Stochastic dynamics, filtering and optimization:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom
Cambridge University Press
2017
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | xxxvii, 709 Seiten Diagramme |
| ISBN: | 9781107182646 |
Internformat
MARC
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| 245 | 1 | 0 | |a Stochastic dynamics, filtering and optimization |c Debasish Roy, G. Visweswara Rao |
| 264 | 1 | |a Cambridge, United Kingdom |b Cambridge University Press |c 2017 | |
| 300 | |a xxxvii, 709 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
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| 650 | 0 | 7 | |a Stochastische Optimierung |0 (DE-588)4057625-5 |2 gnd |9 rswk-swf |
| 653 | 0 | |a Stochastic processes / Mathematical models | |
| 653 | 0 | |a Stochastic differential equations / Numerical solutions | |
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Datensatz im Suchindex
| _version_ | 1804177721950994432 |
|---|---|
| adam_text | Contents
A
Figures XV
Tables xxvii
Preface xxix
Acronyms xxxiii
General Notations xxxv
1 Probability Theory and Random Variables
1.1 Introduction 1
1.2 Probability Space and Basic Definitions 5
1.3 Probability as a Measure 7
L3.1 Caratheodory s extension theorem 10
1.3.2 Uniqueness criterion for measures 16
1.4 Random Variables and Measurable Functions 18
1.4.1 Some properties of random variables 18
1.5 Random Variables and Induced Probability Measures 21
1.5.1 a- algebra generated by a random variable 22
1.6 Probability Distribution and Density Function of a
Random Variable 22
1.6.1 Probability distribution function 2 3
1.6.2 Lebesgue-Stieltjes measure 25
1.6.3 Probability density function 26
1.6.4 Radon-Nikodyn theorem 26
1.7 Vector-valued Random Variables and Joint Probability Distributions 28
1.7.1 Joint probability distributions and density functions 29
1.7.2 Marginal probability distributions and density functions 29
1.8 Integration of Measurable Functions and Expectation of a Random Variable 30
1.8.1 Integration with respect to product measure and Fubini s theorem 31
1.8.2 Monotone convergence theorem 33
1.8.3 Expectation of a random variable 33
ISA Higher order exoectations of a random variable 34
viii Contents
1.8.5 Characteristic and moment generating functions 37
1.9 Independence of Random Variables 37
1.9.1 Independence of events 37
1.9.2 Independence of classes of events 38
1.9.3 Independence of cr-algebras 38
1.9.4 Independence of random variables 38
1.9.5 Independence in terms of CDFs 39
1.9.6 Independence of functions of random variables 40
1.9.7 Independence and expectation of random variables 40
1.9.8 Additional remarks on independence of random variables 41
1.10 Some oft-used Probability Distributions 41
1.10.1 Binomial distribution 42
1.10.2 Poisson distribution 42
1.10.3 Normal distribution 43
1.10.4 Uniform distribution 49
1.10.5 Rayleigh distribution 50
1.11 Transformation of Random Variables 51
1.11.1 Transformation involving a scalar function of vector
random variables 52
1.11.2 Transformation involving vector functions of
random variables 54
1.11.3 Diagonalization of covariance matrix and
transformation to uncorrelated random variables 55
1.11.4 Nataf transformation 57
1.12 Concluding Remarks 60
Exercises 61
Notations 63
2 Random Variables: Conditioning, Convergence and Simulation
2.1 Introduction 65
2.2 Conditional Probability 68
2.2.1 Conditional expectation 70
2.2.2 Change of measure 73
2.2.3 Generalized Bayes formula and conditional probabilities 75
2.2.4 Conditional expectation as the least mean square error estimator 76
2.2.5 Rosenblatt transformation 77
2.3 Convergence of Random Variables 81
2.3.1 Convergence of a sequence of random variables 81
2.3.2 Law of large numbers 84
2.3.3 Central limit theorem (CLT) 85
2.3.4 Random walk and central limit theorem 87
2.4 Some Useful Inequalities in Probability Theory 87
2.5 Monte Carlo (MC) Simulation of Random Variables 91
2.5.1 Random number generation—uniformly distributed random variable 91
2.5.2 Simulation for other distributions 93
2.5.3 Simulation of joint random variables—uncorrelated
and correlated 98
2.5.4 Multidimensional integrals by MC simulation methods 104
2.5.5 Rao-Blackwell theorem and a general approach to variance
reduction techniques 117
2.6 Concluding Remarks 120
Exercises 120
Notations 123
3 An Introduction to Stochastic Processes
3.1 Introduction 125
3.2 Stochastic Process and its Finite Dimensional Distributions 129
3.2.1 Continuity of a stochastic process 130
3.2.2 Version/modification of a stochastic process 131
3.3 Stochastic Processes—Measurability and Filtration 132
3.3.1 Filtration and adapted processes 133
3.3.2 Some basic stochastic processes 133
3.3.3 Stationary stochastic processes 135
3.3.4 Wiener process/Brownian motion 136
3.3.5 Formal definition of a Wiener process 138
3.3.6 Other properties of a Wiener process 139
3.4 Martingales: A General Introduction 150
3.4.1 Doobs decomposition theorem 150
3.4.2 Martingale transform 152
3.4.3 Doob’s upcrossing inequality 154
3.4.4 Martingale convergence theorem 155
3.4.5 Uniform integrability 157
3.5 Stopping Time and Stopped Processes 162
3.5.1 Stopping time 163
3.5.2 Stopped processes 164
3.5.3 Doob’s optional stopping theorem 165
3.5.4 A super-martingale inequality 166
3.5.5 Optional stopping theorem for UI martingales 167
3.6 Some Useful Results for Time-continuous Martingales 172
3.6.1 Doobs and Levy s martingale theorem 172
3.6.2 Martingale convergence theorem 173
3.6.3 Optional stopping theorem 174
3.7 Localization and Local Martingales 182
3.7.1 Definition of a local martingale 182
X Contents
3.8 Concluding Remarks 183
Exercises 183
Notations 186
4 Stochastic Calculus and Diffusion Processes
4.1 Introduction 187
4.2 Stochastic Integral 189
4.2.1 Stochastic integral of a discrete stochastic process 190
4.2.2 Properties of Ito integral of simple adapted processes 193
4.2.3 Ito integral for continuous processes 195
4.3 Ito Processes 198
4.3.1 Larger class of integrands for Ito integral 200
4.4 Stochastic Calculus 201
4.4.1 Integral representation of an SDE 202
4.4.2 Itos formula 203
4.4.3 Itos formula for higher dimensions 215
4.4.4 Dynamical system of higher dimension and application of Itos
formula 223
4.5 Spectral Representations of Stochastic Signals 231
4.5.1 Non-stationary process and evolutionary power spectrum 232
4.5.2 Some interesting aspects of evolutionary power spectrum 244
4.6 Existence and Uniqueness of Solutions to SDEs 247
4.6.1 Locally Lipschitz condition and unique solution to SDE 249
4.6.2 Strong and weak solutions 249
4.6.3 Linear SDEs 250
4.6.4 Markov property of solutions to SDEs 256
4.7 Backward Kolmogorov Equation—Revisiting Evaluation of Expectations 259
4.7.1 Backward Kolmogorov equation 259
4.7.2 Inhomogeneous backward Kolmogorov PDE 262
4.7.3 Adjoint differential operator and forward Kolmogorov PDE 262
4.7.4 Generator Lt 265
4.7.5 Feynman-Kac formula 267
4.8 Solution of PDEs via Corresponding SDEs 267
4.8.1 Solution to elliptic PDEs 270
4.8.2 Exit time distributions from solutions of PDEs 276
4.9 Recurrence and Transience of a Diffusion Process 278
4.10 Girsanov s Theorem and Change of Measure 279
4.10.1 Girsanov s theorem 280
4.10.2 Girsanovs theorem for Brownian motion 281
4.10.3 Girsanovs theorem—Version 1 282
4.10.4 Girsanovs theorem—the general version 286
Contents ХІ
4.11 Martingale Representation Theorem 287
4.11.1 Proof of martingale representation theorem 291
4.12 A Brief Remark on the Martingale Problem 292
4.13 Concluding Remarks 293
Exercises 294
Notations 295
5 Numerical Solutions to Stochastic Differential Equations
5.1 Introduction 299
5.2 Euler-Maruyama (EM) Method for Solving SDEs 301
5.2.1 Order of convergence of EM method 302
5.2.2 Statement of the theorem for global convergence 302
5.3 An Implicit EM Method 315
5.4 Further Issues on Convergence of EM Methods 316
5.5 An introduction to Ito—Taylor Expansion for Stochastic Processes 318
5.6 Derivation of Ito—Taylor Expansion 320
5.6.1 One-step approximations—explicit integration methods 323
5.7 Implementation Issues of the Numerical Integration Schemes 329
5.7.1 Evaluation of MSIs 330
5.8 Stochastic Implicit Methods and Ito-Taylor Expansion 339
5.8.1 Stochastic Newmark method—a two-parameter implicit scheme
for mechanical oscillators 343
5.9 Weak One-step Approximate Solutions of SDEs 351
5.9.1 Statement of the weak convergence theorem 352
5.9.2 Modelling of MSIs and construction of a weak one-step
approximation 356
5.9.3 Stochastic Newmark scheme using weak one-step approximation 364
5.10 Local Linearization Methods for Strong / Weak Solutions of SDEs 370
5.10.1 LTL-based schemes 370
5.11 Concluding Remarks 379
Exercises 380
Notations 384
6 Non-linear Stochastic Filtering and Recursive Monte Carlo Estimation
6.1 Introduction 386
6.2 Objective of Stochastic Filtering 389
6.3 Stochastic Filtering and Kushner-Stratanovitch (KS) Equation 390
6.3.1 Zakai equation 392
6.3.2 KS equation 393
6.3.3 Circularity—the problem of moment closure in non-linear
filtering problems 395
xii Contents
6.3.4 Unnormalized conditional density and Kushner’s theorem 397
6.4 Non-linear Stochastic Filtering and Solution Strategies 400
6.4.1 Extended Kalman filter (EKF) 401
6.4.2 EKF using locally transversal linearization (LTL) 402
6.4.3 EKF applied to parameter estimation 407
6.5 Monte Carlo Filters 411
6.5.1 Bootstrap filter 411
6.5.2 Auxiliary bootstrap filter 418
6.5.3 Ensemble Kalman filter (EnKF) 419
6.6 Concluding Remarks 427
Exercises 428
Notations 429
7 Non-linear Filters with Gain-type Additive Updates
7.1 Introduction 43 2
7.2 Iterated Gain-based Stochastic Filter (IGSF) 432
7.2.1 IGSF scheme 433
7.3 Improved Versions of IGSF 438
7.3.1 Gaussian sum approximation and filter bank 438
7.3.2 Filtering strategy 439
7.3.3 Iterative update scheme for IGSF bank 441
7.3.4 Iterative update scheme for IGSF bank with ADP 442
7.4 KS Filters 444
7.4.1 KS filtering scheme 445
7.5 EnKS Filter—a Variant of KS Filter 451
7.5.1 EnKS filtering scheme 452
7.5.2 EnKS filter—a non-iterative form 453
7.5.3 EnKS filter—an iterative form 457
7.6 Concluding Remarks 464
Notations 465
8 Improved Numerical Solutions to SDEs by Change of Measures
8.1 Introduction 467
8.2 Girsanov Corrected Linearization Method (GCLM) 472
8.2.1 Algorithm for GCLM 477
8.3 Girsanov Corrected Euler-Maruyama (GCEM) Method 491
8.3.1 Additively driven SDEs and the GCEM method 492
8.3.2 Weak correction through a change of measure 493
8.4 Numerical Demonstration of GCEM Method 496
8.5 Concluding Remarks 504
Notations 505
Contents xiii
9 Evolutionary Global Optimization via Change of Measures:
A Martingale Route
9.1 Introduction 507
9.2 Possible Ineffectiveness of Evolutionary Schemes 526
9.3 Global Optimization by Change of Measure and
Martingale Characterization 527
9.4 Local Optimization as a Martingale Problem 528
9.5 The Optimization Scheme—Algorithmic Aspects 530
9.5.1 Discretization of the extremal equation 533
9.5.2 Pseudo codes 541
9.6 Some Applications of the Pseudo Code 2 to Dynamical Systems 543
9.7 Concluding Remarks 552
Notations 553
10 COMBEO-A New Global Optimization Scheme By Change of Measures
10.1 Introduction 556
10.2 COMBEO—Improvements to the Martingale Approach 557
10.2.1 Improvements to the coalescence strategy 557
10.2.2 Improvements to scrambling and introduction of a relaxation
parameter 559
10.2.3 Blending 561
10.3 COMBEO Algorithm 573
10.3.1 Some benchmark problems and solutions by COMBEO 577
10.4 Further Improvements to COMBEO 582
10.4.1 State space splitting (3S) 582
10.4.2 Benchmark problems 585
10.5 Concluding Remarks 589
Notations 589
Appendix A (Chapter 1) 591
Appendix B (Chapter 2) 607
Appendix C (Chapter 3) 614
Appendix D (Chapter 4) 620
Appendix E (Chapter 5) 635
Appendix F (Chapter 6) 642
Appendix G (Chapter 7) 645
Appendix H (Chapter 8) 666
Appendix I (Chapter 9) 668
References 673
Bibliography 694
Index 701
Stochastic Dynamics, Filtering and Optimization
Stochastic processes and probability theory are widely used mathematical tools for modeling
uncertainties of both epistemic and aleatory types. The calculus for diffusion processes, often
used to model such noisy fluctuations, differs fundamentally from that of deterministic
smooth functions. The theory of diffusive stochastic processes is also a ubiquitous ingredient
in modern research strategies for a broad class of optimization problems including stochastic
filtering. This book provides a balanced treatment of theory and applications in this area of
importance.
It covers fundamentals of stochastic processes with applications to dynamical systems in
science, engineering and recursive search algorithms. It discusses fundamental concepts
and theory of stochastic processes, calculus, Ito-Taylor expansion and numerical
integration of stochastic differential equations (SDEs) in detail. Topics such as
Radon-Nikodym derivatives and Girsanov theorems with emphasis on Ito diffusion
processes are comprehensively discussed. The text discusses advances in numerically
integrating dynamical systems, non-linear stochastic filtering and generalized Bayesian
updating theories. It covers many applications of stochastic filtering and global
optimization. MATLAB codes for all the applications discussed here appear on the
weblink www.cambridge.org/9781107182646
Debasish Roy is currently working as Professor, Computational Mechanics Lab,
Department of Civil Engineering, Indian Institute of Science, Bangalore. His research
interests include computational mechanics of non-classical continua, stochastic dynamical
systems and optimization/inverse problems.
G. Visweswara Rao is an Engineering Consultant, Bangalore. His research interests
include structural dynamics specific to earthquake engineering, non-linear and random
vibration, and stochastic structural dynamics.
|
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| id | DE-604.BV044424385 |
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| indexdate | 2024-07-10T07:52:35Z |
| institution | BVB |
| isbn | 9781107182646 |
| language | English |
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| physical | xxxvii, 709 Seiten Diagramme |
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| spelling | Roy, Debasish 1946- Verfasser (DE-588)1044373571 aut Stochastic dynamics, filtering and optimization Debasish Roy, G. Visweswara Rao Cambridge, United Kingdom Cambridge University Press 2017 xxxvii, 709 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Mathematisches Modell Stochastische Optimierung (DE-588)4057625-5 gnd rswk-swf Stochastic processes / Mathematical models Stochastic differential equations / Numerical solutions Mathematical optimization Stochastische Optimierung (DE-588)4057625-5 s DE-604 G., Visweswara Rao Verfasser (DE-588)1044374799 aut Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029825927&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029825927&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Roy, Debasish 1946- G., Visweswara Rao Stochastic dynamics, filtering and optimization Mathematisches Modell Stochastische Optimierung (DE-588)4057625-5 gnd |
| subject_GND | (DE-588)4057625-5 |
| title | Stochastic dynamics, filtering and optimization |
| title_auth | Stochastic dynamics, filtering and optimization |
| title_exact_search | Stochastic dynamics, filtering and optimization |
| title_full | Stochastic dynamics, filtering and optimization Debasish Roy, G. Visweswara Rao |
| title_fullStr | Stochastic dynamics, filtering and optimization Debasish Roy, G. Visweswara Rao |
| title_full_unstemmed | Stochastic dynamics, filtering and optimization Debasish Roy, G. Visweswara Rao |
| title_short | Stochastic dynamics, filtering and optimization |
| title_sort | stochastic dynamics filtering and optimization |
| topic | Mathematisches Modell Stochastische Optimierung (DE-588)4057625-5 gnd |
| topic_facet | Mathematisches Modell Stochastische Optimierung |
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