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E-raamat: Large-Scale Inverse Problems and Quantification of Uncertainty

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Large-Scale Inverse Problems and Quantification of UncertaintySuurem pilt
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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.

List of Contributors
xiii
1 Introduction
1(8)
1.1 Introduction
1(1)
1.2 Statistical Methods
2(2)
1.3 Approximation Methods
4(1)
1.4 Kalman Filtering
5(1)
1.5 Optimization
6(3)
2 A Primer of Frequentist and Bayesian Inference in Inverse Problems
9(24)
P. B. Stark
L. Tenorio
2.1 Introduction
9(1)
2.2 Prior Information and Parameters: What Do You Know, and What Do You Want to Know?
10(6)
2.2.1 The State of the World, Measurement Model, Parameters and Likelihoods
10(2)
2.2.2 Prior and Posterior Probability Distributions
12(4)
2.3 Estimators: What Can You Do with What You Measure?
16(1)
2.4 Performance of Estimators: How Well Can You Do?
17(10)
2.4.1 Bias, Variance
17(3)
2.4.2 Loss and Risk
20(1)
2.4.3 Decision Theory
21(6)
2.5 Frequentist Performance of Bayes Estimators for a BNM
27(3)
2.5.1 MSE of the Bayes Estimator for BNM
27(1)
2.5.2 Frequentist Coverage of the Bayesian Credible Regions for BNM
28(2)
2.5.3 Expected Length of the Bayesian Credible Region for BNM
30(1)
2.6 Summary
30(1)
References
31(2)
3 Subjective Knowledge or Objective Belief? An Oblique Look to Bayesian Methods
33(38)
D. Calvetti
E. Somersalo
3.1 Introduction
33(1)
3.2 Belief, Information and Probability
34(2)
3.3 Bayes' Formula and Updating Probabilities
36(6)
3.3.1 Subjective Nature of the Likelihood
37(2)
3.3.2 Adding Layers: hypermodels
39(3)
3.4 Computed Examples Involving Hypermodels
42(12)
3.5 Dynamic Updating of Beliefs
54(12)
3.6 Discussion
66(2)
References
68(3)
4 Bayesian and Geostatistical Approaches to Inverse Problems
71(16)
P. K. Kitanidis
4.1 Introduction
71(3)
4.2 The Bayesian and Frequentist Approaches
74(3)
4.2.1 Frequentist Approach
74(2)
4.2.2 Bayesian Approach
76(1)
4.3 Prior Distribution
77(4)
4.4 A Geostatistical Approach
81(2)
4.5 Conclusion
83(1)
References
83(4)
5 Using the Bayesian Framework to Combine Simulations and Physical Observations for Statistical Inference
87(20)
D. Higdon
K. Heitmann
E. Lawrence
S. Habib
5.1 Introduction
87(1)
5.2 Bayesian Model Formulation
88(12)
5.2.1 General Formulation
88(1)
5.2.2 Unlimited Simulation Runs
89(3)
5.2.3 Limited Simulation Runs
92(4)
5.2.4 Limited Simulations Runs with Multivariate Output
96(4)
5.3 Application: Cosmic Microwave Background
100(3)
5.4 Discussion
103(1)
References
104(3)
6 Bayesian Partition Models for Subsurface Characterization
107(16)
Y. Efendiev
A. Datta-Gupta
K. Hwang
X. Ma
B. Mallick
6.1 Introduction
107(2)
6.2 Model Equations and Problem Setting
109(2)
6.3 Approximation of the Response Surface Using the Bayesian Partition Model and Two-Stage MCMC
111(4)
6.4 Numerical Results
115(6)
6.5 Conclusions
121(1)
References
121(2)
7 Surrogate and Reduced-Order Modeling: A Comparison of Approaches for Large-Scale Statistical Inverse Problems
123(28)
M. Frangos
Y. Marzouk
K. Willcox
B. van Bloemen Waanders
7.1 Introduction
123(1)
7.2 Reducing the Computational Cost of Solving Statistical Inverse Problems
124(3)
7.2.1 Reducing the Cost of Forward Simulations
125(1)
7.2.2 Reducing the Dimension of the Input Space
126(1)
7.2.3 Reducing the Number of Samples
126(1)
7.3 General Formulation
127(1)
7.4 Model Reduction
128(5)
7.4.1 General Projection Framework
129(1)
7.4.2 Computing the Basis
130(1)
7.4.3 Computing a Basis for Inverse Problem Applications: Sampling the Parameter Space
131(2)
7.5 Stochastic Spectral Methods
133(3)
7.5.1 Surrogate Posterior Distribution
133(2)
7.5.2 Forward Solution Methodologies and Convergence Results
135(1)
7.6 Illustrative Example
136(6)
7.7 Conclusions
142(2)
References
144(7)
8 Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real-Time Bayesian Parameter Estimation
151(28)
N. C. Nguyen
G. Rozza
D. B. P. Huynh
A. T. Patera
8.1 Introduction
152(1)
8.2 Linear Parabolic Equations
152(14)
8.2.1 Reduced Basis Approximation
152(5)
8.2.2 A Posteriori Error Estimation
157(1)
8.2.3 Offline---Online Computational Approach
158(8)
8.3 Bayesian Parameter Estimation
166(7)
8.3.1 Bayesian Approach
166(2)
8.3.2 A Posteriori Bounds for the Expected Value
168(2)
8.3.3 Numerical Example
170(3)
8.4 Concluding Remarks
173(1)
References
173(6)
9 Calibration and Uncertainty Analysis for Computer Simulations with Multivariate Output
179(16)
J. McFarland
L. Swiler
9.1 Introduction
179(1)
9.2 Gaussian Process Models
180(3)
9.2.1 Estimation of Parameters Governing the GP
181(1)
9.2.2 Modeling Time Series Output
182(1)
9.3 Bayesian Model Calibration
183(4)
9.4 Case Study: Thermal Simulation of Decomposing Foam
187(5)
9.4.1 Preliminary Analysis
188(1)
9.4.2 Bayesian Calibration Analysis
189(3)
9.5 Conclusions
192(1)
References
193(2)
10 Bayesian Calibration of Expensive Multivariate Computer Experiments
195(22)
R. D. Wilkinson
10.1 Calibration of Computer Experiments
196(7)
10.1.1 Statistical Calibration Framework
198(2)
10.1.2 Model Error
200(1)
10.1.3 Code Uncertainty
201(2)
10.2 Emulation
203(6)
10.2.1 Bayesian
203(2)
10.2.2 Principal Component
205(4)
10.3 Multivariate Calibration
209(3)
10.4 Summary
212(1)
References
213(4)
11 The Ensemble Kalman Filter and Related Filters
217(30)
I. Myrseth
H. Omre
11.1 Introduction
217(1)
11.2 Model Assumptions
218(5)
11.3 The Traditional Kalman Filter (KF)
223(2)
11.4 The Ensemble Kalman Filter (EnKF)
225(11)
11.4.1 Variable Characteristics
229(1)
11.4.2 Parameter Estimates
230(3)
11.4.3 A Special Case
233(3)
11.5 The Randomized Maximum Likelihood Filter (RMLF)
236(3)
11.6 The Particle Filter (PF)
239(2)
11.7 Closing Remarks
241(2)
References
243(2)
Appendix A Properties of the EnKF Algorithm
245(1)
Appendix B Properties of the RMLF Algorithm
246(1)
12 Using the Ensemble Kalman Filter for History Matching and Uncertainty Quantification of Complex Reservoir Models
247(26)
A. Seiler
G. Evensen
J.-A. Skjervheim
J. Hove
J. G. Vabø
12.1 Introduction
247(2)
12.2 Formulation and Solution of the Inverse Problem
249(3)
12.2.1 Traditional Minimization Methods
249(2)
12.2.2 Sequential Processing of Measurements
251(1)
12.3 EnKF History Matching Workflow
252(6)
12.3.1 Estimation of Relative Permeability
254(2)
12.3.2 Transformed Fault Transmissibility Multipliers
256(1)
12.3.3 State Vector
257(1)
12.3.4 Updating Realizations
257(1)
12.4 Field Case
258(10)
12.4.1 Reservoir Presentation
258(2)
12.4.2 The Initial Ensemble
260(2)
12.4.3 Results
262(6)
12.5 Conclusion
268(2)
References
270(3)
13 Optimal Experimental Design for the Large-Scale Nonlinear Ill-Posed Problem of Impedance Imaging
273(18)
L. Horesh
E. Haber
L. Tenorio
13.1 Introduction
273(2)
13.2 Impedance Tomography
275(1)
13.3 Optimal Experimental Design: Background
276(3)
13.3.1 Optimal Experimental Design for Well-Posed Linear Problems
277(1)
13.3.2 Optimal Experimental Design for Linear Ill-Posed Problems
277(2)
13.4 Optimal Experimental Design for Nonlinear Ill-Posed Problems
279(1)
13.5 Optimization Framework
280(4)
13.5.1 General Scheme
280(2)
13.5.2 Application to Impedance Tomography
282(2)
13.6 Numerical Results
284(2)
13.7 Discussion and Conclusions
286(2)
References
288(3)
4 Solving Stochastic Inverse Problems: A Sparse Grid Collocation Approach
291(30)
N. Zabaras
14.1 Introduction
291(3)
14.2 Mathematical Developments
294(16)
14.2.1 The Stochastic Inverse Problem: Mathematical Problem Definition
295(2)
14.2.2 The Stochastic Metrics and Representation of the Inverse Stochastic Solution q
297(3)
14.2.3 Solving the Direct Stochastic Problem: Adaptivity Sparse Grid Collocation
300(3)
14.2.4 Stochastic Sensitivity Equations and Gradient-Based Optimization Framework
303(4)
14.2.5 Incorporating Correlation Statistics and Investigating Regularization
307(2)
14.2.6 Stochastic Low-Dimensional Modeling
309(1)
14.3 Numerical Examples
310(7)
14.4 Summary
317(1)
References
317(4)
15 Uncertainty Analysis for Seismic Inverse Problems: Two Practical Examples
321(24)
F. Delbos
C. Duffet
D. Sinoquet
15.1 Introduction
321(2)
15.2 Traveltime Inversion for Velocity Determination
323(9)
15.2.1 Characteristics and Formulation
323(2)
15.2.2 Optimization Method
325(1)
15.2.3 Uncertainty Analysis
325(3)
15.2.4 Application
328(4)
15.3 Prestack Stratigraphic Inversion
332(9)
15.3.1 Characteristics and Formulation
333(1)
15.3.2 Optimization Method
334(1)
15.3.3 Uncertainty Analysis
335(4)
15.3.4 Application
339(2)
15.4 Conclusions
341(1)
References
341(4)
16 Solution of Inverse Problems Using Discrete ODE Adjoints
345(22)
A. Sandu
16.1 Introduction
345(3)
16.2 Runge-Kutta Methods
348(4)
16.2.1 Accuracy of the Discrete Adjoint RK Method
349(3)
16.3 Adaptive Steps
352(3)
16.3.1 Efficient Implementation of Implicit RK Adjoints
352(2)
16.3.2 Iterative Solvers
354(1)
16.3.3 Considerations on the Formal Discrete RK Adjoints
354(1)
16.4 Linear Multistep Methods
355(2)
16.4.1 Consistency of Discrete Linear Multistep Adjoints at Intermediate Time Points
356(1)
16.4.2 Consistency of Discrete Linear Multistep Adjoints at the Intital Time
357(1)
16.5 Numerical Results
357(1)
16.6 Application to Data Assimilation
358(4)
16.7 Conclusions
362(1)
References
363(4)
Index 367
Lorenz Biegler, Carnegie Mellon University, USA.

George Biros, Georgia Institute of Technology, USA.

Omar Ghattas, University of Texas at Austin, USA.

Matthias Heinkenschloss, Rice University, USA.

David Keyes, KAUST and Columbia University, USA.

Bani Mallick, Texas A&M University, USA.

Luis Tenorio, Colorado School of Mines, USA.

Bart van Bloemen Waanders, Sandia National Laboratories, USA.

Karen Wilcox, Massachusetts Institute of Technology, USA.

Youssef Marzouk, Massachusetts Institute of Technology, USA.
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