Random Walks Along the Streets and Canals in Compact Cities: Spectral analysis, Dynamical Modularity, Information, and Statistical Mechanics

Different models of random walks on the dual graphs of compact urban structures are considered. Analysis of access times between streets helps to detect the city modularity. The statistical mechanics approach to the ensembles of lazy random walkers is developed. The complexity of city modularity can be measured by an information-like parameter which plays the role of an individual fingerprint of {\it Genius loci}. Global structural properties of a city can be characterized by the thermodynamical parameters calculated in the random walks problem.Comment: 44 pages, 22 figures, 2 table

Constructive solution methodologies to the capacitated newsvendor problem and surrogate extension

The newsvendor problem is a single-period stochastic model used to determine the order quantity of perishable product that maximizes/minimizes the profit/cost of the vendor under uncertain demand. The goal is to fmd an initial order quantity that can offset the impact of backlog or shortage caused by mismatch between the procurement amount and uncertain demand. If there are multiple products and substitution between them is feasible, overstocking and understocking can be further reduced and hence, the vendor\u27s overall profit is improved compared to the standard problem. When there are one or more resource constraints, such as budget, volume or weight, it becomes a constrained newsvendor problem. In the past few decades, many researchers have proposed solution methods to solve the newsvendor problem. The literature is first reviewed where the performance of each of existing model is examined and its contribution is reported. To add to these works, it is complemented through developing constructive solution methods and extending the existing published works by introducing the product substitution models which so far has not received sufficient attention despite its importance to supply chain management decisions. To illustrate this dissertation provides an easy-to-use approach that utilizes the known network flow problem or knapsack problem. Then, a polynomial in fashion algorithm is developed to solve it. Extensive numerical experiments are conducted to compare the performance of the proposed method and some existing ones. Results show that the proposed approach though approximates, yet, it simplifies the solution steps without sacrificing accuracy. Further, this dissertation addresses the important arena of product substitute models. These models deal with two perishable products, a primary product and a surrogate one. The primary product yields higher profit than the surrogate. If the demand of the primary exceeds the available quantity and there is excess amount of the surrogate, this excess quantity can be utilized to fulfill the shortage. The objective is to find the optimal lot sizes of both products, that minimize the total cost (alternatively, maximize the profit). Simulation is utilized to validate the developed model. Since the analytical solutions are difficult to obtain, Mathematical software is employed to find the optimal results. Numerical experiments are also conducted to analyze the behavior of the optimal results versus the governing parameters. The results show the contribution of surrogate approach to the overall performance of the policy. From a practical perspective, this dissertation introduces the applications of the proposed models and methods in different industries such as inventory management, grocery retailing, fashion sector and hotel reservation

A Tutorial on Quantum Master Equations: Tips and tricks for quantum optics, quantum computing and beyond

Quantum master equations are an invaluable tool to model the dynamics of a plethora of microscopic systems, ranging from quantum optics and quantum information processing, to energy and charge transport, electronic and nuclear spin resonance, photochemistry, and more. This tutorial offers a concise and pedagogical introduction to quantum master equations, accessible to a broad, cross-disciplinary audience. The reader is guided through the basics of quantum dynamics with hands-on examples that build up in complexity. The tutorial covers essential methods like the Lindblad master equation, Redfield relaxation, and Floquet theory, as well as techniques like Suzuki-Trotter expansion and numerical approaches for sparse solvers. These methods are illustrated with code snippets implemented in python and other languages, which can be used as a starting point for generalisation and more sophisticated implementations.Comment: 57 pages, 12 figures, 34 code example

Probabilistic Framework for Sensor Management

A probabilistic sensor management framework is introduced, which maximizes the utility of sensor systems with many different sensing modalities by dynamically configuring the sensor system in the most beneficial way. For this purpose, techniques from stochastic control and Bayesian estimation are combined such that long-term effects of possible sensor configurations and stochastic uncertainties resulting from noisy measurements can be incorporated into the sensor management decisions

Adaptive high-resolution finite element schemes

The numerical treatment of flow problems by the finite element method is addressed. An algebraic approach to constructing high-resolution schemes for scalar conservation laws as well as for the compressible Euler equations is pursued. Starting from the standard Galerkin approximation, a diffusive low-order discretization is constructed by performing conservative matrix manipulations. Flux limiting is employed to compute the admissible amount of compensating antidiffusion which is applied in regions, where the solution is sufficiently smooth, to recover the accuracy of the Galerkin finite element scheme to the largest extent without generating non-physical oscillations in the vicinity of steep gradients. A discrete Newton algorithm is proposed for the solution of nonlinear systems of equations and it is compared to the standard fixed-point defect correction approach. The Jacobian operator is approximated by divided differences and an edge-based procedure for matrix assembly is devised exploiting the special structure of the underlying algebraic flux correction (AFC) scheme. Furthermore, a hierarchical mesh adaptation algorithm is designed for the simulation of steady-state and transient flow problems alike. Recovery-based error indicators are used to control local mesh refinement based on the red-green strategy for element subdivision. A vertex locking algorithm is developed which leads to an economical re-coarsening of patches of subdivided cells. Efficient data structures and implementation details are discussed. Numerical examples for scalar conservation laws and the compressible Euler equations in two dimensions are presented to assess the performance of the solution procedure.In dieser Arbeit wird die numerische Simulation von skalaren Erhaltungsgleichungen sowie von kompressiblen Eulergleichungen mit Hilfe der Finite-Elemente Methode behandelt. Dazu werden hochauflösende Diskretisierungsverfahren eingesetzt, welche auf algebraischen Konstruktionsprinzipien basieren. Ausgehend von der Galerkin-Approximation wird eine Methode niedriger Ordnung konstruiert, indem konservative Matrixmodifikationen durchgeführt werden. Anschließend kommt ein sog. Flux-Limiter zum Einsatz, der in Abhängigkeit von der lokalen Glattheit der Lösung den zulässigen Anteil an Antidiffusion bestimmt, die zur Lösung der Methode niedriger Ordnung hinzuaddiert werden kann, ohne dass unphysikalische Oszillationen in der Nähe von steilen Gradienten entstehen. Die resultierenden nichtlinearen Gleichungssysteme können entweder mit Hilfe von Fixpunkt-Defektkorrektur-Techniken oder mittels diskreter Newton-Verfahren gelöst werden. Für letztere wird die Jacobi-Matrix mit dividierten Differenzen approximiert, wobei ein effizienter, kantenbasierter Matrixaufbau aufgrund der speziellen Struktur der zugrunde liegenden Diskretisierung möglich ist. Ferner wird ein hierarchischer Gitteradaptionsalgorithmus vorgestellt, welcher sowohl für die Simulation von stationären als auch zeitabhängigen Strömungen geeignet ist. Die lokale Gitterverfeinerung folgt dem bekannten Rot-Grün Prinzip, wobei rekonstruktionsbasierte Fehlerindikatoren zur Markierung von Elementen zum Einsatz kommen. Ferner erlaubt das sukzessive Sperren von Knoten, die nicht gelöscht werden können, eine kostengünstige Rückvergröberung von zuvor unterteilten Elementen. In der Arbeit wird auf verschiedene Aspekte der Implementierung sowie auf die Wahl von effizienten Datenstrukturen zur Verwaltung der Gitterinformationen eingegangen. Der Nutzen der vorgestellten Simulationswerkzeuge wird anhand von zweidimensionalen Beispielrechnungen für skalare Erhaltungsgleichungen sowie für die kompressiblen Eulergleichungen analysiert