This book focuses on computational methods for large-scale statistical inverse problems and provides an introduction to statistical Bayesian and frequentist methodologies. Recent research advances for approximation methods are discussed, along with Kalman filtering methods and optimization-based approaches to solving inverse problems. The aim is to cross-fertilize the perspectives of researchers in the areas of data assimilation, statistics, large-scale optimization, applied and computational mathematics, high performance computing, and cutting-edge applications.The solution to large-scale inverse problems critically depends on methods to reduce computational cost. Recent research approaches tackle this challenge in a variety of different ways. Many of the computational frameworks highlighted in this book build upon state-of-the-art methods for simulation of the forward problem, such as, fast Partial Differential Equation (PDE) solvers, reduced-order models and emulators of the forward problem, stochastic spectral approximations, and ensemble-based approximations, as well as exploiting the machinery for large-scale deterministic optimization through adjoint and other sensitivity analysis methods.Key Features:• Brings together the perspectives of researchers in areas of inverse problems and data assimilation.• Assesses the current state-of-the-art and identify needs and opportunities for future research.• Focuses on the computational methods used to analyze and simulate inverse problems.• Written by leading experts of inverse problems and uncertainty quantification.Graduate students and researchers working in statistics, mathematics and engineering will benefit from this book.
Author(s): Lorenz Biegler, George Biros, Omar Ghattas, Matthias Heinkenschloss, David Keyes, Bani Mallick, Luis Tenorio, Bart van Bloemen Waanders, Karen Willcox, Youssef Marzouk
Series: Wiley Series in Computational Statistics
Edition: 1
Publisher: Wiley
Year: 2011
Language: English
Pages: 388
Tags: Математика;Вычислительная математика;
Cover Page......Page 1
Wiley Series in Computational Statistics......Page 3
Title: Large-Scale Inverse Problems and Quantification of Uncertainty......Page 4
ISBN 9780470697436......Page 5
Contents......Page 6
List of Contributors......Page 13
1.1 Introduction......Page 16
1.2 Statistical Methods......Page 17
1.3 Approximation Methods......Page 19
1.4 Kalman Filtering......Page 20
1.5 Optimization......Page 21
2.1 Introduction......Page 24
2.2 Prior Information and Parameters: What DoYouKnow,andWhatDoYouWant to Know?......Page 25
2.3 Estimators: What Can You Do with What You Measure?......Page 31
2.4 Performance of Estimators: How Well Can You Do?......Page 32
2.5 Frequentist Performance of Bayes Estimators for a BNM......Page 42
2.6 Summary......Page 45
References......Page 46
3.1 Introduction......Page 48
3.2 Belief, Information and Probability......Page 49
3.3 Bayes’ Formula and Updating Probabilities......Page 51
3.4 Computed Examples Involving Hypermodels......Page 57
3.5 Dynamic Updating of Beliefs......Page 69
3.6 Discussion......Page 81
References......Page 83
4.1 Introduction......Page 86
4.2 The Bayesian and Frequentist Approaches......Page 89
4.3 Prior Distribution......Page 92
4.4 A Geostatistical Approach......Page 96
References......Page 98
5.1 Introduction......Page 102
5.2 Bayesian Model Formulation......Page 103
5.3 Application: Cosmic Microwave Background......Page 115
5.4 Discussion......Page 118
References......Page 119
6.1 Introduction......Page 122
6.2 Model Equations and Problem Setting......Page 124
6.3 Approximation of the Response Surface Using the Bayesian Partition Model and Two-Stage MCMC......Page 126
6.4 Numerical Results......Page 130
References......Page 136
7.1 Introduction......Page 138
7.2 Reducing the Computational Cost of Solving Statistical Inverse Problems......Page 139
7.3 General Formulation......Page 142
7.4 Model Reduction......Page 143
7.5 Stochastic Spectral Methods......Page 148
7.6 Illustrative Example......Page 151
7.7 Conclusions......Page 157
References......Page 159
8 Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real-Time Bayesian Parameter Estimation......Page 166
8.2 Linear Parabolic Equations......Page 167
8.3 Bayesian Parameter Estimation......Page 181
References......Page 188
9.1 Introduction......Page 194
9.2 Gaussian Process Models......Page 195
9.3 Bayesian Model Calibration......Page 198
9.4 Case Study: Thermal Simulation of Decomposing Foam......Page 202
9.5 Conclusions......Page 207
References......Page 208
10 Bayesian Calibration of Expensive Multivariate Computer Experiments......Page 210
10.1 Calibration of Computer Experiments......Page 211
10.2 Emulation......Page 218
10.3 Multivariate Calibration......Page 224
10.4 Summary......Page 227
References......Page 228
11.1 Introduction......Page 232
11.2 Model Assumptions......Page 233
11.3 The Traditional Kalman Filter (KF)......Page 238
11.4 The Ensemble Kalman Filter (EnKF)......Page 240
11.5 The Randomized Maximum Likelihood Filter (RMLF)......Page 251
11.6 The Particle Filter (PF)......Page 254
11.7 Closing Remarks......Page 256
References......Page 258
Appendix A: Properties of the EnKF Algorithm......Page 260
Appendix B: Properties of the RMLF Algorithm......Page 261
12.1 Introduction......Page 262
12.2 Formulation and Solution of the Inverse Problem......Page 264
12.3 EnKF History Matching Work.ow......Page 267
12.4 Field Case......Page 273
12.5 Conclusion......Page 283
References......Page 285
13.1 Introduction......Page 288
13.2 Impedance Tomography......Page 290
13.3 Optimal Experimental Design: Background......Page 291
13.4 Optimal Experimental Design for Nonlinear Ill-Posed Problems......Page 294
13.5 Optimization Framework......Page 295
13.6 Numerical Results......Page 299
13.7 Discussion and Conclusions......Page 301
References......Page 303
14.1 Introduction......Page 306
14.2 Mathematical Developments......Page 309
14.3 Numerical Examples......Page 325
References......Page 332
15.1 Introduction......Page 336
15.2 Traveltime Inversion for Velocity Determination......Page 338
15.3 Prestack Stratigraphic Inversion......Page 347
References......Page 356
16.1 Introduction......Page 360
16.2 Runge-Kutta Methods......Page 363
16.3 Adaptive Steps......Page 367
16.4 Linear Multistep Methods......Page 370
16.5 Numerical Results......Page 372
16.6 Application to Data Assimilation......Page 373
16.7 Conclusions......Page 377
References......Page 378
Index......Page 382
Back Page......Page 388
