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2012 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Stochastic Systems

Uncertainty Quantification and Propagation

verfasst von: Mircea Grigoriu

Verlag: Springer London

Buchreihe : Springer Series in Reliability Engineering

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

Uncertainty 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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
Examples are used to introduce deterministic and stochastic equations and demonstrate the need to describe the state of physical systems by using stochastic equations. While the focus in mathematical studies of stochastic equations is primarily on technical aspects related to the existence and uniqueness of the solutions, the focus in applied studies is on the calculations of these solutions. Graduate courses based on material presented in the book can concentrate on the formulation of stochastic problems, random vibration by Itô’s calculus, and stochastic partial differential equations with applications to random heterogeneous materials.
Mircea Grigoriu
Chapter 2. Essentials of Probability Theory
Abstract
Essentials of probability theory are reviewed and illustrated by examples. The review covers probability spaces, properties of probability measure, measurable functions and random elements, independence for \(\sigma\)-fields, events, and random elements, expectation operator, Fubini’s theorem, convergence concepts, Radon–Nikodym derivative, distribution and density functions, characteristic function, conditional expectation, discrete time martingales, and direct and improved Monte Carlo simulation.
Mircea Grigoriu
Chapter 3. Random Functions
Abstract
General considerations on random functions are followed by essential definitions and properties for these functions. The concepts of weak stationarity and stationarity are defined and illustrated. Mean square continuity, differentiation, and integration as well as spectral and related representations for weakly stationary random functions are discussed extensively. Limitations of second moment calculus are highlighted by the study of sample properties for random functions with finite variance. A broad range of random functions, for example, Gaussian, translation, Markov, martingales, Brownian motion, compound Poisson, and Lévy processes, are examined. Algorithms for generating samples of stationary Gaussian functions, translation models, and non-stationary Gaussian processes conclude our discussion.
Mircea Grigoriu
Chapter 4. Stochastic Integrals
Abstract
Examples are presented to demonstrate the need for considering integrals other than Riemann-Stieltjes integrals when dealing with stochastic integrators. We review essentials of Riemann-Stieltjes integrals and define Itô stochastic integrals with Brownian motion, martingale, and semimartingale integrators. Stratonovich integrals are also defined. Properties of stochastic integrals and related concepts are illustrated by numerous examples.
Mircea Grigoriu
Chapter 5. Itô’s Formula and Applications
Abstract
Itô’s formula is establish for real-valued and \({\mathbb{R}}^d\)-valued continuous and arbitrary semimartingales and its use is illustrated by numerous examples. The relationship between the Itô and Stratonovich integrals is examined prior to presenting a broad range of applications of Itô’s formula. The applications include stochastic differential equations with Gaussian and non-Gaussian white noise, Tanaka’s formula, local solutions for a class of partial differential equations, and improved Monte Carlo estimates based on Girsanov’s theorem.
Mircea Grigoriu
Chapter 6. Probabilistic Models
Abstract
Model construction usually involves two steps. First, a collection of functional forms is proposed for the probability law of a random element. It is common in application to consider a single functional form. Second, the uncertain parameters of a proposed probability law are estimated from the available information. We construct probabilistic models for directional wind speed in hurricanes, uncertain properties of physical systems, inclusions in multi-phase materials, and coefficients of stochastic elliptic equations describing material properties.
Mircea Grigoriu
Chapter 7. Stochastic Ordinary Differential and Difference Equations
Abstract
Summary Methods are developed for solving ordinary differential and difference equations with random coefficients and/or input. Following an introductory section (Sect. 1), we present methods for solving equations with deterministic coefficients and random input (Sect. 2), finite difference equations with random coefficients of arbitrary and small uncertainty (Sect. 3), and ordinary differential equations with random coefficients of arbitrary and small uncertainty (Sect. 4). The methods include Monte Carlo simulation, conditional analysis, stochastic reduced order models, stochastic Galerkin, stochastic collocation, Taylor series, and Neumann series. Applications from stochastic stability, noise induced transitions, random vibration, and reliability of degrading systems conclude the chapter (Sect. 5).
Mircea Grigoriu
Chapter 8. Stochastic Algebraic Equations
Abstract
Methods are discussed for solving approximately linear algebraic equations with random parameters, referred to as stochastic algebraic equations (SAEs). Section 8.1 defines SAEs and outlines potential difficulties related to the solution of these equations. Section 8.2 presents solutions of SAEs with random entries of arbitrary uncertainty by Monte Carlo simulation, stochastic reduced order models, stochastic Galerkin, stochastic collocation, and reliability methods. SAEs with random entries of small uncertainty are solved in Section 8.3 by Taylor series, perturbation, Neumann series, and equivalent linearization methods.
Mircea Grigoriu
Chapter 9. Stochastic Partial Differential Equations
Abstract
A brief discussion on the relevance of stochastic partial differential equations (SPDEs) in Sect. 9.1 is followed by a review of the type of SPDEs studied in the mathematical literature (Sect. 9.2). Section 9.3 shows that SPDEs can be solved by the methods in Chaps. 7 and 8 via time and space discretization. Section 9.4 deals with SPDEs that are frequently encountered in applications. The focus is on stochastic elliptic partial differential equations. We review general concepts on this class of equations (Sect. 9.4.1), and present solutions by Monte Carlo (Sect. 9.4.4), stochastic reduced order models (Sect. 9.4.5), stochastic Galerkin (Sect. 9.4.8), and stochastic collocation (Sect. 9.4.9) methods. The last section of the chapter considers SDEs whose coefficients have small uncertainty, and presents methods for solving these equations by Taylor (Sect. 9.5.1), perturbation (Sect. 9.5.2), and Neumann series (Sect. 9.5.3) representations.
Mircea Grigoriu
Backmatter
Metadaten
Titel
Stochastic Systems
verfasst von
Mircea Grigoriu
Copyright-Jahr
2012
Verlag
Springer London
Electronic ISBN
978-1-4471-2327-9
Print ISBN
978-1-4471-2326-2
DOI
https://doi.org/10.1007/978-1-4471-2327-9

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