Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

Spectral features of matrix-sequences, GLT, symbol, and application in preconditioning Krylov methods, image deblurring, and multigrid algorithms.

The final purpose of any scientific discipline can be regarded as the solution of real-world problems. With this aim, a mathematical modeling of the considered phenomenon is often compulsory. Closed-form solutions of the arising functional equations are usually not available and numerical discretization techniques are required. In this setting, the discretization of an infinite-dimensional linear equation via some linear approximation method, leads to a sequence of linear systems of increasing dimension whose coefficient matrices could inherit a structure from the continuous problem. For instance, the numerical approximation by local methods of constant or nonconstant coefficients systems of Partial Differential Equations (PDEs) over multidimensional domains, gives rise to multilevel block Toeplitz or to Generalized Locally Toeplitz (GLT) sequences, respectively. In the context of structured matrices, the convergence properties of iterative methods, like multigrid or preconditioned Krylov techniques, are strictly related to the notion of symbol, a function whose role relies in describing the asymptotical distribution of the spectrum. This thesis can be seen as a byproduct of the combined use of powerful tools like symbol, spectral distribution, and GLT, when dealing with the numerical solution of structured linear systems. We approach such an issue both from a theoretical and practical viewpoint. On the one hand, we enlarge some known spectral distribution tools by proving the eigenvalue distribution of matrix-sequences obtained as combination of some algebraic operations on multilevel block Toeplitz matrices. On the other hand, we take advantage of the obtained results for designing efficient preconditioning techniques. Moreover, we focus on the numerical solution of structured linear systems coming from the following applications: image deblurring, fractional diffusion equations, and coupled PDEs. A spectral analysis of the arising structured sequences allows us either to study the convergence and predict the behavior of preconditioned Krylov and multigrid methods applied to the coefficient matrices, or to design effective preconditioners and multigrid solvers for the associated linear systems

Probabilistic bounded reachability for stochastic hybrid systems

PhD ThesisStochastic parametric hybrid systems provide a means of formalising automata with continuous nonlinear dynamics, discrete interruptions, and parametric uncertainty (e.g. randomness and/or nondeterminism). They can be used for modelling a vast class of cyber-physical systems – machines comprising physical components orchestrated by a digital control (e.g. medical devices, self-driving cars, and aircraft autopilots). Assuring correct and safe behaviour of such systems is crucial as human lives are often involved. One of the main problems in system verification is reachability analysis. It amounts to determining whether the studied model reaches an unsafe state during its evolution. Introduction of parametric randomness allows the formulation of a quantitative version of the problem – computing the probability of reaching the undesired state. Reachability analysis is a highly challenging problem due to its general undecidability for hybrid systems and undecidability of nonlinear arithmetic (e.g. involving trigonometric functions) over the real numbers. A common approach in this case is to solve a simpler, yet useful, problem. In particular, there are techniques for solving reachability rigorously up to a given numerical precision. The central problem of this research is probabilistic reachability analysis of hybrid systems with random and nondeterministic parameters. In this thesis I have developed two new distinct techniques: a formal approach, based on formal reasoning which provides absolute numerical guarantees; and a statistical one, utilising Monte Carlo sampling that gives statistical guarantees. Namely, the former computes an interval which is guaranteed to contain the exact reachability probability value, while the latter returns an interval containing the probability value with some statistical confidence. By providing weaker guarantees, the statistical approach is capable of handling difficult cases more efficiently than the formal one, which in turn, can be used for parameter set synthesis in the absence of random uncertainty. The latter is one of the key problems in system modelling: identifying sets of parameter values for which a given model satisfies the desired behaviour. I have implemented the described techniques in the publicly available tool ProbReach, which I have then applied to several realistic case studies such as the synthesis of safe and robust controllers for artificial pancreas and the design of UVB treatment for psoriasis.award N00014-13-1-0090 of the US Office of Naval Research

Dynamic Incentives for Optimal Control of Competitive Power Systems

Technologisch herausfordernde Transformationsprozesse wie die Energiewende können durch passende Anreizsysteme entscheidend beschleunigt werden. Ziel solcher Anreize ist es hierbei, ein Umfeld idealerweise so zu schaffen, dass das Zusammenspiel aller aus Sicht der beteiligten Wettbewerber individuell optimalen Einzelhandlungen auch global optimal im Sinne eines übergeordneten Großziels ist. Die vorliegende Dissertation schafft einen regelungstechnischen Zugang zur Frage optimaler Anreizsysteme für heutige und zukünftige Stromnetze im Zieldreieck aus Systemstabilität, ökonomischer Effizienz und Netzdienlichkeit. Entscheidende Neuheit des entwickelten Ansatzes ist die Einführung zeitlich wie örtlich differenzierter Echtzeit-Preissignale, die sich aus der Lösung statischer und dynamischer Optimierungsprobleme ergeben. Der Miteinbezug lokal verfügbarer Messinformationen, die konsequente Mitmodellierung des unterlagerten physikalischen Netzes inklusive resistiver Verluste und die durchgängig zeitkontinuierliche Formulierung aller Teilsysteme ebnen den Weg von einer reinen Anreiz-Steuerung hin zu einer echten Anreiz-Regelung. Besonderes Augenmerk der Arbeit liegt in einer durch das allgemeine Unbundling-Gebot bedingten rigorosen Trennung zwischen Markt- und Netzakteuren. Nach umfangreicher Analyse des hierbei entstehenden geschlossenen Regelkreises erfolgt die beispielhafte Anwendung der Regelungsarchitektur für den Aufbau eines neuartigen Echtzeit-Engpassmanagementsystems. Weitere praktische Vorteile des entwickelten Ansatzes im Vergleich zu bestehenden Konzepten werden anhand zweier Fallstudien deutlich. Die port-basierte Systemmodellierung, der Verzicht auf zentralisierte Regeleingriffe und nicht zuletzt die Möglichkeit zur automatischen, dezentralen Selbstregulation aller Preise über das Gesamtnetz hinweg stellen schließlich die problemlose Erweiterbarkeit um zusätzliche optionale Anreizkomponenten sicher

Dynamic Incentives for Optimal Control of Competitive Power Systems

This work presents a real-time dynamic pricing framework for future electricity markets. Deduced by first-principles analysis of physical, economic, and communication constraints within the power system, the proposed feedback control mechanism ensures both closed-loop system stability and economic efficiency at any given time. The resulting price signals are able to incentivize competitive market participants to eliminate spatio-temporal shortages in power supply quickly and purposively

Dynamic Incentives for Optimal Control of Competitive Power Systems

This work presents a real-time dynamic pricing framework for future electricity markets. Deduced by first-principles analysis of physical, economic, and communication constraints within the power system, the proposed feedback control mechanism ensures both closed-loop system stability and economic efficiency at any given time. The resulting price signals are able to incentivize competitive market participants to eliminate spatio-temporal shortages in power supply quickly and purposively

Diagonal and normal with Toeplitz-block splitting iteration method for space fractional coupled nonlinear Schr\"odinger equations with repulsive nonlinearities

By applying the linearly implicit conservative difference scheme proposed in [D.-L. Wang, A.-G. Xiao, W. Yang. J. Comput. Phys. 2014;272:670-681], the system of repulsive space fractional coupled nonlinear Schr\"odinger equations leads to a sequence of linear systems with complex symmetric and Toeplitz-plus-diagonal structure. In this paper, we propose the diagonal and normal with Toeplitz-block splitting iteration method to solve the above linear systems. The new iteration method is proved to converge unconditionally, and the optimal iteration parameter is deducted. Naturally, this new iteration method leads to a diagonal and normal with circulant-block preconditioner which can be executed efficiently by fast algorithms. In theory, we provide sharp bounds for the eigenvalues of the discrete fractional Laplacian and its circulant approximation, and further analysis indicates that the spectral distribution of the preconditioned system matrix is tight. Numerical experiments show that the new preconditioner can significantly improve the computational efficiency of the Krylov subspace iteration methods. Moreover, the corresponding preconditioned GMRES method shows space mesh size independent and almost fractional order parameter insensitive convergence behaviors

An evaluation of approximate probabilistic reachability techniques for stochastic parametric hybrid systems

Ph. D. ThesisStochastic parametric hybrid systems allow formalising automata with discrete interruptions, continuous nonlinear dynamics and parametric uncertainty (e.g. randomness and/or nondeterminism), and are a useful framework for cyber-physical systems modelling. The problem of designing safe cyber-physical systems is very timely, given that such systems are ubiquitous in modern society, often in safety-critical contexts (e.g., aircraft and cars) with possibly some level of decisional autonomy. Therefore, the verification of cyber-physical systems (and consequently of hybrid systems) is a problem urgently demanding innovative solutions. Unfortunately, this problem is also extremely challenging. Reachability checking is a crucial element of designing safe systems. Given a system model, we specify a set of "goal" states (indicating (un)wanted behaviour) and ask whether the system evolution can reach these states or not. Probabilistic reachability is the corresponding problem for stochastic systems, and it amounts to computing the probability that the system reaches a goal state. The main problem researched in this thesis is probabilistic reachability analysis of hybrid systems with random and/or nondeterministic parameters. For nondeterministic systems, this problem amounts to computing a range of reachability probabilities depending on how nondeterminism is resolved. In this thesis I have investigated and developed three distinct techniques: Statistical methods, involving Monte Carlo, Quasi-Monte Carlo and Randomised Quasi-Monte Carlo sampling with interval estimation techniques which give statistical guarantees; An analytical approximation method, utilising Gaussian Processes that offer a statistical approximation for an (unknown) smooth function over its entire domain; A promising combination of a formal approach, based on formal reasoning which provides absolute numerical guarantees, and the Gaussian Regression method. This research offers contributions on two different levels to the verification of stochastic parametric hybrid systems. From a theoretical point of view, it offers a proof that the reachability probability function is a smooth function of the uncertain parameters of the model, and hence Gaussian Processes techniques can be used to obtain an efficient analytical approximation of the function. From a practical point of view, I have implemented all the above described statistical and approximation techniques as part of the publicly available ProbReach tool, including a Gaussian Process Expectation Propagation algorithm that performs Gaussian Process classification and regression for uni-variate and multiple class labels. My empirical evaluation of the presented techniques to a number of case studies has shown a great Gaussian Process approach advantage with respect to standard statistical model checking techniques.SAgE Doctoral Training Scholarships of Newcastle Universit

A Novel Reinforcement Learning-Optimization Approach for Integrating Wind Energy to Power System with Vehicle-to-Grid Technology

High integration of intermittent renewable energy sources (RES), specifically wind power, has created complexities in power system operations due to their limited controllability and predictability. In addition, large fleets of Electric Vehicles (EVs) are expected to have a large impact on electricity consumption, contributing to the volatility. In this dissertation, a well-coordinated smart charging approach is developed that utilizes the flexibility of EV owners in a way where EVs are used as distributed energy storage units and flexible loads to absorb the fluctuations in the wind power output in a vehicle-to-grid (V2G) setup. Challenges for people participation in V2G, such as battery degradation and insecurity about unexpected trips, are also addressed by using an interactive mechanism in smart grid. First, a static deterministic model is formulated using multi-objective mixed-integer quadratic programming (MIQP) assuming known parameters day ahead of time. Subsequently, a formulation for real-time dynamic schedule is provided using a rolling-horizon with expected value approximation. Simulation experiments demonstrate a significant increase in wind utilization and reduction in charging cost and battery degradation compared to an uncontrolled charging scenario. Formulating the scheduling problem of the EV-wind integrated power system using conventional stochastic programming (SP) approaches is challenging due to the presence of many uncertain parameters with unknown underlying distributions, such as wind, price, and different commuting patterns of EV owners. To alleviate the problem, a model-free Reinforcement Learning (RL) algorithm integrated with deterministic optimization is proposed that can be applied on many multi-stage stochastic problems while mitigating some of the challenges of conventional SP methods (e.g., large scenario tree, computational complexity) as well as the challenges in model-free RL (e.g., slow convergence, unstable learning in dynamic environment). The simulation results of applying the combined approach on the EV scheduling problem demonstrate the effectiveness of the RL-Optimization method in solving the multi-stage EV charge/discharge scheduling problem. The proposed methods perform better than standard RL approaches (e.g., DDQN) in terms of convergence speed and finding the global optima. Moreover, to address the curse of dimensionality issue in RL with large action-state space, a heuristic EV fleet charging/discharging scheme is used combined with RL-optimization approach to solve the EV scheduling problem for a large number of EVs