Mean-field optimal control and optimality conditions in the space of probability measures

We derive a framework to compute optimal controls for problems with states in the space of probability measures. Since many optimal control problems constrained by a system of ordinary differential equations (ODE) modelling interacting particles converge to optimal control problems constrained by a partial differential equation (PDE) in the mean-field limit, it is interesting to have a calculus directly on the mesoscopic level of probability measures which allows us to derive the corresponding first-order optimality system. In addition to this new calculus, we provide relations for the resulting system to the first-order optimality system derived on the particle level, and the first-order optimality system based on $L^2$-calculus under additional regularity assumptions. We further justify the use of the $L^2$-adjoint in numerical simulations by establishing a link between the adjoint in the space of probability measures and the adjoint corresponding to $L^2$-calculus. Moreover, we prove a convergence rate for the convergence of the optimal controls corresponding to the particle formulation to the optimal controls of the mean-field problem as the number of particles tends to infinity

Instantaneous control of interacting particle systems in the mean-field limit

Controlling large particle systems in collective dynamics by a few agents is a subject of high practical importance, e.g., in evacuation dynamics. In this paper we study an instantaneous control approach to steer an interacting particle system into a certain spatial region by repulsive forces from a few external agents, which might be interpreted as shepherd dogs leading sheep to their home. We introduce an appropriate mathematical model and the corresponding optimization problem. In particular, we are interested in the interaction of numerous particles, which can be approximated by a mean-field equation. Due to the high-dimensional phase space this will require a tailored optimization strategy. The arising control problems are solved using adjoint information to compute the descent directions. Numerical results on the microscopic and the macroscopic level indicate the convergence of optimal controls and optimal states in the mean-field limit,i.e., for an increasing number of particles.Comment: arXiv admin note: substantial text overlap with arXiv:1610.0132

Digitale Fremdsprachenkompetenzen? Vorüberlegungen zur Modellierung digitaler Englischkompetenzen im Rahmen des Projekts „Digi_Gap – Digitale Lücken in der Lehrkräftebildung schließen“

In diesem Beitrag werden infolge der Veröffentlichung des Strategiepapiers „Bildung in der digitalen Welt“ der Kultusministerkonferenz Überlegungen zur fachspezifischen Ausgestaltung des dort vorgeschlagenen Rahmenmodells zu den „Kompetenzen in der digitalen Welt“ in Bezug auf das Fach Englisch angestellt. Hierbei handelt es sich um Vorüberlegungen für das Projekt „Digi_Gap – Digitale Lücken in der Lehrkräftebildung schließen“, welches von 2020 bis 2023 als Teil der Qualitätsoffensive Lehrerbildung vom Bundesministerium für Bildung und Forschung (BMBF) an der Goethe-Universität Frankfurt am Main gefördert wird und in dessen Rahmen eine Modellierung digitaler Fremdsprachenkompetenzen Englisch erfolgen soll. Neben den Empfehlungen der Kultusministerkonferenz werden weitere Kompetenzmodelle sowie Ansätze und Konzepte aus dem Bereich der (Fremd-)Sprachenforschung herangezogen, um vorläufige Rückschlüsse bezüglich der avisierten Modellierung zu ziehen, die mit der Perspektive schulischer Bildung im post-digitalen Zeitalter erarbeitet werden soll

Mean-field optimal control for biological pattern formation

We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian approach on the mean-field level. Based on these conditions we propose a gradient descent method to identify relevant parameters such as angle of rotation and force scaling which may be spatially inhomogeneous. We discretize the first-order optimality conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate for the convergence of the controls as the number of particles used for the discretization tends to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility of the approach

Safety and Efficacy of Mistletoe extract (Viscum album) Iscador(R) P in Equine Sarcoid

Equine Sarcoid (ES), a semi malignant skin tumour, is the most common neoplasia of horses (Marti et al.1993). There is no generally effective single treatment existing and the recurrence rate is high (Martens et al.2001). Furthermore as a treatment obstacle, Sarcoids tend to occur as multi-focal tumours or can be found in critical locations (peri-ocular) where success of surgical interventions is very limited (Carstanjen et al.1997). Due to encouraging results in human medicine and small animals practice after adjuvant cancer treatment by mistletoe extracts (Biegel et al.2007) the viscum therapy could hypothetically fill a therapy gap for equine tumour diseases as well. Hence, goal of this clinical trial was to evaluate the safety and efficacy of Viscum album pini (Iscador® P) as a primary therapy of Equine Sarcoid in the context of a prospective, randomized, controlled, double-blind study. Material and Methods: 53 horses with histologically confirmed diagnosis Equine Sarcoid were randomly assigned as Iscador (I; n=32) or Control group (C; n=21). There are 23 mares, 25 geldings and 5 stallions ranged in age from 3 to 17 years with an average of 7.2 years. The horses received scalating dosages of Iscador® P starting at 0.1mg up to 20 mg extract per 1 ml dose or physiological NaCl solution 3 times a week over 105 days sub-cutanously into the pectoral region. The number, localization and morphology of the ES were observed and documented over 12 months. As target variable the positive outcome rate (PO rate) was calculated representing the patients status at the end of the investigation. Results: In the Iscador Group 13 horses (41%) showed an improvement (at least 50% tumour remission). Of these cases a number of 9 patients showed complete cure (28%). In the Control Group only 3 cases (14%; all with complete cure) were classified as improved (p<0.05). Discussion & Conclusions: Iscador® P extract (viscum album pini) represents a safe and efficient therapy alternative in Equine Sarcoids, particularly in cases of multi-focal character. In tumours grown in problematic localizations a therapy attempt with viscum album extracts should be conducted to avoid the risk of damage the intact surrounding tissue, particularly in the eye area. Iscador® P shows significantly better outcomes (cure and improvement) compared to placebo control. Younger horses and horses with more than 2 tumours are responding with higher PO rates than older patients and horses with 1 to 2 ES. This fact could support the control of multiple sarcoids which can be excised only by immense surgery expense. Further investigations to understand factors of response and non-.response have to be conducted in the future. Additionally, due to the good tolerance by the patients and the acceptance of long-term protocol the practicability of Iscador ® P therapy can be assumed. Younger horses responded more often with a positive outcome (PO) than older (Fig. 2a) with PO rates of 57%, 40% and 13% for horses being less than 5 years, 5 to 9 years and more than 9 years, respectively compared to PO 25%, 10% and 0% in control. A significant better therapy course appeared in the Iscador group for horses with >2 ES (Fig. 2b): a number of 12 of 25 horses (48%) showed an improvement, 9 of these a complete remission (36%) including ES localized at the critical areas (head and around the eyes) compared to 6% in the control group. Iscador® P extract (viscum album pini) represents a safe and efficient therapy alternative in Equine Sarcoids, particularly in cases of multi-focal character. In tumours grown in problematic localizations a therapy attempt with viscum album extracts should be conducted to avoid the risk of damage the intact surrounding tissue, particularly in the eye area. Iscador® P shows significantly better outcomes (cure and improvement) compared to placebo control. Younger horses and horses with more than 2 tumours are responding with higher PO rates than older patients and horses with 1 to 2 ES. This fact could support the control of multiple sarcoids which can be excised only by immense surgery expense. Further investigations to understand factors of response and non-.response have to be conducted in the future. Additionally, due to the good tolerance by the patients and the acceptance of long-term protocol the practicability of Iscador ® P therapy can be assumed

US-Gewerkschaften und Arbeitsbedingungen

„Management should manage" - das ist noch immer die Devise der meisten amerikanischen Gewerkschaften. In vielen Tarifverträged ist ausdrücklich festgelegt: ,,Dem Unternehmen ist das ausschließliche Recht vorbehalten, die Geschäfte und Anlagen zu führen und den Arbeitseinsatz zu regeln ... Das beinhaltet das Recht, Arbeitskräfte einzustellen oder begründet zu entlassen, an einen anderen Arbeitsplatz zu versetzen und das Recht, Arbeitnehmer wegen Arbeitsmangels oder aus anderen legitimen Gründen von ihren Pflichten zu entbinden" (1). Die Gewerkschaften verstehen ihre Vertretungsaufgabe vorwiegend dahin, möglichst hohe Löhne und Sozialleistungen zu erkämpfen. Hierbei befinden sie sich im Einklang mit der Mehrheit der amerikanischen Arbeiter - nicht unverständlich in einer Zeit sinkender Reallöhne und in einem Land mit ungenügender Sozialgesetzgebung

Behandlung des equinen Sarkoids beim Pferd mit dem Mistelpräparat Iscador(r)-P.

(Prämierter Beitrag) Bisher gilt kaum eine Therapie des Equinen Sarkoids (ES) als ausreichend wirksam. Ziel der vorliegenden Studie war, im Rahmen einer randomisierten kontrollierten Doppelblindstudie, die Wirksamkeit des Mistelextraktes ISCADOR-P (Weleda) als Primärtherapie beim ES zu evaluieren. Insgesamt wurden 53 Pferde mit der Diagnose ES zufällig einer Verum- (V, n = 32) oder einer PlacebogruppeP, n = 21) zugeordnet. Die Pferde erhielten steigende Dosierungen von ISCADOR-P (0.1 mg bis 20 mg Extrakt/Dosis) oder physiol. NaCl-Lösung 3mal/Woche während 105 Tagen subkutan. Anzahl, Lokalisation und Morphologie der histologisch abgeklärten ES wurden während 12 Monaten protokolliert. In der Verum-Gruppe zeigten 13 Tiere (41%) eine Besserung (mindestens 50%ige Tumorremission), bei 9 (28%) davon war die Heilung vollständig gegenüber 3 Tieren (14%) in der Placebo-Gruppe (P <0.05). Während bei Pferden mit 1–2 ES kein klarer Unterschied zwischen V und P ersichtlich war, zeigte sich ein deutlich besserer Verlauf nach Behandlung mit Verum bei Pferden mit mehr als 2 ES: 12/25 (48%) zeigten eine Besserung, davon 9 eine komplette Remission (inkl. 5 am Kopf lokalisierte ES, von diesen 2 um die Augen); mit Placebo wurde nur 1/15 (6%) vollständig geheilt. ISCADOR-P stellt eine wirksame Alternativtherapie beim ES dar, insbesondere bei multiplem Befall und problematischer Lokalisation