Vector Quantization, a pioneering discretization method based on nearest neighbor search, emerged in the 1950s primarily in signal processing, electrical engineering, and information theory. Later in the 1960s, it evolved into an automatic classification technique for generating prototypes of extensive datasets. In modern terms, it can be recognized as a seminal contribution to unsupervised learning through the k-means clustering algorithm in data science. In contrast, Functional Quantization, a more recent area of study dating back to the early 2000s, focuses on the quantization of continuous-time stochastic processes viewed as random vectors in Banach function spaces. This book distinguishes itself by delving into the quantization of random vectors with values in a Banach space—a unique feature of its content. Its main objectives are twofold: first, to offer a comprehensive and cohesive overview of the latest developments as well as several new results in optimal quantization theory, spanning both finite and infinite dimensions, building upon the advancements detailed in Graf and Luschgy's Lecture Notes volume. Secondly, it serves to demonstrate how optimal quantization can be employed as a space discretization method within probability theory and numerical probability, particularly in fields like quantitative finance. The main applications to numerical probability are the controlled approximation of regular and conditional expectations by quantization-based cubature formulas, with applications to time-space discretization of Markov processes, typically Brownian diffusions, by quantization trees. While primarily catering to mathematicians specializing in probability theory and numerical probability, this monograph also holds relevance for data scientists, electrical engineers involved in data transmission, and professionals in economics and logistics who are intrigued by optimal allocation problems.

This contributed volume convenes a rich selection of works with a focus on innovative mathematical methods with applications in real-world, industrial problems. Studies included in this book are all motivated by a relevant industrial challenge, and demonstrate that mathematics for industry can be extremely rewarding, leading to new mathematical methods and sometimes even to entirely new fields within mathematics. The book is organized into two parts: Computational Sciences and Engineering, and Data Analysis and Finance. In every chapter, readers will find a brief description of why such work fits into this volume; an explanation on which industrial challenges have been instrumental for their inspiration; and which methods have been developed as a result. All these contribute to a greater unity of the text, benefiting not only practitioners and professionals seeking information on novel techniques but also graduate students in applied mathematics, engineering, and related fields.

Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures contains the plenary lectures and papers presented at the 11th International Conference on STRUCTURAL SAFETY AND RELIABILITY (ICOSSAR2013, New York, NY, USA, 16-20 June 2013), and covers major aspects of safety, reliability, risk and life-cycle performance of str

Focusing on fundamental principles, Hydro-Environmental Analysis: Freshwater Environments presents in-depth information about freshwater environments and how they are influenced by regulation. It provides a holistic approach, exploring the factors that impact water quality and quantity, and the regulations, policy and management methods that are necessary to maintain this vital resource. It offers a historical viewpoint as well as an overview and foundation of the physical, chemical, and biological characteristics affecting the management of freshwater environments. The book concentrates on broad and general concepts, providing an interdisciplinary foundation. The author covers the methods of measurement and classification; chemical, physical, and biological characteristics; indicators of ecological health; and management and restoration. He also considers common indicators of environmental health; characteristics and operations of regulatory control structures; applicable laws and regulations; and restoration methods. The text delves into rivers and streams in the first half and lakes and reservoirs in the second half. Each section centers on the characteristics of those systems and methods of classification, and then moves on to discuss the physical, chemical, and biological characteristics of each. In the section on lakes and reservoirs, it examines the characteristics and operations of regulatory structures, and presents the methods commonly used to assess the environmental health or integrity of these water bodies. It also introduces considerations for restoration, and presents two unique aquatic environments: wetlands and reservoir tailwaters. Written from an engineering perspective, the book is an ideal introduction to the aquatic and limnological sciences for students of environmental science, as well as students of environmental engineering. It also serves as a reference for engineers and scientists involved in the management, regulation, or restoration of freshwater environments.

This text introduces engineering students to probability theory and stochastic processes. Along with thorough mathematical development of the subject, the book presents intuitive explanations of key points in order to give students the insights they need to apply math to practical engineering problems. The first five chapters contain the core material that is essential to any introductory course. In one-semester undergraduate courses, instructors can select material from the remaining chapters to meet their individual goals. Graduate courses can cover all chapters in one semester.

The book gives a systematical presentation of stochastic approximation methods for models of American-type options with general pay-off functions for discrete time Markov price processes. Advanced methods combining backward recurrence algorithms for computing of option rewards and general results on convergence of stochastic space skeleton and tree approximations for option rewards are applied to a variety of models of multivariate modulated Markov price processes. The principal novelty of presented results is based on consideration of multivariate modulated Markov price processes and general pay-off functions, which can depend not only on price but also an additional stochastic modulating index component, and use of minimal conditions of smoothness for transition probabilities and pay-off functions, compactness conditions for log-price processes and rate of growth conditions for pay-off functions. The book also contains an extended bibliography of works in the area. This book is the first volume of the comprehensive two volumes monograph. The second volume will present results on structural studies of optimal stopping domains, Monte Carlo based approximation reward algorithms, and convergence of American-type options for autoregressive and continuous time models, as well as results of the corresponding experimental studies.

This textbook provides a self-contained introduction to numerical methods in probability with a focus on applications to finance. Topics covered include the Monte Carlo simulation (including simulation of random variables, variance reduction, quasi-Monte Carlo simulation, and more recent developments such as the multilevel paradigm), stochastic optimization and approximation, discretization schemes of stochastic differential equations, as well as optimal quantization methods. The author further presents detailed applications to numerical aspects of pricing and hedging of financial derivatives, risk measures (such as value-at-risk and conditional value-at-risk), implicitation of parameters, and calibration. Aimed at graduate students and advanced undergraduate students, this book contains useful examples and over 150 exercises, making it suitable for self-study.