@phdthesis{Bartsch2021, author = {Bartsch, Jan}, title = {Theoretical and numerical investigation of optimal control problems governed by kinetic models}, doi = {10.25972/OPUS-24906}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-249066}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2021}, abstract = {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.}, subject = {Optimale Kontrolle}, language = {en} } @article{BartschBorziFanellietal.2021, author = {Bartsch, Jan and Borz{\`i}, Alfio and Fanelli, Francesco and Roy, Souvik}, title = {A numerical investigation of Brockett's ensemble optimal control problems}, series = {Numerische Mathematik}, volume = {149}, journal = {Numerische Mathematik}, number = {1}, doi = {10.1007/s00211-021-01223-6}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-265352}, pages = {1-42}, year = {2021}, abstract = {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.}, language = {en} } @article{ElHelouBiegnerBodeetal.2019, author = {El-Helou, Sabine M. and Biegner, Anika-Kerstin and Bode, Sebastian and Ehl, Stephan R. and Heeg, Maximilian and Maccari, Maria E. and Ritterbusch, Henrike and Speckmann, Carsten and Rusch, Stephan and Scheible, Raphael and Warnatz, Klaus and Atschekzei, Faranaz and Beider, Renata and Ernst, Diana and Gerschmann, Stev and Jablonka, Alexandra and Mielke, Gudrun and Schmidt, Reinhold E. and Sch{\"u}rmann, Gesine and Sogkas, Georgios and Baumann, Ulrich H. and Klemann, Christian and Viemann, Dorothee and Bernuth, Horst von and Kr{\"u}ger, Renate and Hanitsch, Leif G. and Scheibenbogen, Carmen M. and Wittke, Kirsten and Albert, Michael H. and Eichinger, Anna and Hauck, Fabian and Klein, Christoph and Rack-Hoch, Anita and Sollinger, Franz M. and Avila, Anne and Borte, Michael and Borte, Stephan and Fasshauer, Maria and Hauenherm, Anja and Kellner, Nils and M{\"u}ller, Anna H. and {\"U}lzen, Anett and Bader, Peter and Bakhtiar, Shahrzad and Lee, Jae-Yun and Heß, Ursula and Schubert, Ralf and W{\"o}lke, Sandra and Zielen, Stefan and Ghosh, Sujal and Laws, Hans-Juergen and Neubert, Jennifer and Oommen, Prasad T. and H{\"o}nig, Manfred and Schulz, Ansgar and Steinmann, Sandra and Klaus, Schwarz and D{\"u}ckers, Gregor and Lamers, Beate and Langemeyer, Vanessa and Niehues, Tim and Shai, Sonu and Graf, Dagmar and M{\"u}glich, Carmen and Schmalzing, Marc T. and Schwaneck, Eva C. and Tony, Hans-Peter and Dirks, Johannes and Haase, Gabriele and Liese, Johannes G. and Morbach, Henner and Foell, Dirk and Hellige, Antje and Wittkowski, Helmut and Masjosthusmann, Katja and Mohr, Michael and Geberzahn, Linda and Hedrich, Christian M. and M{\"u}ller, Christiane and R{\"o}sen-Wolff, Angela and Roesler, Joachim and Zimmermann, Antje and Behrends, Uta and Rieber, Nikolaus and Schauer, Uwe and Handgretinger, Rupert and Holzer, Ursula and Henes, J{\"o}rg and Kanz, Lothar and Boesecke, Christoph and Rockstroh, J{\"u}rgen K. and Schwarze-Zander, Carolynne and Wasmuth, Jan-Christian and Dilloo, Dagmar and H{\"u}lsmann, Brigitte and Sch{\"o}nberger, Stefan and Schreiber, Stefan and Zeuner, Rainald and Ankermann, Tobias and Bismarck, Philipp von and Huppertz, Hans-Iko and Kaiser-Labusch, Petra and Greil, Johann and Jakoby, Donate and Kulozik, Andreas E. and Metzler, Markus and Naumann-Bartsch, Nora and Sobik, Bettina and Graf, Norbert and Heine, Sabine and Kobbe, Robin and Lehmberg, Kai and M{\"u}ller, Ingo and Herrmann, Friedrich and Horneff, Gerd and Klein, Ariane and Peitz, Joachim and Schmidt, Nadine and Bielack, Stefan and Groß-Wieltsch, Ute and Classen, Carl F. and Klasen, Jessica and Deutz, Peter and Kamitz, Dirk and Lassy, Lisa and Tenbrock, Klaus and Wagner, Norbert and Bernbeck, Benedikt and Brummel, Bastian and Lara-Villacanas, Eusebia and M{\"u}nstermann, Esther and Schneider, Dominik T. and Tietsch, Nadine and Westkemper, Marco and Weiß, Michael and Kramm, Christof and K{\"u}hnle, Ingrid and Kullmann, Silke and Girschick, Hermann and Specker, Christof and Vinnemeier-Laubenthal, Elisabeth and Haenicke, Henriette and Schulz, Claudia and Schweigerer, Lothar and M{\"u}ller, Thomas G. and Stiefel, Martina and Belohradsky, Bernd H. and Soetedjo, Veronika and Kindle, Gerhard and Grimbacher, Bodo}, title = {The German national registry of primary immunodeficiencies (2012-2017)}, series = {Frontiers in Immunology}, volume = {10}, journal = {Frontiers in Immunology}, doi = {10.3389/fimmu.2019.01272}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-226629}, year = {2019}, abstract = {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.}, language = {en} }