Stochastic Systems: Uncertainty Quantification and Propagation

✍ Scribed by Mircea Grigoriu

• Publisher: Springer
• Year: 2012
• Tongue: English
• Leaves: 534
• Series: Springer Series in Reliability Engineering
• Edition: 2012
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover......Page 1
Stochastic Systems......Page 3
Contents......Page 6
A.1 Parametric Models......Page 11
A.2 Quantizers......Page 15
A.3 Stochastic Reduced Order Models (SROMs)......Page 16
References......Page 17
2.2.1 Sample Space......Page 18
2.2.2 σ-Field......Page 19
2.2.3 Probability Measure......Page 20
2.2.4 Construction of Probability Spaces......Page 23
2.3 Measurable Functions and Random Elements......Page 25
2.4 Independence......Page 27
2.5 Sequence of Events......Page 30
2.6 Expectation......Page 31
2.7 Convergence of Sequences of Random Variables......Page 36
2.8 Radon--Nikodym Derivative......Page 40
2.9 Distribution and Density Functions......Page 41
2.10 Characteristic Function......Page 45
2.11 Conditional Expectation......Page 47
2.12 Discrete Time Martingales......Page 51
2.13 Monte Carlo Simulation......Page 56
2.13.2 Non-Gaussian Variables......Page 57
2.13.3 Estimators......Page 59
2.14 Exercises......Page 62
References......Page 66
3.1 Introduction......Page 67
3.2 Finite Dimensional Distributions......Page 70
3.3 Sample Properties......Page 72
3.4 Second Moment Properties......Page 75
3.5.1 mathbb R-Valued Stochastic Processes......Page 78
3.5.2 mathbb Rd-Valued Stochastic Processes......Page 80
3.5.3 mathbb R-Valued Random Fields......Page 81
3.6.1 Continuity......Page 84
3.6.2 Differentiability......Page 85
3.6.3 Integration......Page 87
3.6.4 Spectral Representation......Page 89
3.6.5 Karhunen--Loève Expansion......Page 91
3.7 Classes of Stochastic Processes......Page 93
3.7.2 Translation Random Functions......Page 94
3.7.3 Ergodic Random Functions......Page 96
3.7.4 Markov Random Functions......Page 98
3.7.5 Processes with Independent Increments......Page 101
3.7.6 Continuous Time Martingales......Page 103
3.8.1 Stationary Gaussian Random Functions......Page 117
3.8.2 Translation Vector Processes......Page 121
3.8.3 Non-Stationary Gaussian Processes......Page 127
3.9 Exercises......Page 132
References......Page 134
4.1 Introduction......Page 136
4.2 Riemann-Stieltjes Integrals......Page 137
4.3 Stochastic Integrals int B dB and int N dN......Page 138
4.4 Stochastic Integrals with Brownian Motion Integrators......Page 143
4.4.1 Integrands in fancyscript H02......Page 144
4.4.2 Integrands in fancyscript H2......Page 145
4.4.3 Integrands in fancyscript H......Page 148
4.5 Stochastic Integrals with Martingale Integrators......Page 149
4.6 Stochastic Integrals with Semimartingale Integrators......Page 153
4.7 Quadratic Variation and Covariation Processes......Page 155
4.8 Exercises......Page 159
References......Page 161
5.2 Itô's Formula for mathbb R-Valued Semimartingales......Page 162
5.2.1 Continuous Semimartingales......Page 163
5.2.2 Arbitrary Semimartingales......Page 165
5.3 Itô's Formula for mathbb Rd-Valued Semimartingales......Page 169
5.4 Itô and Stratonovich Integrals......Page 171
5.5 Applications......Page 172
5.5.1 Stochastic Differential Equations......Page 173
5.5.2 Tanaka's Formula......Page 188
5.5.3 Random Walk Method......Page 191
5.5.4 Girsanov's Theorem......Page 200
5.6 Exercises......Page 204
References......Page 205
6.1 Introduction......Page 207
6.2.1 Gaussian Variables......Page 209
6.2.2 Translation Variables......Page 212
6.2.3 Bounded Variables......Page 216
6.2.4 Directional Wind Speed for Hurricanes......Page 217
6.3 Random Functions......Page 220
6.3.1 Systems with Uncertain Parameters......Page 221
6.3.2 Inclusions in Multi-Phase Materials......Page 223
6.3.3 Probabilistic Models for Microstructures......Page 227
6.4 Exercises......Page 239
References......Page 241
7.1 Introduction......Page 243
7.2.1 Discrete Time Linear Systems......Page 245
7.2.2 Continuous Time Linear Systems......Page 247
7.2.3 Continuous Time Nonlinear Systems......Page 254
7.3 Stochastic Difference Equations with Random Coefficients......Page 260
7.3.1 General Considerations......Page 261
7.3.2 Monte Carlo Simulation......Page 266
7.3.3 Conditional Analysis......Page 267
7.3.4 Stochastic Reduced Order Models......Page 271
7.3.6 Taylor Series......Page 273
7.3.7 Perturbation Series......Page 275
7.4.1 General Considerations......Page 277
7.4.2 Monte Carlo Simulation......Page 282
7.4.3 Conditional Analysis......Page 283
7.4.4 Conditional Monte Carlo Simulation......Page 284
7.4.5 State Augmentation......Page 292
7.4.6 Stochastic Reduced Order Models......Page 295
7.4.7 Stochastic Galerkin Method......Page 296
7.4.8 Stochastic Collocation Method......Page 301
7.4.9 Taylor, Perturbation, and Neumann Series......Page 306
7.5 Applications......Page 309
7.5.1 Stochastic Stability......Page 310
7.5.2 Noise Induced Transitions......Page 317
7.5.3 Solution of Uncertain Dynamic Systems by SROMs......Page 323
7.5.4 Degrading Systems......Page 333
7.6 Exercises......Page 337
References......Page 339
8.1 Introduction......Page 342
8.2 SAEs with Arbitrary Uncertainty......Page 343
8.2.1 General Considerations......Page 344
8.2.2 Monte Carlo Method......Page 347
8.2.3 Stochastic Reduced Order Model Method......Page 349
8.2.4 Stochastic Galerkin Method......Page 362
8.2.5 Stochastic Collocation Method......Page 368
8.2.6 Reliability Method......Page 372
8.3 SAEs with Small Uncertainty......Page 373
8.3.1 Taylor Series......Page 374
8.3.2 Perturbation Series......Page 376
8.3.3 Neumann Series......Page 378
8.3.4 Equivalent Linearization......Page 379
8.4 Exercises......Page 380
References......Page 382
9.1 Introduction......Page 384
9.2 Stochastic Partial Differential Equations......Page 385
9.3 Discrete Approximations of SPDEs......Page 391
9.4 Applied SPDEs: Arbitrary Uncertainty......Page 398
9.4.1 General Considerations......Page 399
9.4.2 Deterministic Boundary Value Problems......Page 401
9.4.3 Stochastic Boundary Value Problems......Page 403
9.4.4 Monte Carlo Simulation......Page 408
9.4.6 Stochastic Reduced Order Models......Page 419
9.4.7 Extended Stochastic Reduced Order Models......Page 432
9.4.8 Stochastic Galerkin Method......Page 436
9.4.9 Stochastic Collocation Method......Page 445
9.5 Applied SPDEs: Small Uncertainty......Page 451
9.5.1 Taylor Series......Page 452
9.5.2 Perturbation Series......Page 453
9.5.3 Neumann Series......Page 454
9.6 Exercises......Page 456
References......Page 457
A.1.1 Karhunen--Loève Expansion......Page 460
A.1.2 Spectral Representation......Page 463
A.1.3 Sampling Theorem......Page 464
B.1Metric Spaces......Page 474
B.1.1Topology Generated by a Metric......Page 482
B.1.2Closed, Complete, and Compact Sets......Page 484
B.1.3 Sequences......Page 486
B.1.4 Contraction......Page 487
B.2Linear or Vector Spaces......Page 488
B.3Normed Linear Spaces......Page 489
B.3.2 Basis and Separability......Page 492
B.3.3 Operators......Page 493
B.4Hilbert Spaces......Page 495
B.4.1 Basis and Fourier Representations......Page 496
B.4.2 Linear Functionals......Page 501
B.4.3 Weak Convergence......Page 502
B.4.4 Bounden Linear Operators......Page 503
B.4.5 Spectral Theory......Page 506
B.5.1Useful Inequalities......Page 508
B.5.2 Lp Spaces as Normed Spaces......Page 509
B.6Orthogonal Polynomials......Page 511
B.6.1Hermite Polynomials......Page 512
B.6.2Homogeneous Chaos......Page 515
B.6.3Multiple Wiener--Itô Integrals......Page 520
Index......Page 524