Bounding inferences for large-scale continuous-time Markov chains : a new approach based on lumping and imprecise Markov chains

If the state space of a homogeneous continuous-time Markov chain is too large, making inferences becomes computationally infeasible. Fortunately, the state space of such a chain is usually too detailed for the inferences we are interested in, in the sense that a less detailed—smaller—state space suffices to unambiguously formalise the inference. However, in general this so-called lumped state space inhibits computing exact inferences because the corresponding dynamics are unknown and/or intractable to obtain. We address this issue by considering an imprecise continuous-time Markov chain. In this way, we are able to provide guaranteed lower and upper bounds for the inferences of interest, without suffering from the curse of dimensionality

First Steps Towards an Imprecise Poisson Process

The Poisson process is the most elementary continuous-time stochastic process that models a stream of repeating events. It is uniquely characterised by a single parameter called the rate. Instead of a single value for this rate, we here consider a rate interval and let it characterise two nested sets of stochastic processes. We call these two sets of stochastic process imprecise Poisson processes, explain why this is justified, and study the corresponding lower and upper (conditional) expectations. Besides a general theoretical framework, we also provide practical methods to compute lower and upper (conditional) expectations of functions that depend on the number of events at a single point in time.Comment: Extended pre-print of a paper accepted for presentation at ISIPTA 201

Imprecise continuous-time Markov chains : efficient computational methods with guaranteed error bounds

Imprecise continuous-time Markov chains are a robust type of continuous-time Markov chains that allow for partially specified time-dependent parameters. Computing inferences for them requires the solution of a non-linear differential equation. As there is no general analytical expression for this solution, efficient numerical approximation methods are essential to the applicability of this model. We here improve the uniform approximation method of Krak et al. (2016) in two ways and propose a novel and more efficient adaptive approximation method. For ergodic chains, we also provide a method that allows us to approximate stationary distributions up to any desired maximal error

First steps towards an imprecise Poisson process

The Poisson process is the most elementary continuous-time stochastic process that models a stream of repeating events. It is uniquely characterised by a single parameter called the rate. Instead of a single value for this rate, we here consider a rate interval and let it characterise two nested sets of stochastic processes. We call these two sets of stochastic process imprecise Poisson processes, explain why this is justified, and study the corresponding lower and upper (conditional) expectations. Besides a general theoretical framework, we also provide practical methods to compute lower and upper (conditional) expectations of functions that depend on the number of events at a single point in time

Generation and analysis of plasmas with centrally reduced helicity in full-tungsten ASDEX Upgrade

Die vielversprechendsten Konzepte zur Nutzbarmachung der Kernfusion sind toroidale Maschinen wie Tokamaks, in denen ein Plasma durch helikal gewundene Magnetfeldlinien eingeschlossen wird. Um die Feldlinien zu verdrillen wird im Tokamak ein toroidaler Plasmastrom benötigt, welcher größtenteils durch einen Transformator induziert wird, weswegen konventionelle Tokamaks auf gepulsten Betrieb beschränkt sind. Außerdem macht dieser Strom das Plasma anfällig für eine Vielzahl von den Einschluss verringernden magnetohydrodynamischen (MHD) Instabilitäten, die an Orten mit rationaler Feldlinienhelizität $q$ auftreten können --- Sägezähne bei 1/1, neoklassische Tearingmoden (NTMs) beispielsweise bei 3/2 oder 2/1. Diese Arbeit präsentiert Untersuchungen von Plasmen mit durch externen Stromtrieb zentral angehobenen $q$-Profilen unter stationären Bedingungen im voll-Wolfram Tokamak ASDEX Upgrade. Ohne das üblicherweise monotone $q$-profil verschwinden Instabilitäten mit niedriger Helizität, was die Plasmastabilität verbessert. Außerdem erhöht das Anheben von $q$ den sog. Bootstrap-Strom, den das Plasma in der Gegenwart von Druckgradienten selbst treibt, was es unabhängiger vom Transformator macht. Im besten Fall könnte ein ,,Advanced Tokamak`` (AT) stationär betrieben werden. Zusätzlich sollte ein angehobenes und dadurch flaches/leicht invertiertes $q$-Profil auch den Einschluss durch Behinderung von Turbulenz verbessern. Die Rekonstruktion des erzeugten $q$-Profiles wird mit dem neuen Gleichgewichtscode IDE und Messungen einer Schlüsseldiagnostik, die auf dem Motional Stark Effect aufbaut, bewerkstelligt. Während dieser Arbeit wurde entdeckt, dass die Diagnostik durch polarisiertes Hintergrundlicht gestört wird. Ein Prototyp zur Korrektur wurde erfolgreich getestet. Außerdem wurden Nichtlinearitäten im optischen System der Diagnostic gefunden und ein Kalibrationsschema zur Kompensation entworfen. Es wurden sowohl die konventionelle Herangehensweise an AT-Studien, bei denen ein Plasma früh geheizt wird um die Relaxation des anfangs angehobenen $q$-Profils zu verzögern, als auch ein neuer Ansatz bei dem ein relaxiertes $q$-Profil verändert wird, untersucht. Ersteres wurde als für nichtinduktiven Betrieb geeignet befunden, blieb aber anfällig für 2/1-NTMs. Letzteres erlaubte den nichtinduktive Betrieb bei hohem Plasmadruck unter reaktorrelevanten Bedingungen ohne Verringerung des Einschlusses oder starke MHD Instabilitäten. Der Wärmetransport im Plasma wurde in stationären Phasen mit erhöhtem $q$ mithilfe des numerischen Simulationscodes TGLF modelliert. Ohne Scherflüsse durch ein radiales elektrisches Feld sind die Vorhersagen von TGLF für den Wärmetransport jenseits der experimentellen Beobachtungen. Mit solchen Flüssen werden die experimentell beobachteten Temperaturen signifikant übertroffen, der Transport also unterschätzt. Die Ergebnisse dieser Arbeit zeigen, dass der attraktive nichtinduktive Betrieb mit hohem Einschluss in ASDEX Upgrade erreicht werden kann. Dementsprechend können weitere AT-Studien durchgeführt werden.The most promising concepts for harnessing nuclear fusion are toroidal devices like tokamaks, where a plasma is confined by helically twisted magnetic field lines. To provide the twisting of the field lines, a tokamak relies on a toroidal current in the plasma, which is largely generated by a transformer. As such, conventional tokamaks are limited to pulsed operation. Moreover, this current makes tokamak plasmas prone to numerous confinement degrading magnetohydrodynamic (MHD) instabilities that can emerge at locations where the field line helicity $q$ takes on rational values like 1/1, 3/2 or 2/1, i.e.~sawteeth or neoclassical tearing modes (NTMs). This thesis presents studies of plasmas with centrally elevated $q$-profiles created by external electron-cyclotron and neutral beam current drive (ECCD/NBCD) under steady-state conditions in the full-tungsten tokamak ASDEX Upgrade. Without the usually monotonic $q$-profile, instabilities of low helicity disappear, thereby improving the plasma stability. Furthermore, elevating $q$ increases the amount of so-called (toroidal) bootstrap current, which the plasma drives by itself in the presence of pressure gradients, thereby reducing the reliance on the transformer. In the best case, an advanced tokamak (AT) could thus run in steady state. Additionally, an elevated and thus flat/slightly reversed $q$-profile is thought to improve confinement by impeding turbulent transport. Reconstruction of the tailored $q$-profile is accomplished with the new integrated data equilibrium (IDE) code and information from a key diagnostic that is based on the Motional Stark Effect (MSE). During the course of this work it was discovered that the MSE diagnostic suffers from interference from polarised background light. A prototype mitigation system was successfully tested. Also, non-linearities in the diagnostic's optical relay system were found and a calibration scheme devised to take them into account. Both the conventional approach of AT-studies where the plasma is heated early in the discharge to delay the relaxation of the initially elevated $q$-profiles as well as a new approach of manipulating an already equilibrated $q$-profile have been explored. The former was found to be suitable for non-inductive operation but remained susceptible to 2/1-NTMs. The latter allowed non-inductive operation at high plasma pressure under reactor-relevant conditions to be achieved without confinement degradation or any strong MHD instabilities. The plasma heat transport in steady-state phases with elevated $q$ was modelled using the numerical simulation code TGLF. Without shear flows due to a radial electric field TGLF's predictions for the ion heat transport are beyond the experimental observation, which may be due to TGLF not considering stabilising effects of fast ions. With such shear flows the TGLF temperature predictions exceed the experimental observations significantly, i.e.~heat transport is underestimated. The findings of this work show that the attractive non-inductive operation with high confinement can be achieved in ASDEX Upgrade. As such, further AT-studies are possible

Imprecise Markov Models for Scalable and Robust Performance Evaluation of Flexi-Grid Spectrum Allocation Policies

The possibility of flexibly assigning spectrum resources with channels of different sizes greatly improves the spectral efficiency of optical networks, but can also lead to unwanted spectrum fragmentation.We study this problem in a scenario where traffic demands are categorised in two types (low or high bit-rate) by assessing the performance of three allocation policies. Our first contribution consists of exact Markov chain models for these allocation policies, which allow us to numerically compute the relevant performance measures. However, these exact models do not scale to large systems, in the sense that the computations required to determine the blocking probabilities---which measure the performance of the allocation policies---become intractable. In order to address this, we first extend an approximate reduced-state Markov chain model that is available in the literature to the three considered allocation policies. These reduced-state Markov chain models allow us to tractably compute approximations of the blocking probabilities, but the accuracy of these approximations cannot be easily verified. Our main contribution then is the introduction of reduced-state imprecise Markov chain models that allow us to derive guaranteed lower and upper bounds on blocking probabilities, for the three allocation policies separately or for all possible allocation policies simultaneously.Comment: 16 pages, 7 figures, 3 table