La confiance dans les systèmes complexes est aujourd'hui primordiale. Parmi les langages de modélisation dysfonctionnelle des systèmes, les chaînes de Markov sont un bon compromis entre la calculabilité des modèles et le pouvoir d'expression qu'elles apportent. Cependant, comme les chaînes de Markov rendent compte des différents états du système, leur taille est confrontée à l'explosion combinatoire. Il y a deux obstacles majeurs induits par cette explosion : la difficulté d'écrire des chaînes pour les grands systèmes à la main, et les besoins en ressources calculatoires pour leur résolution. Le premier obstacle est dépassé facilement en compilant les chaînes de Markov depuis un modèle de plus haut niveau (dans ces travaux, AltaRica 3.0 est utilisé).Dans cette thèse, nous nous sommes concentrés sur la génération partielle de chaînes de Markov, afin de dépasser le problème d'explosion combinatoire. La méthode est fondée sur l'observation que les systèmes réparables, même les plus grands, passent leur temps dans un petit nombre d'états proches de l'état nominal du système. La génération partielle utilise l'algorithme de Dijkstra, auquel est combiné un facteur de pertinence, qui permet la sélection des états les plus probables du système. Il est possible d'encadrer les valeurs des indicateurs de sûreté obtenus avec la chaîne partielle grâce à l'introduction d'une chaîne partielle avec puits.La méthode de génération partielle est entièrement implémentée et fait partie du projet AltaRica 3.0. Il est ainsi possible de calculer les indicateurs de sûreté des systèmes directement depuis un modèle AltaRica. Divers expériences ont été menées pour illustrer la faisabilité de la méthode, son passage à l'échelle, ainsi que ses points forts et ses limites. / Trustworthiness in systems is of paramount importance. Among safety modeling languages, Markov chains are a good tradeoff between the safety concepts that can be modeled and the ease of calculation. However, as they model the different states of the systems, they suffer from the state space explosion. This explosion has two drawbacks: it makes Markov chains very difficult to write by hand for large systems, and large Markov chain calculation is resource consuming. The first drawback is easily tackled by generating Markov chains from higher-level languages (such as AltaRica 3.0).In this thesis, we focused on the partial generation of Markov chains, to tackle the state space explosion of the models. This idea is based on the observation that even large repairable systems spent most of their times in a few number of states, that are close to the nominal state of the system. The partial generation is based on Dijkstra's algorithm and on a so-called relevance factor to generate only the most probable states of the Markov chain. The reliability indicators obtained with such a partial chain can be bounded with a slightly different Markov chain.The partial generation method is fully implemented in the AltaRica 3.0 project to automatically calculate the reliability indicators of a system modeled in AltaRica. Different experiments illustrate the practability of the method, as well as its strengths and weaknesses.

The use of networks to model the spread of epidemics through structured populations is widespread. However, epidemics on networks lead to intractable exact systems with the need to coarse grain and focus on some average quantities. Often, the underlying stochastic processes are Markovian and so are the resulting mean-field models constructed as systems of ordinary differential equations (ODEs). However, the lack of memory (or memorylessness) does not accurately describe real disease dynamics. For instance, many epidemiological studies have shown that the true distribution of the infectious period is rather centred around its mean, whereas the memoryless assumption imposes an exponential distribution on the infectious period. Assumptions such as these greatly affect the predicted course of an epidemic and can lead to inaccurate predictions about disease spread. Such limitations of existing approaches to modelling epidemics on networks motivated my efforts to develop non-Markovian models which would be better suited to capture essential realistic features of disease dynamics. In the first part of my thesis I developed a pairwise, multi-stage SIR (susceptible-infected-recovered) model. Each infectious node goes through some K 2 N infectious stages, which for K > 1 means that the infectious period is gamma-distributed. Analysis of the model provided analytic expressions for the epidemic threshold and the expected final epidemic size. Using available epidemiological data on the infectious periods of various diseases, I demonstrated the importance of considering the shape of the infectious period distribution. The second part of the thesis expanded the framework of non-Markovian dynamics to networks with heterogeneous degree distributions with non-negligible levels of clustering. These properties are ubiquitous in many real-world networks and make model development and analysis much more challenging. To this end, I have derived and analysed a compact pairwise model with the number of equations being independent of the range of node degrees, and investigated the effects of clustering on epidemic dynamics. My thesis culminated with the third part where I explored the relationships between several different modelling methodologies, and derived an original non-Markovian Edge-Based Compartmental Model (EBCM) which allows both transmission and recovery to be arbitrary independent stochastic processes. The major result is a rigorous mathematical proof that the message passing (MP) model and the EBCM are equivalent, and thus, the EBCM is statistically exact on the ensemble of configuration model networks. From this consideration I derived a generalised pairwise-like model which I then used to build a model hierarchy, and to show that, given corresponding parameters and initial conditions, these models are identical to MP model or EBCM. In the final part of my thesis I considered the important problem of coupling epidemic dynamics with changes in network structure in response to the perceived risk of the epidemic. This was framed as a susceptible-infected-susceptible (SIS) model on an adaptive network, where susceptible nodes can disconnect from infected neighbours and, after some fixed time delay, connect to a random susceptible node that they are not yet connected to. This model assumes that nodes have perfect information on the state of all other nodes. Robust oscillations were found in a significant region of the parameter space, including an enclosed region known as an 'endemic bubble'. The major contribution of this work was to show that oscillations can occur in a wide region of the parameter space, this is in stark contrast with most previous research where oscillations were limited to a very narrow region of the parameter space. Any mathematical model is a simplification of reality where assumptions must be made. The models presented here show the importance of interrogating these assumptions to ensure that they are as realistic as possible while still being amenable to analysis.

The purpose of this PhD thesis is to study mesoscopic and macroscopic fluctuations in Interacting Particle Systems. The thesis is split into two main parts. In the first part, we consider a system of Ising spins interacting via Kac potential evolving with Glauber dynamics and study the macroscopic motion of an one-dimensional interface under forced displacement as the result of large scale fluctuations. In the second part, we consider a diffusive system modelled by a Simple Symmetric Exclusion Process (SSEP) which is driven out of equilibrium by the action of current reservoirs at the boundary and study the non-equilibrium fluctuations around the hydrodynamic limit for the SSEP with current reservoirs. We give a brief summary of the first part. In recent years, there has been significant effort to derive deterministic models describing two-phase materials and their dynamical properties. In this context, we investigate the law that governs the power needed to force a motion of a one dimensional macroscopic interface between two different phases of a given ferromagnetic sample with a prescribed speed V at low temperature. We show that given the mesoscopic deterministic non-local evolution equation for the magnetisation (a non local version of the Allen-Cahn equation), we consider a stochastic Ising spin system with Glauber dynamics and Kac interaction (the underlying microscopic stochastic process) whose mesoscopic scaling limit (intermediate scale between microscale and macroscale) is the given PDE, and we calculate the corresponding large deviations functional which would provide the action functional. We obtain that by deriving upper and lower bounds of the large deviation cost functional. Concepts from statistical mechanics such as contours, free energy, local equilibrium allow a better understanding of the structure of the cost functional. Then we characterise the limiting behaviour of the action functional under a parabolic rescaling, by proving that for small values of the ratio between the distance and the time, the interface moves with a constant speed, while for larger values the occurrence of nucleations is the preferred way to make the transition. This led to a production of two published papers [12] and [14] with my supervisor D. Tsagkarogiannis and N. Dirr. In the second part we study the non-equilibrium fluctuations of a system modelled by SSEP with current reservoirs around its hydrodynamic limit. In particular, we prove that, in the limit, the appropriately scaled fluctuation field is given by a Generalised Ornstein- Uhlenbeck process. For the characterisation of the limiting fluctuation field we implement the Holley-Stroock theory. This is not straightforward due to the boundary terms coming from the nature of the model. Hence, by following a martingale approach (martingale decomposition) and the derivation of the equation of the variance for this model combined with “good” enough correlation estimates (the so-called v-estimates), we reduce the problem to a form whose Holley-Stroock result in [45] is now applicable. This is work in progress jointly with my supervisor and P. Gonçalves, [13].

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 109-112).

No description available.

Diss. (sammanfattning) Uppsala : Uppsala universitet, 2005. / Härtill 4 uppsatser.

Orientador: Simão Nicolau Stelmastchuk / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T21:13:11Z (GMT). No. of bitstreams: 1 Vieira_FranciscoZuiltonGoncalves_M.pdf: 2460011 bytes, checksum: bb34e809ab256fe3bb3b1bd74fc35eec (MD5) Previous issue date: 2011 / Resumo: Esta dissertação tem como tema o estudo das cadeias de Markov discretas com valores em um espaço de estados enumerável. Cadeias de Markov são processos estocásticos no seguinte sentido: dado o momento presente, o futuro não depende do passado, mas somente do momento presente. Nosso estudo é realizado sobre cadeias de Markov homogêneas (CMH) discretas. Inicialmente, introduzimos a definição e conceitos básicos das CMH discretas. Tais estudos nos conduzem ao conceito de topologia das matrizes de Transição associada as CMH. A topologia de tais cadeias é a ferramenta necessária para o estudo dos conjuntos recorrentes e transcientes, os quais são de grande importância nesta teoria. O estudo de estados estacionários e a propriedade forte de Markov também são abordados. Esta última propriedade serve para construção do conceito de estado recorrente. A partir deste último conceito trabalhamos com os conceitos de positivo e nulo recorrente. Por fim, estudamos o importante conceito de tempo absorção, o qual é entendido como o tempo que algum estado é absorvido a um conjunto recorrente / Abstract: This dissertation deals with the study of discrete Markov chains with values in a countable state space. Markov chains are processes stochastic in the following sense: given the present moment, the future does not depend on the past, but only in the present moment. Our study is conducted on homogeneous Markov chains (HMC) discrete. Initially, we introduced the definition and the basic concepts of discrete HMC. Such studies lead us to understand the concept of topology Transition matrices associated to HMC. The topology of these chains is a necessary tool for the study of the recurrent and transient sets, which are of great importance in this theory. The study of steady states and the strong Markov properties are also addressed. This latter property serves to build the concept of recurrent state. From this latter concept we work with the concepts of positive and null recurrent. Finally, we studied the important concept of absorption time, which is understood as the time that some state is absorbed to a set recurrent / Mestrado / Matematica / Mestre em Matemática

Accurate terrain models provide the chassis designer with a powerful tool to make informed design decisions early in the design process. During this stage, engineers are challenged with predicting vehicle loads through modeling and simulation. The accuracy of these simulation results depends not only on the fidelity of the model, but also on the excitation to the model. It is clear that the terrain is the main excitation to the vehicle [1]. The inputs to these models are often based directly on physical measurements (terrain profiles); therefore, the terrain measurements must be as accurate as possible. A collection of novel methods can be developed to aid in the study and application of 3D terrain measurements, which are dense and non-uniform, including efficient gridding, stochastic modeling, and compact characterization. Terrain measurements are not collected with uniform spacing, which is necessary for efficient data storage and simulation. Many techniques are developed to help effectively grid dense terrain point clouds in a curved regular grid (CRG) format, including center and random vehicle paths, sorted gridding methods, and software implementation. In addition, it is beneficial to characterize the terrain as a realization of an underlying stochastic process and to develop a mathematical model of that process. A method is developed to represent a continuous-state Markov chain as a collection of univariate distributions, to be applied to terrain road profiles. The resulting form is extremely customizable and significantly more compact than a discrete-state Markov chain, yet it still provides a viable alternative for stochastically modeling terrain. Many new simulation techniques take advantage of 3D gridded roads along with traditional 2D terrain profiles. A technique is developed to model and synthesize 3D terrain surfaces by applying a variety of 2D stochastic models to the topological components of terrain, which are also decomposed into frequency bandwidths and down-sampled. The quality of the synthetic surface is determined using many statistical tests, and the entire work is implemented into a powerful software suite. Engineers from many disciplines who work with terrain surfaces need to describe the overall physical characteristics compactly and consistently. A method is developed to characterize terrain surfaces with a few coefficients by performing a principal component analysis, via singular value decomposition (SVD), to the parameter sets that define a collection of surface models. / Master of Science

Integrated Vehicle Health Management (IVHM) systems use models and algorithmic techniques to process Condition-based Data (CBD) to offer prognostic information and actionable imperatives in support of Condition-based Maintenance (CBM) for the system. IVHM technology was first introduced by NASA to gather data, diagnose, detect, and predict faults, and support operational and post-maintenance activities in space vehicles. Eventually, it expanded to other vehicle types such as aircraft, ships, and land vehicles [1]. In recent years, the United States Army has been implementing a policy of CBM to transition from preventive to predictive maintenance [2]. One of the many challenges faced by the Army is the lack of accurate methods to assess ground vehicle reliability using modeling and/or simulation. This study aims at developing a Markov Chain-based algorithm that can detect anomalies and that is capable of accurately predicting the operational states of military ground vehicles. Several different Markov Chain Models (MCMs) have been developed and tested in their ability to predict the next state of a vehicle, given its current state (diagnostics and prognostics), and to examine how well a given model can detect unknown measurements (anomaly detection). A target of 90% Correct Classification (PCC) was established for all the vehicle performance data. The results suggest that it is possible to predict at a high level of accuracy the likely operational states of the military vehicles using MCMs. The anomaly detection test results revealed that MCMs can clearly distinguish a change in the performance data, that does not match the expected performance.

In max-plus algebra we work with the max-plus semi-ring which is the set ℝ_max=[-∞)∪ℝ together with operations 𝑎⊕𝑏 = max(𝑎,𝑏) and 𝑎⊗𝑏= 𝑎+𝑏.  The additive and multiplicative identities are taken to be ε=-∞ and ε=0 respectively. Max-plus algebra is one of many idempotent semi-rings which have been considered in various fields of mathematics. Max-plus algebra is becoming more popular not only because its operations are associative, commutative and distributive as in conventional algebra but because it takes systems that are non-linear in conventional algebra and makes them linear. Max-plus algebra also arises as the algebra of asymptotic growth rates of functions in conventional algebra which will play a significant role in several aspects of this thesis. This thesis is a survey of max-plus algebra that will concentrate on max-plus linear algebra results. We will then consider from a max-plus perspective several results by Wentzell and Freidlin for finite state Markov chains with an asymptotic dependence. / Master of Science