Challenges in Optimal Control of Nonlinear PDE-Systems

The workshop focussed on various aspects of optimal control problems for systems of nonlinear partial differential equations. In particular, discussions around keynote presentations in the areas of optimal control of nonlinear/non-smooth systems, optimal control of systems involving nonlocal operators, shape and topology optimization, feedback control and stabilization, sparse control, and associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, also aspects of control of fluid structure interaction problems as well as problems arising in the optimal control of quantum systems were considered

Random neural networks for rough volatility

We construct a deep learning-based numerical algorithm to solve path-dependent partial differential equations arising in the context of rough volatility. Our approach is based on interpreting the PDE as a solution to an SPDE, building upon recent insights by Bayer, Qiu and Yao, and on constructing a neural network of reservoir type as originally developed by Gonon, Grigoryeva, Ortega. The reservoir approach allows us to formulate the optimisation problem as a simple least-square regression for which we prove theoretical convergence properties.Comment: 33 pages, 3 figure

Role of non-ideality for the ion transport in porous media: derivation of the macroscopic equations using upscaling

This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. We first consider equilibrium solutions in the absence of external forces. In such a case, the velocity and diffusive fluxes vanish and the equilibrium electrostatic potential is the solution of a variant of Poisson-Boltzmann equation coupled with algebraic equations. Contrary to the ideal case, this nonlinear equation has no monotone structure. However, based on invariant region estimates for Poisson-Boltzmann equation and for small characteristic value of the solute packing fraction, we prove existence of at least one solution. To our knowledge this existence result is new at this level of generality. When the motion is governed by a small static electric field and a small hydrodynamic force, we generalize O'Brien's argument to deduce a linearized model. Our second main result is the rigorous homogenization of these linearized equations and the proof that the effective tensor satisfies Onsager properties, namely is symmetric positive definite. We eventually make numerical comparisons with the ideal case. Our numerical results show that the MSA model confirms qualitatively the conclusions obtained using the ideal model but there are quantitative differences arising that can be important at high charge or high concentrations.Comment: 46 page

Vorschlag für neue objektorientierte gleichungsbasierte Modellbibliotheken für thermodynamische Systeme

This thesis proposes two new model libraries for fluid properties and for components that can be used for the simulation of thermodynamic systems such as refrigeration, air-conditioning, and heat-pump systems. The new fluid property library is written in C/C++ and can be interfaced from various software tools and programming languages. The new component model library is written in the object-oriented equation-based modeling language Modelica. Furthermore, tools for the automated generation of class diagrams and the visualization of the solution process as well as the numerical results in relevant diagrams are presented. Both new libraries are based on a thorough object-oriented analysis. A set of general design rules for the development of object-oriented component model libraries is formulated to ensure that the resulting library can be used by the entire spectrum of possible users from experienced developers to design engineers. The new object-based fluid property library is based on a generalized approach to include external fluid property computation codes in Modelica. It is simple to extend to additional external fluid property computation codes. It allows for a numerically efficient handling of fluid properties in Modelica and in a number of software tools. The new model library for components and systems was developed based on the newly introduced design rules. It features a structure that is simple to understand and flexible to allow for extensions. All balance equations are formulated in an easy and comprehensible way in base components. The new component model library contains models with different levels of detail to allow for a problem-dependent model selection. Two applications are presented to demonstrate the capabilities of the two new libraries and to demonstrate their extendibility and multidisciplinarity.Diese Arbeit beschreibt zwei neue Modellbibliotheken mit Stoffdaten- und Komponentenmodellen zur Simulation thermodynamischer Systeme wie zum Beispiel Kälteanlagen, Klimaanlagen und Wärmepumpen. Die neue Stoffdatenbibliothek ist basierend auf C/C++ entwickelt worden und verfügt über Schnittstellen für unterschiedliche Anwendungsprogramme und Programmiersprachen. Die neue Komponentenbibliothek ist in der objektorientierten gleichungsbasierten Modellierungssprache Modelica geschrieben. Es werden außerdem Werkzeuge vorgestellt, die die automatische Generierung von Klassenstrukturdiagrammen und die Visualisierung des Lösungsprozesses sowie der numerischen Ergebnisse in thermodynamischen Diagrammen ermöglichen. Beide neuen Bibliotheken basieren auf einer gründlichen objektorientierten Analyse. Es werden des Weiteren allgemeingültige Richtlinien für die Erstellung von objektorientierten Modellbibliotheken entwickelt, die sicherstellen, dass die entwickelte Bibliothek das gesamte Spektrum möglicher Benutzer vom Code-Entwickler bis hin zum Anwender unterstützt. Die neue objektbasierte Stoffdatenbibliothek basiert auf einem generalisierten Ansatz zur Einbindung externer Stoffdatenbibliotheken in Modelica. Der beschriebene Ansatz lässt sich einfach auf weitere externe Bibliotheken zur Stoffdatenberechnung erweitern. Er erlaubt die numerisch effektive Behandlung der eingebundenen Bibliotheken in Modelica sowie in Anwendungsprogrammen. Die neue Modellbibliothek für Komponenten und Systeme verfügt über eine Struktur, die einfach verständlich und flexibel erweiterbar ist. Alle Erhaltungsgleichungen sind in einfach verständlicher Form in Basiskomponenten formuliert. Die neue Komponentenmodellbibliothek enthält Modelle unterschiedlicher Modellierungstiefe, um eine problemabhängige Auswahl von Modellen zu erlauben. Zwei Anwendungen für die beiden neuen Modellbibliotheken werden präsentiert, um die Erweiterbarkeit und die Multidisziplinarität der neuen Bibliotheken zu demonstrieren