Large-scale Inverse Problems and

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.

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.

Introduction
Introduction
Statistical Methods
Approximation Methods
Kalman Filtering
Optimization
A Primer of Frequentist and Bayesian Inference in Inverse Problems
Introduction
Prior Information and Parameters: What do you know, and what do you want to know?
Estimators: What can you do with what you measure?
Performance of estimators: How well can you do?
Frequentist performance of Bayes estimators for a BNM
Summary
Bibliography
Subjective Knowledge or Objective Belief? An Oblique Look to Bayesian Methods
Introduction
Belief, information and probability
Bayes' formula and updating probabilities
Computed examples involving hypermodels
Dynamic updating of beliefs
Discussion
Bibliography
Bayesian and Geostatistical Approaches to Inverse Problems
Introduction
The Bayesian and Frequentist Approaches
Prior Distribution
A Geostatistical Approach
Concluding
Bibliography
Using the Bayesian Framework to Combine Simulations and Physical Observations for Statistical Inference
Introduction
Bayesian Model Formulation
Application: Cosmic Microwave Background
Discussion
Bibliography
Bayesian Partition Models for Subsurface Characterization
Introduction
Model equations and problem setting
Approximation of the response surface using the Bayesian Partition Model and two-stage
MCMC
Numerical results
Conclusions
Bibliography
Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems
Introduction
Reducing the computational cost of solving statistical inverse problems
General formulation
Model reduction
Stochastic spectral methods
Illustrative example
Conclusions
Bibliography
Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs; Application to real-time Bayesian parameter estimation
Introduction
Linear Parabolic Equations
Bayesian Parameter Estimation
Concluding Remarks
Bibliography
Calibration and Uncertainty Analysis for Computer Simulations with Multivariate
Output
Introduction
Gaussian Process Models
Bayesian Model Calibration
Case Study: Thermal Simulation of Decomposing Foam
Conclusions
Bibliography
Bayesian Calibration of Expensive Multivariate Computer Experiments
Calibration of computer experiments
Principal component emulation
Multivariate calibration
Summary
Bibliography
The Ensemble Kalman Filter and Related Filters
Introduction
Model Assumptions
The Traditional Kalman Filter (KF)
The Ensemble Kalman Filter (EnKF)
The Randomized Maximum Likelihood Filter (RMLF)
The Particle Filter (PF)
Closing Remarks
Appendix A: Properties of the EnKF Algorithm
Appendix B: Properties of the RMLF Algorithm
Bibliography
Using the ensemble Kalman Filter for history matching and uncertainty quantification of complex reservoir models
Introduction
Formulation and solution of the inverse problem
EnKF history matching workflow
Field Case
Conclusion
Bibliography
Optimal Experimental Design for the Large-Scale Nonlinear Ill-posed Problem of Impedance Imaging
Introduction
Impedance Tomography
Optimal Experimental Design - Background
Optimal Experimental Design for Nonlinear Ill-Posed Problems
Optimization Framework
Numerical Results
Discussion and Conclusions
Bibliography
Solving Stochastic Inverse Problems: A Sparse Grid Collocation Approach
Introduction
Mathematical developments
Numerical Examples
Summary
Bibliography
Uncertainty analysis for seismic inverse problems: two practical examples
Introduction
Traveltime inversion for velocity determination
Prestack stratigraphic inversion
Conclusions
Bibliography
Solution of inverse problems using discrete ODE adjoints
Introduction
Runge-Kutta Methods
Adaptive Steps
Linear Multistep Methods
Numerical Results
Application to Data Assimilation
Conclusions
Bibliography
TBD
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Large-scale Inverse Problems and Quantification of Uncertainty

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