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Stochastic Geometry Analysis of Cellular Networks
by Martin Haenggi Bartłomiej Błaszczyszyn Paul Keeler Sayandev MukherjeeAchieve faster and more efficient network design and optimization with this comprehensive guide. Some of the most prominent researchers in the field explain the very latest analytic techniques and results from stochastic geometry for modeling the signal-to-interference-plus-noise ratio (SINR) distribution in heterogeneous cellular networks. This book will help you understand the effects of combining different system deployment parameters on such key performance indicators as coverage and capacity, enabling efficient allocation of simulation resources. In addition to covering results for network models based on the Poisson point process, this book presents recent results for when non-Poisson base station configurations appear Poisson due to random propagation effects such as fading and shadowing, as well as non-Poisson models for base station configurations, with a focus on determinantal point processes and tractable approximation methods. Theoretical results are illustrated with practical long-term evolution (LTE) applications and compared with real-world deployment results.
Stochastic Geometry Analysis of Multi-Antenna Wireless Networks
by Xianghao Yu Chang Li Jun Zhang Khaled B. LetaiefThis book presents a unified framework for the tractable analysis of large-scale, multi-antenna wireless networks using stochastic geometry. This mathematical analysis is essential for assessing and understanding the performance of complicated multi-antenna networks, which are one of the foundations of 5G and beyond networks to meet the ever-increasing demands for network capacity. Describing the salient properties of the framework, which makes the analysis of multi-antenna networks comparable to that of their single-antenna counterparts, the book discusses effective design approaches that do not require complex system-level simulations. It also includes various application examples with different multi-antenna network models to illustrate the framework’s effectiveness.
Stochastic Geometry Analysis of Space-Air-Ground Networks
by Minwei Shi Kai Yang He ZhouThis book presents a comprehensive framework for the theoretical analysis of space-air-ground networks using stochastic geometry. This analytical approach is indispensable for evaluating the performance of large-scale space-air-ground networks, which serve as critical facilitators for the advancement of the sixth-generation wireless communication aimed at providing high-speed broadband coverage for remote areas. By incorporating the features of topology and channel model in different tiers, this book investigates the key performance metrics in terms of load balancing, coverage assessment, and mobility management. The developed mechanisms provide effective design insights for space-air-ground networks while obviating the need for complex system-level simulations.
Stochastic Geometry for Image Analysis (Wiley-iste Ser.)
by Xavier DescombesThis book develops the stochastic geometry framework for image analysis purpose. Two main frameworks are described: marked point process and random closed sets models. We derive the main issues for defining an appropriate model. The algorithms for sampling and optimizing the models as well as for estimating parameters are reviewed. Numerous applications, covering remote sensing images, biological and medical imaging, are detailed. This book provides all the necessary tools for developing an image analysis application based on modern stochastic modeling.
Stochastic H2/H ∞ Control: A Nash Game Approach
by Weihai Zhang Lihua Xie Bor-Sen ChenThe H∞ control has been one of the important robust control approaches since the 1980s. This book extends the area to nonlinear stochastic H2/H∞ control, and studies more complex and practically useful mixed H2/H∞ controller synthesis rather than the pure H∞ control. Different from the commonly used convex optimization method, this book applies the Nash game approach to give necessary and sufficient conditions for the existence and uniqueness of the mixed H2/H∞ control. Researchers will benefit from our detailed exposition of the stochastic mixed H2/H∞ control theory, while practitioners can apply our efficient algorithms to address their practical problems.
Stochastic Hybrid Systems (Automation and Control Engineering)
by Christos G. Cassandras John LygerosBecause they incorporate both time- and event-driven dynamics, stochastic hybrid systems (SHS) have become ubiquitous in a variety of fields, from mathematical finance to biological processes to communication networks to engineering. Comprehensively integrating numerous cutting-edge studies, Stochastic Hybrid Systems presents a captivating treatment of some of the most ambitious types of dynamic systems.Cohesively edited by leading experts in the field, the book introduces the theoretical basics, computational methods, and applications of SHS. It first discusses the underlying principles behind SHS and the main design limitations of SHS. Building on these fundamentals, the authoritative contributors present methods for computer calculations that apply SHS analysis and synthesis techniques in practice. The book concludes with examples of systems encountered in a wide range of application areas, including molecular biology, communication networks, and air traffic management. It also explains how to resolve practical problems associated with these systems. Stochastic Hybrid Systems achieves an ideal balance between a theoretical treatment of SHS and practical considerations. The book skillfully explores the interaction of physical processes with computerized equipment in an uncertain environment, enabling a better understanding of sophisticated as well as everyday devices and processes.
Stochastic Large-Scale Engineering Systems
by Spyros G. TzafestasThis book focuses on the class of large-scale stochastic systems, which has dominated the attention of many academic and research groups. It discusses distributed-sensor networks, decentralized detection theory, and econometric models with integrated and decentralized policymakers.
Stochastic Modeling and Analysis of Telecom Networks
by Laurent Decreusefond Pascal MoyalThis book addresses the stochastic modeling of telecommunication networks, introducing the main mathematical tools for that purpose, such as Markov processes, real and spatial point processes and stochastic recursions, and presenting a wide list of results on stability, performances and comparison of systems. The authors propose a comprehensive mathematical construction of the foundations of stochastic network theory: Markov chains, continuous time Markov chains are extensively studied using an original martingale-based approach. A complete presentation of stochastic recursions from an ergodic theoretical perspective is also provided, as well as spatial point processes. Using these basic tools, stability criteria, performance measures and comparison principles are obtained for a wide class of models, from the canonical M/M/1 and G/G/1 queues to more sophisticated systems, including the current “hot topics” of spatial radio networking, OFDMA and real-time networks. Contents 1. Introduction. Part 1: Discrete-time Modeling 2. Stochastic Recursive Sequences. 3. Markov Chains. 4. Stationary Queues. 5. The M/GI/1 Queue. Part 2: Continuous-time Modeling 6. Poisson Process. 7. Markov Process. 8. Systems with Delay. 9. Loss Systems. Part 3: Spatial Modeling 10. Spatial Point Processes.
Stochastic Modeling for Medical Image Analysis
by Ayman El-Baz Georgy Gimel’farb Jasjit S. SuriStochastic Modeling for Medical Image Analysis provides a brief introduction to medical imaging, stochastic modeling, and model-guided image analysis.Today, image-guided computer-assisted diagnostics (CAD) faces two basic challenging problems. The first is the computationally feasible and accurate modeling of images from different modalities to obt
Stochastic Modeling for Reliability
by Maxim Finkelstein Ji Hwan ChaFocusing on shocks modeling, burn-in and heterogeneous populations, Stochastic Modeling for Reliability naturally combines these three topics in the unified stochastic framework and presents numerous practical examples that illustrate recent theoretical findings of the authors. The populations of manufactured items in industry are usually heterogeneous. However, the conventional reliability analysis is performed under the implicit assumption of homogeneity, which can result in distortion of the corresponding reliability indices and various misconceptions. Stochastic Modeling for Reliability fills this gap and presents the basics and further developments of reliability theory for heterogeneous populations. Specifically, the authors consider burn-in as a method of elimination of 'weak' items from heterogeneous populations. The real life objects are operating in a changing environment. One of the ways to model an impact of this environment is via the external shocks occurring in accordance with some stochastic point processes. The basic theory for Poisson shock processes is developed and also shocks as a method of burn-in and of the environmental stress screening for manufactured items are considered. Stochastic Modeling for Reliability introduces and explores the concept of burn-in in heterogeneous populations and its recent development, providing a sound reference for reliability engineers, applied mathematicians, product managers and manufacturers alike.
Stochastic Models for Prices Dynamics in Energy and Commodity Markets: An Infinite-Dimensional Perspective (Springer Finance)
by Fred Espen Benth Paul KrühnerThis monograph presents a theory for random field models in time and space, viewed as stochastic processes with values in a Hilbert space, to model the stochastic dynamics of forward and futures prices in energy, power, and commodity markets. In this book, the well-known Heath–Jarrow–Morton approach from interest rate theory is adopted and extended into an infinite-dimensional framework, allowing for flexible modeling of price stochasticity across time and along the term structure curve. Various models are introduced based on stochastic partial differential equations with infinite-dimensional Lévy processes as noise drivers, emphasizing random fields described by low-dimensional parametric covariance functions instead of classical high-dimensional factor models. The Filipović space, a separable Hilbert space of Sobolev type, is found to be a convenient state space for the dynamics of forward and futures term structures. The monograph provides a classification of important operators in this space, covering covariance operators and the stochastic modeling of volatility term structures, including the Samuelson effect. Fourier methods are employed to price many derivatives of interest in energy, power, and commodity markets, and sensitivity 'delta' expressions can be derived. Additionally, the monograph covers forward curve smoothing, the connection between forwards with fixed delivery and delivery period, as well as the classical theory of forward and futures pricing. This monograph will appeal to researchers and graduate students interested in mathematical finance and stochastic analysis applied in the challenging markets of energy, power, and commodities. Practitioners seeking sophisticated yet flexible and analytically tractable risk models will also find it valuable.
Stochastic Models in Reliability
by Terje Aven Uwe JensenThis book provides a comprehensive up-to-date presentation of some of the classical areas of reliability, based on a more advanced probabilistic framework using the modern theory of stochastic processes. This framework allows analysts to formulate general failure models, establish formulae for computing various performance measures, as well as determine how to identify optimal replacement policies in complex situations. In this second edition of the book, two major topics have been added to the original version: copula models which are used to study the effect of structural dependencies on the system reliability; and maintenance optimization which highlights delay time models under safety constraints. Terje Aven is Professor of Reliability and Risk Analysis at University of Stavanger, Norway. Uwe Jensen is working as a Professor at the Institute of Applied Mathematics and Statistics of the University of Hohenheim in Stuttgart, Germany. Review of first edition: "This is an excellent book on mathematical, statistical and stochastic models in reliability. The authors have done an excellent job of unifying some of the stochastic models in reliability. The book is a good reference book but may not be suitable as a textbook for students in professional fields such as engineering. This book may be used for graduate level seminar courses for students who have had at least the first course in stochastic processes and some knowledge of reliability mathematics. It should be a good reference book for researchers in reliability mathematics." --Mathematical Reviews (2000)
Stochastic Networked Control Systems
by Tamer Başar Serdar YükselNetworked control systems are increasingly ubiquitous today, with applications ranging from vehicle communication and adaptive power grids to space exploration and economics. The optimal design of such systems presents major challenges, requiring tools from various disciplines within applied mathematics such as decentralized control, stochastic control, information theory, and quantization. A thorough, self-contained book, Stochastic Networked Control Systems: Stabilization and Optimization under Information Constraints aims to connect these diverse disciplines with precision and rigor, while conveying design guidelines to controller architects. Unique in the literature, it lays a comprehensive theoretical foundation for the study of networked control systems, and introduces an array of concrete tools for work in the field. Salient features included: · Characterization, comparison and optimal design of information structures in static and dynamic teams. Operational, structural and topological properties of information structures in optimal decision making, with a systematic program for generating optimal encoding and control policies. The notion of signaling, and its utilization in stabilization and optimization of decentralized control systems. · Presentation of mathematical methods for stochastic stability of networked control systems using random-time, state-dependent drift conditions and martingale methods. · Characterization and study of information channels leading to various forms of stochastic stability such as stationarity, ergodicity, and quadratic stability; and connections with information and quantization theories. Analysis of various classes of centralized and decentralized control systems. · Jointly optimal design of encoding and control policies over various information channels and under general optimization criteria, including a detailed coverage of linear-quadratic-Gaussian models. · Decentralized agreement and dynamic optimization under information constraints. This monograph is geared toward a broad audience of academic and industrial researchers interested in control theory, information theory, optimization, economics, and applied mathematics. It could likewise serve as a supplemental graduate text. The reader is expected to have some familiarity with linear systems, stochastic processes, and Markov chains, but the necessary background can also be acquired in part through the four appendices included at the end. · Characterization, comparison and optimal design of information structures in static and dynamic teams. Operational, structural and topological properties of information structures in optimal decision making, with a systematic program for generating optimal encoding and control policies. The notion of signaling, and its utilization in stabilization and optimization of decentralized control systems. · Presentation of mathematical methods for stochastic stability of networked control systems using random-time, state-dependent drift conditions and martingale methods. · Characterization and study of information channels leading to various forms of stochastic stability such as stationarity, ergodicity, and quadratic stability; and connections with information and quantization theories. Analysis of various classes of centralized and decentralized control systems. · Jointly optimal design of encoding and control policies over various information channels and under general optimization criteria, including a detailed coverage of linear-quadratic-Gaussian models. · Decentralized agreement and dynamic optimization under information constraints. This monograph is geared toward a broad audience of academic and industrial researchers interested in control theory, information theory, optimization, economics, and applied mathematics. It could likewise serve as a supplemental graduate text. The reader is expected to have some familiarity with linear systems, stochastic processes, and Markov chai...
Stochastic Networks
by Frank Kelly Elena YudovinaIn the past decade the proliferation of local and global communication networks for computer and human communication, the development of parallel computers with large numbers of processors, and the design of flexible and robust manufacturing systems have spurred major advances in our understanding of queuing volume reviews recent progress. While research on queuing networks uses many of the traditional queuing theory insights, it is more concerned with how network components interact than with detailed models of how an individual queue behaves. In the last few years there have been some surprises, in particular with regard to the conditions for stability of multiclass queuing networks, which is covered in this book. It also covers the challenges reflected Brownian motion has set both as a mathematical object and as a modelling paradigm; the usefulness of ideas from the interacting particle system world; the application of large deviation theory; and the developing connections with optimization and dynamical systems theory.
Stochastic Optimal Control of Structures
by Jie Li Yongbo PengThis book proposes, for the first time, a basic formulation for structural control that takes into account the stochastic dynamics induced by engineering excitations in the nature of non-stationary and non-Gaussian processes. Further, it establishes the theory of and methods for stochastic optimal control of randomly-excited engineering structures in the context of probability density evolution methods, such as physically-based stochastic optimal (PSO) control. By logically integrating randomness into control gain, the book helps readers design elegant control systems, mitigate risks in civil engineering structures, and avoid the dilemmas posed by the methods predominantly applied in current practice, such as deterministic control and classical linear quadratic Gaussian (LQG) control associated with nominal white noises.
Stochastic Optimization for Distributed Energy Resources in Smart Grids
by Yuanxiong Guo Yuguang Fang Pramod P. KhargonekarThis brief focuses on stochastic energy optimization for distributed energy resources in smart grids. Along with a review of drivers and recent developments towards distributed energy resources, this brief presents research challenges of integrating millions of distributed energy resources into the grid. The brief then proposes a novel three-level hierarchical architecture for effectively integrating distributed energy resources into smart grids. Under the proposed hierarchical architecture, distributed energy resource management algorithms at the three levels (i. e. , smart home, smart neighborhood, and smart microgrid) are developed in this brief based on stochastic optimization that can handle the involved uncertainties in the system.
Stochastic Optimization Methods: Applications in Engineering and Operations Research
by Kurt MartiThis book examines optimization problems that in practice involve random model parameters. It outlines the computation of robust optimal solutions, i.e., optimal solutions that are insensitive to random parameter variations, where appropriate deterministic substitute problems are needed. Based on the probability distribution of the random data and using decision theoretical concepts, optimization problems under stochastic uncertainty are converted into corresponding deterministic problems.Due to the probabilities and expectations involved, the book also shows how to apply approximative solution techniques. Several deterministic and stochastic approximation methods are provided: Taylor expansion methods, regression and response surface methods (RSM), probability inequalities, multiple linearization of survival/failure domains, discretization methods, convex approximation/deterministic descent directions/efficient points, stochastic approximation and gradient procedures, and differentiation formulas for probabilities and expectations.The fourth edition of this classic text has been carefully and thoroughly revised. It includes new chapters on the solution of stochastic linear programs by discretization of the underlying probability distribution, and on solving deterministic optimization problems by means of controlled random search methods and multiple random search procedures. It also presents a new application of stochastic optimization methods to machine learning problems with different loss functions. For the computation of optimal feedback controls under stochastic uncertainty, besides the open-loop feedback procedures, a new method based on Taylor expansions with respect to the gain parameters is presented. The book is intended for researchers and graduate students who are interested in stochastics, stochastic optimization, and control. It will also benefit professionals and practitioners whose work involves technical, economicand/or operations research problems under stochastic uncertainty.
Stochastic Process Variation in Deep-Submicron CMOS
by Amir ZjajoOne of the most notable features of nanometer scale CMOS technology is the increasing magnitude of variability of the key device parameters affecting performance of integrated circuits. The growth of variability can be attributed to multiple factors, including the difficulty of manufacturing control, the emergence of new systematic variation-generating mechanisms, and most importantly, the increase in atomic-scale randomness, where device operation must be described as a stochastic process. In addition to wide-sense stationary stochastic device variability and temperature variation, existence of non-stationary stochastic electrical noise associated with fundamental processes in integrated-circuit devices represents an elementary limit on the performance of electronic circuits. In an attempt to address these issues, Stochastic Process Variation in Deep-Submicron CMOS: Circuits and Algorithms offers unique combination of mathematical treatment of random process variation, electrical noise and temperature and necessary circuit realizations for on-chip monitoring and performance calibration. The associated problems are addressed at various abstraction levels, i. e. circuit level, architecture level and system level. It therefore provides a broad view on the various solutions that have to be used and their possible combination in very effective complementary techniques for both analog/mixed-signal and digital circuits. The feasibility of the described algorithms and built-in circuitry has been verified by measurements from the silicon prototypes fabricated in standard 90 nm and 65 nm CMOS technology.
Stochastic Processes
by Toshio NakagawaReliability theory is of fundamental importance for engineers and managers involved in the manufacture of high-quality products and the design of reliable systems. In order to make sense of the theory, however, and to apply it to real systems, an understanding of the basic stochastic processes is indispensable. As well as providing readers with useful reliability studies and applications, Stochastic Processes also gives a basic treatment of such stochastic processes as: the Poisson process,the renewal process,the Markov chain,the Markov process, andthe Markov renewal process.Many examples are cited from reliability models to show the reader how to apply stochastic processes. Furthermore, Stochastic Processes gives a simple introduction to other stochastic processes such as the cumulative process, the Wiener process, the Brownian motion and reliability applications. Stochastic Processes is suitable for use as a reliability textbook by advanced undergraduate and graduate students. It is also of interest to researchers, engineers and managers who study or practise reliability and maintenance.
Stochastic Reachability Analysis of Hybrid Systems
by Luminita Manuela BujorianuStochastic reachability analysis (SRA) is a method of analyzing the behavior of control systems which mix discrete and continuous dynamics. For probabilistic discrete systems it has been shown to be a practical verification method but for stochastic hybrid systems it can be rather more. As a verification technique SRA can assess the safety and performance of, for example, autonomous systems, robot and aircraft path planning and multi-agent coordination but it can also be used for the adaptive control of such systems. Stochastic Reachability Analysis of Hybrid Systems is a self-contained and accessible introduction to this novel topic in the analysis and development of stochastic hybrid systems. Beginning with the relevant aspects of Markov models and introducing stochastic hybrid systems, the book then moves on to coverage of reachability analysis for stochastic hybrid systems. Following this build up, the core of the text first formally defines the concept of reachability in the stochastic framework and then treats issues representing the different faces of SRA: * stochastic reachability based on Markov process theory; * martingale methods; * stochastic reachability as an optimal stopping problem; and * dynamic programming. The book is rounded off by an appendix providing mathematical underpinning on subjects such as ordinary differential equations, probabilistic measure theory and stochastic modeling, which will help the non-expert-mathematician to appreciate the text. Stochastic Reachability Analysis of Hybrid Systems characterizes a highly interdisciplinary area of research and is consequently of significant interest to academic researchers and graduate students from a variety of backgrounds in control engineering, applied mathematics and computer science. The Communications and Control Engineering series reports major technological advances which have potential for great impact in the fields of communication and control. It reflects research in industrial and academic institutions around the world so that the readership can exploit new possibilities as they become available.
Stochastic Reliability and Maintenance Modeling
by Toshio Nakagawa Tadashi DohiIn honor of the work of Professor Shunji Osaki, Stochastic Reliability and Maintenance Modeling provides a comprehensive study of the legacy of and ongoing research in stochastic reliability and maintenance modeling. Including associated application areas such as dependable computing, performance evaluation, software engineering, communication engineering, distinguished researchers review and build on the contributions over the last four decades by Professor Shunji Osaki. Fundamental yet significant research results are presented and discussed clearly alongside new ideas and topics on stochastic reliability and maintenance modeling to inspire future research. Across 15 chapters readers gain the knowledge and understanding to apply reliability and maintenance theory to computer and communication systems. Stochastic Reliability and Maintenance Modeling is ideal for graduate students and researchers in reliability engineering, and workers, managers and engineers engaged in computer, maintenance and management works.
Stochastic Structural Optimization
by Makoto Yamakawa Makoto OhsakiStochastic Structural Optimization presents a comprehensive picture of robust design optimization of structures, focused on nonparametric stochastic-based methodologies. Good practical structural design accounts for uncertainty, for which reliability-based design offers a standard approach, usually incorporating assumptions on probability functions which are often unknown. By comparison, a worst-case approach with bounded support used as a robust design offers simplicity and a lower level of sensitivity. Linking structural optimization with these two approaches by a unified framework of non-parametric stochastic methodologies provides a rigorous theoretical background and high level of practicality. This text shows how to use this theoretical framework in civil and mechanical engineering practice to design a safe structure which accounts for uncertainty. Connects theory with practice in the robust design optimization of structures Advanced enough to support sound practical designs This book provides comprehensive coverage for engineers and graduate students in civil and mechanical engineering.Makoto Yamakawa is a Professor at Tokyo University of Science, and a member of the Advisory Board of the 2020 Asian Congress of Structural and Multidisciplinary Optimization. Makoto Ohsaki is a Professor at Kyoto University, Japan, treasurer of the International Association for Shell & Spatial Structures and former President of the Asian Society for Structural and Multidisciplinary Optimization.
Stochastic Systems
by Mircea GrigoriuUncertainty is an inherent feature of both properties of physical systems and the inputs to these systems that needs to be quantified for cost effective and reliable designs. The states of these systems satisfy equations with random entries, referred to as stochastic equations, so that they are random functions of time and/or space. The solution of stochastic equations poses notable technical difficulties that are frequently circumvented by heuristic assumptions at the expense of accuracy and rigor. The main objective of Stochastic Systems is to promoting the development of accurate and efficient methods for solving stochastic equations and to foster interactions between engineers, scientists, and mathematicians. To achieve these objectives Stochastic Systems presents: A clear and brief review of essential concepts on probability theory, random functions, stochastic calculus, Monte Carlo simulation, and functional analysis Probabilistic models for random variables and functions needed to formulate stochastic equations describing realistic problems in engineering and applied sciences Practical methods for quantifying the uncertain parameters in the definition of stochastic equations, solving approximately these equations, and assessing the accuracy of approximate solutions Stochastic Systems provides key information for researchers, graduate students, and engineers who are interested in the formulation and solution of stochastic problems encountered in a broad range of disciplines. Numerous examples are used to clarify and illustrate theoretical concepts and methods for solving stochastic equations. The extensive bibliography and index at the end of the book constitute an ideal resource for both theoreticians and practitioners.
Stochastic Teams, Games, and Control under Information Constraints (Systems & Control: Foundations & Applications)
by Tamer Başar Serdar YükselThis monograph presents a mathematically rigorous and accessible treatment of the interaction between information, decision, control, and probability in single-agent and multi-agent systems. The book provides a comprehensive and unified theory of information structures for stochastic control, stochastic teams, stochastic games, and networked control systems.Part I of the text is concerned with a general mathematical theory of information structures for stochastic teams, leading to systematic characterizations and classifications, geometric and topological properties, implications on existence, approximations and relaxations, their comparison, and regularity of optimal solutions in information. Information structures in stochastic games are then considered in Part II, and the dependence of equilibrium solutions and behavior on information is demonstrated. Part III studies information design through information theory in networked control systems – both linear and nonlinear – and discusses optimality and stability criteria. Finally, Part IV introduces information and signaling games under several solution concepts, with applications to prior mismatch, cost mismatch and privacy, reputation games and jamming. This text will be a valuable resource for researchers and graduate students interested in control theory, information theory, statistics, game theory, and applied mathematics. Readers should be familiar with the basics of linear systems theory, stochastic processes, and Markov chains.
Stochastic Tools in Mathematics and Science
by Alexandre J. Chorin Ole H Hald"Stochastic Tools in Mathematics and Science" covers basic stochastic tools used in physics, chemistry, engineering and the life sciences. The topics covered include conditional expectations, stochastic processes, Brownian motion and its relation to partial differential equations, Langevin equations, the Liouville and Fokker-Planck equations, as well as Markov chain Monte Carlo algorithms, renormalization, basic statistical mechanics, and generalized Langevin equations and the Mori-Zwanzig formalism. The applications include sampling algorithms, data assimilation, prediction from partial data, spectral analysis, and turbulence. The book is based on lecture notes from a class that has attracted graduate and advanced undergraduate students from mathematics and from many other science departments at the University of California, Berkeley. Each chapter is followed by exercises. The book will be useful for scientists and engineers working in a wide range of fields and applications. For this new edition the material has been thoroughly reorganized and updated, and new sections on scaling, sampling, filtering and data assimilation, based on recent research, have been added. There are additional figures and exercises. Review of earlier edition: "This is an excellent concise textbook which can be used for self-study by graduate and advanced undergraduate students and as a recommended textbook for an introductory course on probabilistic tools in science." Mathematical Reviews, 2006