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E-raamat: Stochastic Dynamics, Filtering and Optimization

, (Indian Institute of Science, Bangalore)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 04-May-2017
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316996195
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Stochastic Dynamics, Filtering and OptimizationSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 04-May-2017
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781316996195
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Targeted at graduate students, researchers and practitioners in the field of science and engineering, this book gives a self-contained introduction to a measure-theoretic framework in laying out the definitions and basic concepts of random variables and stochastic diffusion processes. It then continues to weave into a framework of several practical tools and applications involving stochastic dynamical systems. These include tools for the numerical integration of such dynamical systems, nonlinear stochastic filtering and generalized Bayesian update theories for solving inverse problems and a new stochastic search technique for treating a broad class of non-convex optimization problems. MATLAB® codes for all the applications are uploaded on the companion website.

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This book introduces essential concepts in stochastic processes that interface seamlessly with applications of interest in science and engineering.
Figures xv
Tables xxvii
Preface xxix
Acronyms xxxiii
General Notations xxxv
1 Probability Theory and Random Variables
1.1 Introduction
1(4)
1.2 Probability Space and Basic Definitions
5(2)
1.3 Probability as a Measure
7(11)
1.3.1 Caratheodory's extension theorem
10(6)
1.3.2 Uniqueness criterion for measures
16(2)
1.4 Random Variables and Measurable Functions
18(3)
1.4.1 Some properties of random variables
18(3)
1.5 Random Variables and Induced Probability Measures
21(1)
1.5.1 sigma-algebra generated by a random variable
22(1)
1.6 Probability Distribution and Density Function of a Random Variable
22(6)
1.6.1 Probability distribution function
23(2)
1.6.2 Lebesgue-Stieltjes measure
25(1)
1.6.3 Probability density function
26(1)
1.6.4 Radon-Nikodyn theorem
26(2)
1.7 Vector-valued Random Variables and Joint Probability Distributions
28(2)
1.7.1 Joint probability distributions and density functions
29(1)
1.7.2 Marginal probability distributions and density functions
29(1)
1.8 Integration of Measurable Functions and Expectation of a Random Variable
30(7)
1.8.1 Integration with respect to product measure and Fubini's theorem
31(2)
1.8.2 Monotone convergence theorem
33(1)
1.8.3 Expectation of a random variable
33(1)
1.8.4 Higher order expectations of a random variable
34(3)
1.8.5 Characteristic and moment generating functions
37(1)
1.9 Independence of Random Variables
37(4)
1.9.1 Independence of events
37(1)
1.9.2 Independence of classes of events
38(1)
1.9.3 Independence of a-algebras
38(1)
1.9.4 Independence of random variables
38(1)
1.9.5 Independence in terms of CDFs
39(1)
1.9.6 Independence of functions of random variables
40(1)
1.9.7 Independence and expectation of random variables
40(1)
1.9.8 Additional remarks on independence of random variables
41(1)
1.10 Some oft-used Probability Distributions
41(10)
1.10.1 Binomial distribution
42(1)
1.10.2 Poisson distribution
42(1)
1.10.3 Normal distribution
43(6)
1.10.4 Uniform distribution
49(1)
1.10.5 Rayleigh distribution
50(1)
1.11 Transformation of Random Variables
51(9)
1.11.1 Transformation involving a scalar function of vector random variables
52(2)
1.11.2 Transformation involving vector functions of random variables
54(1)
1.11.3 Diagonalization of covariance matrix and transformation to uncorrelated random variables
55(2)
1.11.4 Nataf transformation
57(3)
1.12 Concluding Remarks
60(1)
Exercises
61(2)
Notations
63(2)
2 Random Variables: Conditioning, Convergence and Simulation
2.1 Introduction
65(3)
2.2 Conditional Probability
68(13)
2.2.1 Conditional expectation
70(3)
2.2.2 Change of measure
73(2)
2.2.3 Generalized Bayes' formula and conditional probabilities
75(1)
2.2.4 Conditional expectation as the least mean square error estimator
76(1)
2.2.5 Rosenblatt transformation
77(4)
2.3 Convergence of Random Variables
81(6)
2.3.1 Convergence of a sequence of random variables
81(3)
2.3.2 Law of large numbers
84(1)
2.3.3 Central limit theorem (CLT)
85(2)
2.3.4 Random walk and central limit theorem
87(1)
2.4 Some Useful Inequalities in Probability Theory
87(4)
2.5 Monte Carlo (MC) Simulation of Random Variables
91(29)
2.5.1 Random number generation-uniformly distributed random variable
91(2)
2.5.2 Simulation for other distributions
93(5)
2.5.3 Simulation of joint random variables-uncorrelated and correlated
98(6)
2.5.4 Multidimensional integrals by MC simulation methods
104(13)
2.5.5 Rao-Blackwell theorem and a general approach to variance reduction techniques
117(3)
2.6 Concluding Remarks
120(1)
Exercises
120(3)
Notations
123(2)
3 An Introduction to Stochastic Processes
3.1 Introduction
125(4)
3.2 Stochastic Process and its Finite Dimensional Distributions
129(3)
3.2.1 Continuity of a stochastic process
130(1)
3.2.2 Version/modification of a stochastic process
131(1)
3.3 Stochastic Processes-Measurability and Filtration
132(18)
3.3.1 Filtration and adapted processes
133(1)
3.3.2 Some basic stochastic processes
133(2)
3.3.3 Stationary stochastic processes
135(1)
3.3.4 Wiener process/ Brownian motion
136(2)
3.3.5 Formal definition of a Wiener process
138(1)
3.3.6 Other properties of a Wiener process
139(11)
3.4 Martingales: A General Introduction
150(12)
3.4.1 Doob's decomposition theorem
150(2)
3.4.2 Martingale transform
152(2)
3.4.3 Doob's uperossing inequality
154(1)
3.4.4 Martingale convergence theorem
155(2)
3.4.5 Uniform integrability
157(5)
3.5 Stopping Time and Stopped Processes
162(10)
3.5.1 Stopping time
163(1)
3.5.2 Stopped processes
164(1)
3.5.3 Doob's optional stopping theorem
165(1)
3.5.4 A super-martingale inequality
166(1)
3.5.5 Optional stopping theorem for UI martingales
167(5)
3.6 Some Useful Results for Time-continuous Martingales
172(10)
3.6.1 Doob's and Levy's martingale theorem
172(1)
3.6.2 Martingale convergence theorem
173(1)
3.6.3 Optional stopping theorem
174(8)
3.7 Localization and Local Martingales
182(1)
3.7.1 Definition of a local martingale
182(1)
3.8 Concluding Remarks
183(1)
Exercises
183(3)
Notations
186(1)
4 Stochastic Calculus and Diffusion Processes
4.1 Introduction
187(2)
4.2 Stochastic Integral
189(9)
4.2.1 Stochastic integral of a discrete stochastic process
190(3)
4.2.2 Properties of Ito integral of simple adapted processes
193(2)
4.2.3 Ito integral for continuous processes
195(3)
4.3 Ito Processes
198(3)
4.3.1 Larger class of integrands for Ito integral
200(1)
4.4 Stochastic Calculus
201(30)
4.4.1 Integral representation of an SDE
202(1)
4.4.2 Ito's formula
203(12)
4.4.3 Ito's formula for higher dimensions
215(8)
4.4.4 Dynamical system of higher dimension and application of Ito's formula
223(8)
4.5 Spectral Representations of Stochastic Signals
231(16)
4.5.1 Non-stationary process and evolutionary power spectrum
232(12)
4.5.2 Some interesting aspects of evolutionary power spectrum
244(3)
4.6 Existence and Uniqueness of Solutions to SDEs
247(12)
4.6.1 Locally Lipschitz condition and unique solution to SDE
249(1)
4.6.2 Strong and weak solutions
249(1)
4.6.3 Linear SDEs
250(6)
4.6.4 Markov property of solutions to SDEs
256(3)
4.7 Backward Kolmogorov Equation-Revisiting Evaluation of Expectations
259(8)
4.7.1 Backward Kolmogorov equation
259(3)
4.7.2 Inhomogeneous backward Kolmogorov PDE
262(1)
4.7.3 Adjoint differential operator and forward Kolmogorov PDE
262(3)
4.7.4 Generator Lt
265(2)
4.7.5 Feynman-Kac formula
267(1)
4.8 Solution of PDEs via Corresponding SDEs
267(11)
4.8.1 Solution to elliptic PDEs
270(6)
4.8.2 Exit time distributions from solutions of PDEs
276(2)
4.9 Recurrence and Transience of a Diffusion Process
278(1)
4.10 Girsanov's Theorem and Change of Measure
279(8)
4.10.1 Girsanov's theorem
280(1)
4.10.2 Girsanov's theorem for Brownian motion
281(1)
4.10.3 Girsanov's theorem-Version 1
282(4)
4.10.4 Girsanov's theorem-the general version
286(1)
4.11 Martingale Representation Theorem
287(5)
4.11.1 Proof of martingale representation theorem
291(1)
4.12 A Brief Remark on the Martingale Problem
292(1)
4.13 Concluding Remarks
293(1)
Exercises
294(1)
Notations
295(4)
5 Numerical Solutions to Stochastic Differential Equations
5.1 Introduction
299(2)
5.2 Euler-Maruyama (EM) Method for Solving SDEs
301(14)
5.2.1 Order of convergence of EM method
302(1)
5.2.2 Statement of the theorem for global convergence
302(13)
5.3 An Implicit EM Method
315(1)
5.4 Further Issues on Convergence of EM Methods
316(2)
5.5 An introduction to Ito-Taylor Expansion for Stochastic Processes
318(2)
5.6 Derivation of Ito-Taylor Expansion
320(9)
5.6.1 One-step approximations-explicit integration methods
323(6)
5.7 Implementation Issues of the Numerical Integration Schemes
329(10)
5.7.1 Evaluation of MSIs
330(9)
5.8 Stochastic Implicit Methods and Ito-Taylor Expansion
339(12)
5.8.1 Stochastic Newmark method-a two-parameter implicit scheme for mechanical oscillators
343(8)
5.9 Weak One-step Approximate Solutions of SDEs
351(19)
5.9.1 Statement of the weak convergence theorem
352(4)
5.9.2 Modelling of MSIs and construction of a weak one-step approximation
356(8)
5.9.3 Stochastic Newmark scheme using weak one-step approximation
364(6)
5.10 Local Linearization Methods for Strong / Weak Solutions of SDEs
370(9)
5.10.1 LTL-based schemes
370(9)
5.11 Concluding Remarks
379(1)
Exercises
380(4)
Notations
384(2)
6 Non-linear Stochastic Filtering and Recursive Monte Carlo Estimation
6.1 Introduction
386(3)
6.2 Objective of Stochastic Filtering
389(1)
6.3 Stochastic Filtering and Kushner-Stratanovitch (KS) Equation
390(10)
6.3.1 Zakai equation
392(1)
6.3.2 KS equation
393(2)
6.3.3 Circularity-the problem of moment closure in non-linear filtering problems
395(2)
6.3.4 Unnormalized conditional density and Kushner's theorem
397(3)
6.4 Non-linear Stochastic Filtering and Solution Strategies
400(11)
6.4.1 Extended Kalman filter (EKF)
401(1)
6.4.2 EKF using locally transversal linearization (LTL)
402(5)
6.4.3 EKF applied to parameter estimation
407(4)
6.5 Monte Carlo Filters
411(16)
6.5.1 Bootstrap filter
411(7)
6.5.2 Auxiliary bootstrap filter
418(1)
6.5.3 Ensemble Kalman filter (EnKF)
419(8)
6.6 Concluding Remarks
427(1)
Exercises
428(1)
Notations
429(3)
7 Non-linear Filters with Gain-type Additive Updates
7.1 Introduction
432(1)
7.2 Iterated Gain-based Stochastic Filter (IGSF)
432(6)
7.2.1 IGSF scheme
433(5)
7.3 Improved Versions of IGSF
438(6)
7.3.1 Gaussian sum approximation and filter bank
438(1)
7.3.2 Filtering strategy
439(2)
7.3.3 Iterative update scheme for IGSF bank
441(1)
7.3.4 Iterative update scheme for IGSF bank with ADP
442(2)
7.4 KS Filters
444(7)
7.4.1 KS filtering scheme
445(6)
7.5 EnKS Filter-a Variant of KS Filter
451(13)
7.5.1 EnKS filtering scheme
452(1)
7.5.2 EnKS filter-a non-iterative form
453(4)
7.5.3 EnKS filter-an iterative form
457(7)
7.6 Concluding Remarks
464(1)
Notations
465(2)
8 Improved Numerical Solutions to SDEs by Change of Measures
8.1 Introduction
467(5)
8.2 Girsanov Corrected Linearization Method (GCLM)
472(19)
8.2.1 Algorithm for GCLM
477(14)
8.3 Girsanov Corrected Euler-Maruyama (GCEM) Method
491(5)
8.3.1 Additively driven SDEs and the GCEM method
492(1)
8.3.2 Weak correction through a change of measure
493(3)
8.4 Numerical Demonstration of GCEM Method
496(8)
8.5 Concluding Remarks
504(1)
Notations
505(2)
9 Evolutionary Global Optimization via Change of Measures: A Martingale Route
9.1 Introduction
507(19)
9.2 Possible Ineffectiveness of Evolutionary Schemes
526(1)
9.3 Global Optimization by Change of Measure and Martingale Characterization
527(1)
9.4 Local Optimization as a Martingale Problem
528(2)
9.5 The Optimization Scheme-Algorithmic Aspects
530(13)
9.5.1 Discretization of the extremal equation
533(8)
9.5.2 Pseudo codes
541(2)
9.6 Some Applications of the Pseudo Code 2 to Dynamical Systems
543(9)
9.7 Concluding Remarks
552(1)
Notations
553(3)
10 COMBEO-A New Global Optimization Scheme By Change of Measures
10.1 Introduction
556(1)
10.2 COMBEO-Improvements to the Martingale Approach
557(16)
10.2.1 Improvements to the coalescence strategy
557(2)
10.2.2 Improvements to scrambling and introduction of a relaxation parameter
559(2)
10.2.3 Blending
561(12)
10.3 COMBEO Algorithm
573(9)
10.3.1 Some benchmark problems and solutions by COMBEO
577(5)
10.4 Further Improvements to COMBEO
582(7)
10.4.1 State space splitting (3S)
582(3)
10.4.2 Benchmark problems
585(4)
10.5 Concluding Remarks
589(1)
Notations
589(2)
Appendix A (Chapter 1) 591(16)
Appendix B (Chapter 2) 607(7)
Appendix C (Chapter 3) 614(6)
Appendix D (Chapter 4) 620(15)
Appendix E (Chapter 5) 635(7)
Appendix F (Chapter 6) 642(3)
Appendix G (Chapter 7) 645(21)
Appendix H (Chapter 8) 666(2)
Appendix I (Chapter 9) 668(5)
References 673(21)
Bibliography 694(7)
Index 701
Debasish Roy is currently working as Professor in the Computational Mechanics Laboratory at the Indian Institute of Science, Bangalore. He obtained his Ph.D. from the Indian Institute of Science, followed by post-doctoral research at the University of Innsbruck, Austria. Besides being a fellow of the Indian National Academy of Engineering, he has also held an Honorary Professorship in the School of Engineering, University of Aberdeen, and a distinguished visiting fellowship of the Royal Academy of Engineering, London. His areas of research include computational mechanics of non-classical continua, stochastic dynamical systems and optimization/inverse problems. He has published over 120 papers in journals of international repute, delivered keynote/invited lectures at many international conferences and served on editorial boards. G. Visweswara Rao was Technical Advisor in ACS Design Consulting Private Limited, Bangalore. He received his Ph.D. from the Indian Institute of Science, Bangalore, in 1989. He has published several research papers in the areas of structural dynamics specific to earthquake engineering, nonlinear and random vibration, and structural control, and co-authored Elements of Structural Dynamics: A New Perspective (2012) with D. Roy. His areas of research include non-linear and stochastic structural dynamics.
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