TY - THES A1 - Bartsch, Jan T1 - Theoretical and numerical investigation of optimal control problems governed by kinetic models T1 - Theoretische und numerische Untersuchung von Optimalsteuerungsproblemen mit kinetischen Modellen N2 - This thesis is devoted to the numerical and theoretical analysis of ensemble optimal control problems governed by kinetic models. The formulation and study of these problems have been put forward in recent years by R.W. Brockett with the motivation that ensemble control may provide a more general and robust control framework for dynamical systems. Following this formulation, a Liouville (or continuity) equation with an unbounded drift function is considered together with a class of cost functionals that include tracking of ensembles of trajectories of dynamical systems and different control costs. Specifically, $L^2$, $H^1$ and $L^1$ control costs are taken into account which leads to non--smooth optimization problems. For the theoretical investigation of the resulting optimal control problems, a well--posedness theory in weighted Sobolev spaces is presented for Liouville and related transport equations. Specifically, existence and uniqueness results for these equations and energy estimates in suitable norms are provided; in particular norms in weighted Sobolev spaces. Then, non--smooth optimal control problems governed by the Liouville equation are formulated with a control mechanism in the drift function. Further, box--constraints on the control are imposed. The control--to--state map is introduced, that associates to any control the unique solution of the corresponding Liouville equation. Important properties of this map are investigated, specifically, that it is well--defined, continuous and Frechet differentiable. Using the first two properties, the existence of solutions to the optimal control problems is shown. While proving the differentiability, a loss of regularity is encountered, that is natural to hyperbolic equations. This leads to the need of the investigation of the control--to--state map in the topology of weighted Sobolev spaces. Exploiting the Frechet differentiability, it is possible to characterize solutions to the optimal control problem as solutions to an optimality system. This system consists of the Liouville equation, its optimization adjoint in the form of a transport equation, and a gradient inequality. Numerical methodologies for solving Liouville and transport equations are presented that are based on a non--smooth Lagrange optimization framework. For this purpose, approximation and solution schemes for such equations are developed and analyzed. For the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov--Tadmor method, a Runge--Kutta scheme, and a Strang splitting method are discussed. Stability and second--order accuracy of these resulting schemes are proven in the discrete $L^1$ norm. In addition, conservation of mass and positivity preservation are confirmed for the solution method of the Liouville model. As numerical optimization strategy, an adapted Krylow--Newton method is applied. Since the control is considered to be an element of $H^1$ and to obey certain box--constraints, a method for calculating a $H^1$ projection is presented. Since the optimal control problem is non-smooth, a semi-smooth adaption of Newton's method is taken into account. Results of numerical experiments are presented that successfully validate the proposed deterministic framework. After the discussion of deterministic schemes, the linear space--homogeneous Keilson--Storer master equation is investigated. This equation was originally developed for the modelling of Brownian motion of particles immersed in a fluid and is a representative model of the class of linear Boltzmann equations. The well--posedness of the Keilson--Storer master equation is investigated and energy estimates in different topologies are derived. To solve this equation numerically, Monte Carlo methods are considered. Such methods take advantage of the kinetic formulation of the Liouville equation and directly implement the behaviour of the system of particles under consideration. This includes the probabilistic behaviour of the collisions between particles. Optimal control problems are formulated with an objective that is constituted of certain expected values in velocity space and the $L^2$ and $H^1$ costs of the control. The problems are governed by the Keilson--Storer master equation and the control mechanism is considered to be within the collision kernel. The objective of the optimal control of this model is to drive an ensemble of particles to acquire a desired mean velocity and to achieve a desired final velocity configuration. Existence of solutions of the optimal control problem is proven and a Keilson--Storer optimality system characterizing the solution of the proposed optimal control problem is obtained. The optimality system is used to construct a gradient--based optimization strategy in the framework of Monte--Carlo methods. This task requires to accommodate the resulting adjoint Keilson--Storer model in a form that is consistent with the kinetic formulation. For this reason, we derive an adjoint Keilson--Storer collision kernel and an additional source term. A similar approach is presented in the case of a linear space--inhomogeneous kinetic model with external forces and with Keilson--Storer collision term. In this framework, a control mechanism in the form of an external space--dependent force is investigated. The purpose of this control is to steer the multi--particle system to follow a desired mean velocity and position and to reach a desired final configuration in phase space. An optimal control problem using the formulation of ensemble controls is stated with an objective that is constituted of expected values in phase space and $H^1$ costs of the control. For solving the optimal control problems, a gradient--based computational strategy in the framework of Monte Carlo methods is developed. Part of this is the denoising of the distribution functions calculated by Monte Carlo algorithms using methods of the realm of partial differential equations. A standalone C++ code is presented that implements the developed non--linear conjugated gradient strategy. Results of numerical experiments confirm the ability of the designed probabilistic control framework to operate as desired. An outlook section about optimal control problems governed by non--linear space--inhomogeneous kinetic models completes this thesis. N2 - Diese Arbeit widmet sich der numerischen und theoretischen Analyse von Proble- men der optimalen Kontrolle von Ensembles, die durch kinetische Modelle gesteuert werden. Die Formulierung und Untersuchung von Ensemble–Kontrollproblemen wur- den in den letzten Jahren von R.W. Brockett vorgeschlagen und vorangetrieben, mit der Motivation, dass Ensemblekontrolle einen allgemeineren und robusteren Rahmen für die Kontrolle von dynamischen Systemen bieten kann. In Anlehnung an diese Formulierung der Ensemble–Steuerung werden eine Liouville– (oder Kontinuitäts– ) Gleichung mit unbeschränkter Driftfunktion und eine Klasse von Kostenfunk- tionalen miteinbezogen, die das Nachverfolgen der Ensembles und verschiedener Kon- trollkosten berücksichtigen. Insbesondere werden L2, H1 und L1 Kontrollkosten be- trachtet. Für die theoretische Untersuchung der resultierenden Optimalsteuerungs- problemen wird eine Gutgestelltheitstheorie in gewichteten Sobolev–Räumen für die Liouville– und Transportgleichungen vorgestellt. Insbesondere werden Existenz– und Eindeutigkeitsresultate sowie Energieabschätzungen in geeigneten Normen präsen- tiert; insbesondere in gewichteten Sobolev–Räumen. Dann wird eine Klasse von nicht–glatten Optimalsteuerungsproblemen formuliert mit der Liouville–Gleichung als Nebenbedingung und einem Kontrollmechanismus in der Driftfunktion. Weiter- hin werden Box–Einschränkungen angenommen. ... KW - Optimale Kontrolle KW - Optimierung / Nebenbedingung KW - Liouville and transport equations KW - Ensemble optimal control Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-249066 ER - TY - JOUR A1 - Bartsch, Jan A1 - Borzì, Alfio A1 - Fanelli, Francesco A1 - Roy, Souvik T1 - A numerical investigation of Brockett’s ensemble optimal control problems JF - Numerische Mathematik N2 - This paper is devoted to the numerical analysis of non-smooth ensemble optimal control problems governed by the Liouville (continuity) equation that have been originally proposed by R.W. Brockett with the purpose of determining an efficient and robust control strategy for dynamical systems. A numerical methodology for solving these problems is presented that is based on a non-smooth Lagrange optimization framework where the optimal controls are characterized as solutions to the related optimality systems. For this purpose, approximation and solution schemes are developed and analysed. Specifically, for the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov–Tadmor method, a Runge–Kutta scheme, and a Strang splitting method are discussed. The resulting optimality system is solved by a projected semi-smooth Krylov–Newton method. Results of numerical experiments are presented that successfully validate the proposed framework. KW - numerical analysis KW - Brockett KW - ensemble optimal control problems Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-265352 VL - 149 IS - 1 ER - TY - JOUR A1 - El-Helou, Sabine M. A1 - Biegner, Anika-Kerstin A1 - Bode, Sebastian A1 - Ehl, Stephan R. A1 - Heeg, Maximilian A1 - Maccari, Maria E. A1 - Ritterbusch, Henrike A1 - Speckmann, Carsten A1 - Rusch, Stephan A1 - Scheible, Raphael A1 - Warnatz, Klaus A1 - Atschekzei, Faranaz A1 - Beider, Renata A1 - Ernst, Diana A1 - Gerschmann, Stev A1 - Jablonka, Alexandra A1 - Mielke, Gudrun A1 - Schmidt, Reinhold E. A1 - Schürmann, Gesine A1 - Sogkas, Georgios A1 - Baumann, Ulrich H. A1 - Klemann, Christian A1 - Viemann, Dorothee A1 - Bernuth, Horst von A1 - Krüger, Renate A1 - Hanitsch, Leif G. A1 - Scheibenbogen, Carmen M. A1 - Wittke, Kirsten A1 - Albert, Michael H. A1 - Eichinger, Anna A1 - Hauck, Fabian A1 - Klein, Christoph A1 - Rack-Hoch, Anita A1 - Sollinger, Franz M. A1 - Avila, Anne A1 - Borte, Michael A1 - Borte, Stephan A1 - Fasshauer, Maria A1 - Hauenherm, Anja A1 - Kellner, Nils A1 - Müller, Anna H. A1 - Ülzen, Anett A1 - Bader, Peter A1 - Bakhtiar, Shahrzad A1 - Lee, Jae-Yun A1 - Heß, Ursula A1 - Schubert, Ralf A1 - Wölke, Sandra A1 - Zielen, Stefan A1 - Ghosh, Sujal A1 - Laws, Hans-Juergen A1 - Neubert, Jennifer A1 - Oommen, Prasad T. A1 - Hönig, Manfred A1 - Schulz, Ansgar A1 - Steinmann, Sandra A1 - Klaus, Schwarz A1 - Dückers, Gregor A1 - Lamers, Beate A1 - Langemeyer, Vanessa A1 - Niehues, Tim A1 - Shai, Sonu A1 - Graf, Dagmar A1 - Müglich, Carmen A1 - Schmalzing, Marc T. A1 - Schwaneck, Eva C. A1 - Tony, Hans-Peter A1 - Dirks, Johannes A1 - Haase, Gabriele A1 - Liese, Johannes G. A1 - Morbach, Henner A1 - Foell, Dirk A1 - Hellige, Antje A1 - Wittkowski, Helmut A1 - Masjosthusmann, Katja A1 - Mohr, Michael A1 - Geberzahn, Linda A1 - Hedrich, Christian M. A1 - Müller, Christiane A1 - Rösen-Wolff, Angela A1 - Roesler, Joachim A1 - Zimmermann, Antje A1 - Behrends, Uta A1 - Rieber, Nikolaus A1 - Schauer, Uwe A1 - Handgretinger, Rupert A1 - Holzer, Ursula A1 - Henes, Jörg A1 - Kanz, Lothar A1 - Boesecke, Christoph A1 - Rockstroh, Jürgen K. A1 - Schwarze-Zander, Carolynne A1 - Wasmuth, Jan-Christian A1 - Dilloo, Dagmar A1 - Hülsmann, Brigitte A1 - Schönberger, Stefan A1 - Schreiber, Stefan A1 - Zeuner, Rainald A1 - Ankermann, Tobias A1 - Bismarck, Philipp von A1 - Huppertz, Hans-Iko A1 - Kaiser-Labusch, Petra A1 - Greil, Johann A1 - Jakoby, Donate A1 - Kulozik, Andreas E. A1 - Metzler, Markus A1 - Naumann-Bartsch, Nora A1 - Sobik, Bettina A1 - Graf, Norbert A1 - Heine, Sabine A1 - Kobbe, Robin A1 - Lehmberg, Kai A1 - Müller, Ingo A1 - Herrmann, Friedrich A1 - Horneff, Gerd A1 - Klein, Ariane A1 - Peitz, Joachim A1 - Schmidt, Nadine A1 - Bielack, Stefan A1 - Groß-Wieltsch, Ute A1 - Classen, Carl F. A1 - Klasen, Jessica A1 - Deutz, Peter A1 - Kamitz, Dirk A1 - Lassy, Lisa A1 - Tenbrock, Klaus A1 - Wagner, Norbert A1 - Bernbeck, Benedikt A1 - Brummel, Bastian A1 - Lara-Villacanas, Eusebia A1 - Münstermann, Esther A1 - Schneider, Dominik T. A1 - Tietsch, Nadine A1 - Westkemper, Marco A1 - Weiß, Michael A1 - Kramm, Christof A1 - Kühnle, Ingrid A1 - Kullmann, Silke A1 - Girschick, Hermann A1 - Specker, Christof A1 - Vinnemeier-Laubenthal, Elisabeth A1 - Haenicke, Henriette A1 - Schulz, Claudia A1 - Schweigerer, Lothar A1 - Müller, Thomas G. A1 - Stiefel, Martina A1 - Belohradsky, Bernd H. A1 - Soetedjo, Veronika A1 - Kindle, Gerhard A1 - Grimbacher, Bodo T1 - The German national registry of primary immunodeficiencies (2012-2017) JF - Frontiers in Immunology N2 - Introduction: The German PID-NET registry was founded in 2009, serving as the first national registry of patients with primary immunodeficiencies (PID) in Germany. It is part of the European Society for Immunodeficiencies (ESID) registry. The primary purpose of the registry is to gather data on the epidemiology, diagnostic delay, diagnosis, and treatment of PIDs. Methods: Clinical and laboratory data was collected from 2,453 patients from 36 German PID centres in an online registry. Data was analysed with the software Stata® and Excel. Results: The minimum prevalence of PID in Germany is 2.72 per 100,000 inhabitants. Among patients aged 1-25, there was a clear predominance of males. The median age of living patients ranged between 7 and 40 years, depending on the respective PID. Predominantly antibody disorders were the most prevalent group with 57% of all 2,453 PID patients (including 728 CVID patients). A gene defect was identified in 36% of patients. Familial cases were observed in 21% of patients. The age of onset for presenting symptoms ranged from birth to late adulthood (range 0-88 years). Presenting symptoms comprised infections (74%) and immune dysregulation (22%). Ninety-three patients were diagnosed without prior clinical symptoms. Regarding the general and clinical diagnostic delay, no PID had undergone a slight decrease within the last decade. However, both, SCID and hyper IgE-syndrome showed a substantial improvement in shortening the time between onset of symptoms and genetic diagnosis. Regarding treatment, 49% of all patients received immunoglobulin G (IgG) substitution (70%-subcutaneous; 29%-intravenous; 1%-unknown). Three-hundred patients underwent at least one hematopoietic stem cell transplantation (HSCT). Five patients had gene therapy. Conclusion: The German PID-NET registry is a precious tool for physicians, researchers, the pharmaceutical industry, politicians, and ultimately the patients, for whom the outcomes will eventually lead to a more timely diagnosis and better treatment. KW - registry for primary immunodeficiency KW - primary immunodeficiency (PID) KW - German PID-NET registry KW - PID prevalence KW - European Society for Immunodeficiencies (ESID) KW - IgG substitution therapy KW - CVID Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-226629 VL - 10 ER -