.11.19 at 14:00

WIAS ESH

Ana Djurdjevac FU Berlin, Germany

Random PDEs on moving hypersurfaces

Abstract.

It is well-known that in a variety of applications, especially in the biological modeling, PDEs that appear can be better formulated on evolving curved domains. ... Our theoretical convergence rates are confirmed by numerical experiments. This work is supported by DFG through project AA1-3 of MATH+.

Optimization

Wed, 23.10.19 at 11:00

WIAS HVP11A R 3.13

Martin Holler University Graz, Austria

A variational model for learning convolutional image atoms from incomplete data

Abstract.

Using lifting and relaxation strategies, we present a convex variational model for learning a convolutional sparse representation of image data via a few basic atoms. ... such an atom-based representation is computed from incomplete, noisy and blurry data.

Optimization

Thu, 26.09.19 at 15:00

WIAS R406

Martin Brokate Technical University Munich, Germany

Sensitivity in rate independent evolutions

Abstract.

As a topic in science, rate independent evolutions have appeared more than 100 years ago; their study as a mathematical subject in its own began in the 1960's. ... we will present in particular the question of differential sensitivity, that is, whether the associated solution operators possess weak derivatives.

Optimization

Fri, 02.08.19 at 15:15

WIAS ESH

Ya-xiang Yuan Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Efficient optimization algorithms for large scale data analysis

Abstract.

In this talk, two classes of problems in large scale data analysis and their optimization algorithms will be discussed. ... Numerical results of applications, e.g., electronic structure calculations, l1-regularized logistic regression problems, Lasso problems and Hartree-Fock total energy minimization problems, will be highlighted.

Optimization

Thu, 11.07.19 at 15:00

WIAS-405-406

Georg Stadler Courant Institute of Mathematical Sciences, New York university

Sparse optimal control of PDEs with uncertain coefficients

Abstract.

I will discuss sparse solutions of optimal control problems governed by elliptic PDEs with uncertain coefficients. Sparsity of controls is achieved by incorporating the L^1-norm of the mean of the pointwise squared controls in the objective. The main focus is on stochastic controls that share the same sparsity structure, i.e., controls that depend on the realization of the random parameters but have identical support. We propose an iterative norm reweighting formulation, which iterates over functions defined over the physical space only and thus avoids approximation of the random space. Combining a Newton method with low-rank operator approximations, this results in an efficient solution method that avoids approximation of the uncertain parameter random space. The qualitative structure of the optimal controls and the performance of the solution algorithm are studied numerically using control problems governed by the Laplace and Helmholtz equations. This is joint work with Chen Li (NYU).

Optimization

Wed, 10.07.19 at 13:00

WIAS ESH

Georg Stadler Courant Institute of Mathematical Sciences, New York university

Estimation of extreme event probabilities by combining large deviation theory and PDE-constrained optimization, with application to tsunami waves

Abstract.

Tsunami waves are caused by a sudden change of ocean depth (bathymetry) after an earthquake below the ocean floor. ... Preliminary numerical results with the 1D inviscid shallow water equation are presented. This is joint work with Shanyin Tong and Eric Vanden-Eijnden (both NYU).

Optimization

Mon, 03.06.19 at 11:30

WIAS ESH

Guozhi Dong Humboldt Universität zu Berlin

Quantitative magnetic resonance imaging: From fingerprinting to integrated physics-based models

Abstract.

In this talk, we introduce a novel method for quantitative MRI. ... The efficiency of our new method is proved theoretically and also verified by numerical examples.

Optimization

Mon, 03.06.19 at 11:00

WIAS ESH

Selma Metzer PTB - TU Berlin

Approximate large-scale Bayesian inference with application to magnetic resonance fingerprinting

Abstract.

A class of nonlinear, large-scale regression problems is considered where the parameters model the spatial distribution of some property. ... Finally, the approach is applied to MRF. In using simulated data with known ground truth, it is shown that by using the prior knowledge of smoothness in the spatial distribution of the sought parameters, the results are significantly better than those achieved through maximum likelihood estimation.

Optimization

Wed, 23.01.19 at 13:00

WIAS ESH

Maria Soledad Aronna EMAp/FGV - TU Berlin

On second order optimality conditions for control-affine problems: the finite and infinite dimensional case

Abstract.

In this talk I will present the main features of first and second order optimality conditions for optimal control problems of ordinary differential equations that are affine with respect to the control. ... Finally, if time allows it, I will briefly discuss the state-constrained case.

Optimization

Wed, 16.01.19 at 13:00

WIAS ESH

Tobias Keil WIAS Berlin

Optimal control of a coupled Cahn-Hilliard-Navier-Stokes system with variable fluid densities

Abstract.

This talk is concerned with the optimal control of two immiscible fluids with non-matched densities. ... The method is based on an adaptation of a bundle-free implicit programming approach for MPECs in function space presented by Hintermüller and Surowiec in 2016.

Optimization

Wed, 19.12.18 at 11:00

WIAS ESH

O. Huber WIAS

Using reformulations in nonsmooth mathematical programming: the examples of friction contact problems and optimal value functions

Abstract.

In this talk we illustrate how reformulations can be used to deal with nonsmoothness. In particular, we look at two specific structures arising from applications: the first one concerns finite dimensional variational inequalities with a nonsmooth functional. This is motivated by the study of contact problems with Coulomb friction. The second example involves optimization models with Optimal Value Functions (OVF) in the problem data. The OVF concept subsumes regularizers from fitting problems and coherent risk measures from Stochastic Optimization. Each reformulation scheme rely on tools from convex and variational analysis. In both instances, we obtain equivalent problems that are solvable by off-the-shelf solvers. Finally, numerical results are presented and discussed: in the OVF case, the example is a risk-averse equilibrium problem from the electricity market.

Optimization

Fri, 30.11.18 at 13:00

WIAS ESH

S.-M. Stengl WIAS

Uncertainty Quantification of the Ambrosio--Tortorelli approximation in image segmentation

Abstract.

In this talk we want to deal with quantification of uncertainties in image segmentation based on the Mumford--Shah model. The aim is to address the error propagation of noise and other error types in the original image to the segmentation and especially the edges. We analyze therefore the in the literature well-known Ambrosio--Tortorelli approximation and discuss the existence of measurable selections of its solutions as well as sampling-based methods. We end the talk with numerical examples.

Optimization

Mon, 10.09.18 at 13:30

WIAS ESH

K. Ito North Carolina State University

Semismooth Newton method for variational inequalities and MPEC

Optimization

Wed, 20.06.18 at 13:00

WIAS-Mo 39, 4th f...

L. Goubergrits Charité - Universitätsmedizin Berlin

Modelling and simulation for treatment planning: CFD methods for valve treatment

Abstract.

In aging population, the prevalence of heart valve diseases and heart failure with more than 8 per cent of the population in their 70s (valve disease) and 10 per cent in their 80s (heart failure) is one of the most relevant diseases for the healthcare system in industrial countries. Both diseases are chronic and can amplify each other. Wrongly treated valve diseases can lead to severe heart failure and vice versa. The prognosis of heart failure is still very poor (6-year mortality rate more than 67 per cent). CFD approach promises precise diagnosis without invasive procedures. Furthermore, CFD allows predictiv modelling allowing to support clinicians with treatment decision as well as treatment planing and optimization. Finally CFD approach promises risk and cost minimization. Current CFD abilities, challenges and requirements for CFD translation into the clinical practice are presented and discussed.

Optimization

Mon, 18.06.18 at 15:00

WIAS-Mo 39, 4th f...

J. Pfefferer TU Munich

hp-finite elements for fractional diffusion

Abstract.

In this talk we introduce and analyze a numerical scheme based on hp-finite elements to solve boundary value problems involving the spectral fractional Laplacian. The approach is based on a reformulation of the problem posed on a semi-infinite cylinder in one more spatial dimension. After a suitable truncation of this cylinder, the resulting problem is discretized with linear finite elements in the original domain and with hp-finite elements in the extended direction. The proposed approach yields a reduction of the computational complexity in terms of degrees of freedom and even has slightly improved convergence properties compared to the state-of-the-art discretization using linear finite elements for both the original domain and the extended direction. The performance of the method is illustrated by numerical experiments.

Optimization

Wed, 13.06.18 at 13:00

WIAS ESH

K. Welker U Trier

Constrained shape optimization problems in shape spaces

Abstract.

Shape optimization problems arise frequently in technological processes which are modelled in the form of partial differential equations (PDEs) or variational inequalities (VIs). In many practical circumstances, the shape under investigation is parametrized by finitely many parameters, which on the one hand allows the application of standard optimization approaches, but on the other hand limits the space of reachable shapes unnecessarily. In this talk, the theory of shape optimization is connected to the differential-geometric structure of shape spaces. In particular, efficient algorithms in terms of shape spaces and the resulting framework from infinite dimensional Riemannian geometry are presented. In this context, the space of H1/2-shapes is defined. The H1/2-shapes are a generalization of smooth shapes and arise naturally in shape optimization problems. Moreover, VI constrained shape optimization problems are treated from an analytical and numerical point of view in order to formulate approaches aiming at semi-smooth Newton methods on shape vector bundles. Shape optimization problems constrained by VIs are very challenging because of the necessity to operate in inherently non-linear and non-convex shape spaces. In classical VIs, there is no explicit dependence on the domain, which adds an unavoidable source of non-linearity and non-convexity due to the non-linear and non-convex nature of shape spaces.

Optimization

Mon, 28.05.18 at 14:00

WIAS ESH

F. J. Romero Hinrichsen ETH Zurich

Dynamical super-resolution with applications to ultrafast ultrasound

Abstract.

Recently there has been a successful development in ultrasound imaging, increasing significantly the sampling rate and therefore enhancing this imaging's capacities. In particular, for vessel imaging, the use of microbubble tracking allows us to super-resolve blood vessels, and by estimating the particles' speeds inside them, it is possible to calculate the vessels' diameters. In this context, we model the microbubble tracking problem, formulating it in terms of a sparse spike recovery problem in the phase space (the position and velocity space), that allows us to obtain simultaneously the speed of the microbubbles and their location. This leads to an L1 minimization algorithm for point source tracking, that promises to be faster than current alternatives.

Optimization

Wed, 23.05.18 at 13:00

WIAS ESH

A. Alphonse WIAS

Directional differentiability for elliptic QVIs of obstacle type

Abstract.

Quasi-variational inequalities (QVIs) are generalisations of variational inequalities where the associated constraint set is no longer explicitly given but it depends on the solution itself. In this talk, we present some work on the directional differentiability of the multi-valued mapping that takes the source term of a QVI onto the set of solutions. This result represents a first step in the study of differential sensitivity of QVIs in infinite dimensions. We also discuss an application to thermoforming and show some numerical experiments.

Optimization

Wed, 16.05.18 at 13:00

WIAS ESH

L. Banz U Salzburg

A posteriori error estimates for $hp$-dual mixed finite elements by implicit reconstruction

Abstract.

A posteriori error estimates are derived for dual mixed methods of $hp$-adaptive finite elements for variational equations as well as for variational inequalities. The error control relies on the use of a special, but never computed, $H^1$-reconstruction of the non-smooth discrete potential. Thus, no post-processing reconstruction (and therewith no additional computation) is needed. Moreover, the use of the discrete potential instead of its reconstruction improves significantly the error estimation in terms of the numerical efficiency indices which are nearly constant and close to one in the numerical experiments. Numerical experiments demonstrate the convergence rates and the efficiency indices of these a posteriori error estimates in $h$- and $hp$-adaptivity.

Optimization

Wed, 18.04.18 at 15:00

WIAS-Mo 39, 4th f...

P. Dvurechensky WIAS

First-order optimization methods with inexact information about the objective function value and its gradient

Abstract.

In this talk I will discuss first-order methods with inexact oracle for finite-dimensional optimization. Oracle model of optimization methods assumes that, given a point, the oracle returns some information on the objective function at this point. In the case of first-order optimization methods, this information is the function value and its gradient at this point. I will start with convex problems, inexact oracle defined in the work by O. Devolder, F. Glineur, Yu. Nesterov, Math. Prog., 2014, and convergence rates for gradient descent and accelerated gradient descent in this case. I will also describe an extension for non-convex problems. Then I will discuss some ideas on how these methods potentially can be extended and applied for infinite-dimensional problems. If time allows, I will cover other optimization problems and methods, which I work with. Among others, optimal transport problem and an accelerated gradient descent for its solution, randomized optimization methods, such as random coordinate descent and random derivative-free method, variational inequalities, saddle-point problems and first-order methods for their solution.

Optimization

Fri, 02.02.18 at 14:30

WIAS ESH

Th. Ruf U Augsburg

On a variational approach to the nonlinear wave equation

Abstract.

In a 2012 paper, E. Serra and P. Tilli proved a conjecture by E. de Giorgi stating that global weak solutions of wave equations such as w'' - Delta w + w|w|^(p-2) = 0 on R^+ times R^d can be obtained as limits of minimizers of suitable variational functionals. We generalize this proof to equations of the form w'' - Delta w + f_w (t; x; w) = 0 with p-growth conditions on f, also replacing R^d by an arbitrary open set O subset R^d and suitable boundary conditions.

Optimization

Fri, 19.01.18 at 14:00

WIAS ESH

S. Bartels Albert-Ludwigs-Universität Freiburg

Finite element methods for nonsmooth problems and application to a problem in optimal insulation

Abstract.

Nonsmooth problems arise in the mathematical modeling of contact and obstacle problems, the description of plastic material behavior, and mathematical image processing. The unknown functions are typically characterized as minimizers of nondifferentiable functionals. Numerical schemes approximately solve these problems either via duality methods or classically by making use of appropriate regularizations. In the talk we discuss the discretization and iterative solution of a model problem defined on functions of bounded variation. The numerical analysis of finite element discretizations leads to reduced convergence rates which can be improved using adaptive mesh refinement. Suitable iterative solution procedures are ADMM schemes, for which we propose an automatic step size adjustment strategy, and gradient flows, for which we demonstrate the unconditional stability of a semi-implicit time discretization. The methods are applicable in the numerical determination of optimal insulating films for heat conducting bodies. Below a critical value of available insulation mass an unexpected break of symmetry occurs.

Optimization

Thu, 11.01.18 at 10:00

WIAS ESH

T. Kluth Universität Bremen

Model-based magnetic particle imaging

Abstract.

Magnetic particle imaging (MPI) is a tracer-based imaging modality developed to detect the concentration of superparamagnetic iron oxide nanoparticles. It is highly sensitive to the nanoparticle's nonlinear response to a dynamic applied magnetic field. Model-based reconstruction techniques are still not able to reach the quality of data-based approaches in which the linear system function is determined by a time-consuming measurement process. Possible reasons include the relaxation behavior of nanoparticles in fast changing magnetic fields. However, the equilibrium model described by the Langevin function is still used to predict the system behavior. In this context we discuss the ill-posedness of the imaging problem. We further focus on the model-based MPI reconstruction problem incorporating deviations in the forward operator. This is illustrated by initial results from real data using a regularized total least squares approach.

Optimization

Thu, 21.12.17 at 14:15

WIAS, HVP11A, 4.13

Guozhi Dong HU Berlin

Regularization methods and nonlinear PDEs for solving inverse and imaging problems

Abstract.

In this talk, I will present some recent developments on regularization methods in image sciences. I shall show a tiny background and also mention some of the state of the art in this area. The focus will be then to discuss some non-convex regularization models for specific problems which suffer from displacement errors. The non-convex energy functionals reveal to have tight connections to some nonlinear (geometric) partial differential equations, e.g. mean curvature flows. Finally, I will show some numerical results with some discussions on the algorithms.

Optimization

Mon, 27.11.17 at 09:30

WIAS, HVP11A, 4.13

Andrea Ceretani HU Berlin

Anomalous diffusion with free boundaries

Abstract.

Recently, time fractional Stefan-like problems have been used to model anomalous diffusion with free boundaries and long memory retention. Nevertheless, the physical foundations of this usage is still unclear. We present a model for acid water neutralization with anomalous and fast diffusion. Though this problem presents short memory retention, it is a first step in deriving mathematical models for anomalous diffusion under memory effects based on commonly accepted physical laws. The problem consists in the neutralization of an acid solution in which the hydrogen ions are transported according to Cattaneo's diffusion, and we consider the specific case of a marble slab reacting with a sulphuric acid solution in a one-dimensional geometry. The mathematical problem is reduced to a hyperbolic free boundary problem where the consumption of the slab is described by a nonlinear differential equation. We prove global well-posedness and present some numerical simulations.

Optimization

Thu, 12.10.17 at 10:30

HU Adlershof, roo...

Stephan Schmidt University of Wuerzburg

SQP Methods for shape optimization based on weak shape Hessians

Abstract.

Many PDE constrained optimization problems fall into the category of shape optimization, meaning the geometry of the domain is the unknown to be found. Most natural applications are drag minimization in fluid dynamics, but many tomography and image reconstruction problems also fall into this category. The talk introduces shape optimization as a special sub-class of PDE constraint optimization problems. The main focus here will be on generating Newton-like methods for large scale applications. The key for this endeavor is the derivation of the shape Hessian, that is the second directional derivative of a cost functional with respect to geometry changes in a weak form based on material derivatives instead of classical local shape derivatives. To avoid human errors, a computer aided derivation system is also introduced. The methodologies are tested on problem from fluid dynamics and geometric inverse problems.

Optimization

Wed, 26.07.17 at 13:15

WIAS ESH

Hasnaa Zidani ENSTA ParisTech

Multi-objective control problems under state constraints

Abstract.

In this talk, we shall present a new approach, based on Hamilton-Jacobi theory, for characterizing the Pareto front for multi-objective optimal control problems in presence of state constraints. We define an auxiliary control problem for an augmented dynamical system and show that the pareto front is a subset of the zero level set of the auxiliary value function. This characterization allows to deduce an efficient numerical procedure for computing the entire Pareto front and the corresponding optimal trajectories. Moreover, the approach allows to consider objective functions of different structures (minimum time cost, Bolza cost and/or infinite horizon objective). A numerical example will be considered to show the relevance of this approach.

Optimization

Thu, 22.06.17 at 09:30

HU Berlin, Adlers...

Caroline Löbhard WIAS

An adaptive space-time discretization for parabolic optimal control problem with state constraints

Abstract.

We present a space-time discretization which is based on a reformulation of the stationarity conditions of a Moreau-Yosida regularized parabolic optimal control problem, which involves only the state variable. The resulting nonlinear partial differential equation is of fourth order in space, and of second order in time. In order to cope with the disbalance of regularity of the respective solutions, we develop a taylored discontinuous Galerkin scheme and derive convergence rates in the mesh size as well as an integrated update strategy for the regularization parameter related to the state constraints. We also propose an adaptive mesh refinement strategy and illustrate the performance of our method in numerical test cases.

Optimization

Wed, 21.06.17 at 15:30

WIAS, HVP11A, 4.13

Mattia Bongini Barcelona Graduate School of Economics

Optimal Control Problems in Transport Dynamics

Abstract.

In this talk we shall discuss several results concerning the “indirect control of populations”, i.e., how to influence a group of individuals by means of external agents with a directly controlled dynamics. By using the general theory of functionals defined on spaces of measures, we give sufficient conditions for the existence of optimal control strategies and then we present a Pontryagin Maximum Principle for such controls in the form of an Hamiltonian flow in the Wasserstein space of probability measure. Finally, we present an application of the above framework to the evacuation problem of a crowd from an unknown environment with the help of undercover stewards.

Optimization

Wed, 31.05.17 at 13:15

WIAS, HVP11A, 4.13

Jo Andrea Brüggemann HU Berlin / WIAS

Solution methods for a contact model motivated by the human heart's pericardium

Abstract.

Introducing a function space description of the contact within the human heart's pericardium, in this talk, we will study a relaxation of the latter. The quasi-variational inequality (QVI) in the focus of interest is attacked with a fixed-point approach and the hereby arising sequence of variational problems can be efficiently solved with a path-following semi-smooth Newton method.

Optimization

Wed, 24.05.17 at 13:15

WIAS ESH

Hongpeng Sun Renmin University of China

Weak convergence of proximal ADMM and its relaxations in Hilbert spaces

Abstract.

ADMM (Alternating Direction Method of Multipliers) is a popular first order method for mathematical imaging and inverse problems. However, the weak convergence of ADMM in infinite dimensional spaces is not clear yet, which is different from the classical Augmented Lagrangian Method. We will discuss the weak convergence of ADMM and its proximal variants with relaxations in Hilbert spaces.

Optimization

Tue, 09.05.17 at 15:30

WIAS, HVP11A, 4.13

Kazufumi Ito North Carolina State University, USA

Value function calculus and applications

Abstract.

In this talk the sensitivity analysis is discussed for the parameter-dependent optimization. The sensitivity of the optimal value function with respect to the change in parameters plays a significant role in the optimization theory, including economics, finance, the Hamilton-Jacobi theory, the inf-sup duality and the structural design and the bi-level optimization. We develop the calculus for the value function and present its applications in the variational calculus, the bi-level optimization and the optimal control and optimal design and inverse problems.

Optimization

Fri, 05.05.17 at 14:15

WIAS ESH

Behzad Azmi Univ. Graz, Austria

On the stabilizability of infinite dimensional systems via receding horizon control

Abstract.

One efficient strategy for dealing with optimal control problems on an infinite time horizon is the receding horizon framework. In this approach, an infinite horizon optimal control problem is approximated by a sequence of finite horizon problems in a receding horizon fashion. Stability is not generally ensured due to the use of a finite prediction horizon. Thus, in order to ensure the asymptotic stability of the controlled system, additional terminal cost functions and/or terminal constraints are often needed to add to the finite horizon problems. In this presentation, we are concerned with the stabilization of several classes of infinite-dimensional controlled systems by means of a Receding Horizon Control (RHC) scheme. In this scheme, no terminal costs or terminal constraints are used to ensured the stability. The key assumption is the stabilizability of the underlying system. Based on this condition the suboptimality and stability of RHC are investigated. To justify the applicability of this framework, we consider controlled systems governed by different types of partial differential equations. Numerical examples are presented as well.

Optimization

Thu, 04.05.17 at 09:30

HU Berlin, Adlers...

Axel Kröner HU Berlin / CMAP, Ecole Polytechnique, Paris-Saclay

Optimal control of infinite dimensional systems

Abstract.

In this talk we analyze second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup operator, the control entering linearly in the cost function. We derive first and second order optimality conditions, taking advantage of the Goh transform. The general framework allows the application to heat, wave, and Schrödinger equation. This is joint work with S. Aronna und F. Bonnans.

Optimization

Wed, 29.03.17 at 13:15

WIAS ESH

Christian Clason Universität Duisburg-Essen

Convex relaxation of hybrid discrete-continuous control problems

Abstract.

We consider control problems for partial differential equations where the distributed control should take on values only from a given discrete and hence non-convex set. Such problems occur for example in parameter identification or topology optimization. Similar to their use in sparse optimization, L1-type norms can be used to formulate a convex relaxation which can be solved by semi-smooth Newton methods. We illustrate this approach using linear model problems and discuss the extension to vector-valued and nonlinear control problems.

Optimization

Tue, 07.02.17 at 10:15

WIAS ESH

Sara Merino-Aceituno Imperial College London, UK

Kinetic theory to study emergent phenomena in biology: an example on swarming

Abstract.

Classical methods in kinetic theory are challenged when studying emergent phenomena in the biological and social sciences. New methodologies are needed to study the problems at hand which typically involve many agents that interact locally. The aim of this talk is to introduce the general framework of kinetic theory and some of the new challenges of applying it to biological systems. We illustrate it in the case of the so-called collective motion of self-propelled particles, like swarming of birds. Particularly, based on the Vicsek model, we study systems of agents (birds) that move at a constant speed while trying to align their body orientation with those of their neighbors. Starting from a particle description, we find the macroscopic dynamics.

Optimization

Tue, 17.01.17 at 10:15

WIAS ESH

Tim Sullivan FU Berlin

Well-Posedness of Bayesian Inverse Problems - Stable Priors on Quasi-Banach Spaces

Abstract.

The Bayesian perspective on inverse problems has attracted much mathematical attention in recent years, and particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an infinite-dimensional space, a typical example being a scalar or tensor field coupled to some observed data via an ordinary or partial differential equation. Numerical solution of such infinite-dimensional BIPs must necessarily be performed in an approximate manner on a finite-dimensional subspace, but it is profitable to delay discretisation to the last possible moment and consider the original infinite-dimensional problem as the primary object of study, since infinite-dimensional well-posedness results and algorithms descend to any finite-dimensional subspace in a discretisation-independent way, whereas careless early discretisation may lead to a sequence of well-posed finite-dimensional BIPs or algorithms whose stability properties degenerate as the discretisation dimension increases. This presentation will give an introduction to the framework of well-posed BIPs in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451-559, 2010) and others. Recently, this framework has been extended to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen-Loeve expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger and total variation metrics upon perturbations of the misfit function and observed data.

Optimization

Tue, 10.01.17 at 10:15

WIAS ESH

Andrew Lam Uni Regensburg

Diffuse interface models of tumor growth and optimizing cancer treatment times

Abstract.

There has been a recent focus in modeling tumor growth with diffuse interface models, due to their ability to capture topological transitions and the nature of the equations allows for further mathematical treatment. In the first part of this talk I will introduce a class of Cahn-Hilliard systems that are used to capture the basic dynamics of tumor growth. Then, we will discuss an optimal control problem for chemotherapy, which is a cancer treatment using drugs to eliminate tumor cells. Treatments are usually conducted in cycles, and long treatment times may cause harm to the patient. Thus, it is important to optimize both the treatment time and drug dosage to minimize patient suffering. In the second part of this talk, we will analyze an optimal control problem with an objective functional depending on a free time variable, which represents the unknown treatment time to be optimized.

Optimization

Tue, 13.12.16 at 10:15

WIAS ESH

Amal Alphonse WIAS

Existence for a fractional porous medium equation on an evolving surface

Abstract.

In this talk, which is based on joint work with Prof. Charlie Elliott, I will present an existence theory for a porous medium equation with a fractional diffusion on an evolving surface. The nonlocal nature of the fractional diffusion (which in our case is the square root of the Laplacian) in combination with the nonlinearity and the moving domain makes the problem interesting. After defining the fractional Laplacian and giving a Dirichlet-to-Neumann characterisation of it in a general setting of closed Riemannian manifolds, I will define what we mean by a weak solution and then proceed with the proof of existence. This will involve harmonic extensions on semi-infinite and truncated cylinders, convergence/decay estimates and some technical results in order to deal with the time-evolving surface. I will finish by discussing some ideas for further work.

Optimization

Tue, 06.12.16 at 10:15

WIAS ESH

Luca Calatroni Ecole Polytechnique, Paris

Infimal convolution of data discrepancies for mixed noise removal in images

Abstract.

In several real-world imaging applications such as microscopy, astronomy and medical imaging, transmission and/or acquisition faults result in a combination of multiple noise statistics in the observed image. Variational data discrepancy models designed to deal with such mixtures linearly combine standard data fidelities used for single-noise removal or make use of either approximated and cheap or exact but computationally expensive log-likelihood functionals. Via a joint MAP estimation, we derive a statistically consistent variational model combining data fidelities associated to single noise distributions in a handy infimal convolution fashion by which individual noise components corrupting the data are modeled appropriately and decomposed from each other. After showing the well-posedness of the model in suitable function spaces, we propose a semi-smooth Newton-type scheme to compute its numerical solution efficiently.

Optimization

Tue, 29.11.16 at 10:15

WIAS, HVP11A 4.13

Martin Kanitsar University of Graz, Austria

Numerical shape optimization for an industrial application

Abstract.

For industrial applications in fluid dynamics, finding an optimal shape with respect to some cost functional is important. Moreover, constraints due to inflow and outflow boundaries as well as a respecified construction space need to be taken care of. For the implementation of a gradient descent scheme, the shape sensitivity calculus is used to derive the shape gradient of the cost functional with respect to changes in the shape. The underlying physics are described by the stationary Navier-Stokes equation, the primal equation, and an adjoint equation is used for calculating the shape derivative. Highlighting some details on the numerical realization for a 3D application is the main part of this talk, but results on existence of an optimal shape will be pointed out too. In addition to illustrating interests of the industry, hints are given for appropriate usage of the software OpenFOAM and Star-CD CCM+.

Optimization

Tue, 22.11.16 at 10:15

WIAS ESH

Soheil Hajian WIAS/HU Berlin

Total variation diminishing RK methods for the optimal control of conservation laws

Abstract.

Optimal control problems subject to conservation laws are among challenging optimization problems due to difficulties that arise when shocks are present in the solution. Such optimization problems have application, for instance, in gas networks. Beside theoretical difficulties at the continuous level, discretization of such optimal control problems should be done with care in order to guarantee convergence. In this talk, we present stability results of the total variation diminishing (TVD) Runge-Kutta (RK) methods for such optimal control problems. In particular we show that enforcing strong stability preserving (SSP) for both forward and adjoint problem results to a first order time-discretization. However, requiring SSP only for forward problem is sufficient to obtain a stable discrete adjoint. We also present order-conditions for the TVD-RK methods in the optimal control context.

Optimization

Tue, 15.11.16 at 10:15

WIAS ESH

M. Hintermüller WIAS/HU Berlin

Bilevel optimization with applications to image processing

Optimization

Fri, 21.10.16 at 15:00

WIAS ESH

M. Holler Karl-Franzens-Universität Graz, Austria

Higher order regularization and applications to medical image processing and data decompression

Abstract.

Variational methods are a powerful tool for tackling ill-posed problems in image processing. As such, they rely heavily on appropriate regularization terms which render a stable recovery possible and strongly influence qualitative solution properties. In this talk, we consider regularization concepts for both static and dynamic data that are based on higher order differentiation. Beginning with the static setting, we first discuss analytical properties of Total Generalized Variation (TGV) regularization which allow for well-posedness results for standard inverse problems. We then consider the application of TGV in the context of a variational model for image decompression, being in particular applicable to JPEG or JPEG 2000 compressed images. As second application, we introduce a nuclear-norm-based vectorial TGV functional for joint MR-PET reconstruction that exploits structural similarities between the two modalities. Moving to the dynamic setting, we motivate and introduce a suitable extension of derivative based regularization for spatio-temporal data. After establishing essential analytical properties, we deal with applications to the reconstruction of highly subsampled dynamic MR data and the decompression of MPEG compressed movies.

Optimization

Wed, 12.10.16 at 14:00

WIAS-HVP11A 4.01

C. Geldhauser WIAS

Scaling limits of interacting diffusions

Abstract.

In this talk we consider a system of N coupled stochastic differential equations, which we interpret as a system of N particles evolving according to the dynamics given by the SDEs. Due to the properties of the driving force and the noise, the limit as N goes to infinity does not lead in general to a well-posed equation. We develop conditions on the interaction strength between the particles to ensure existence of solutions to the limiting stochastic PDE. Moreover, we investigate the long-time behaviour of the solution. This is joint work with Anton Bovier.

Research Seminar on Mathematical Optimization / Non-smooth Variational Problems and Operator Equations 📅