Large-Scale Inverse Problems and Quantification of Uncertainty

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.

Table of Contents:
1 Introduction 1.1 Introduction 1.2 Statistical Methods 1.3 Approximation Methods 1.4 Kalman Filtering 1.5 Optimization 2 A Primer of Frequentist and Bayesian Inference in Inverse Problems 2.1 Introduction 2.2 Prior Information and Parameters: What do you know, and what do you want to know? 2.3 Estimators: What can you do with what you measure? 2.4 Performance of estimators: How well can you do? 2.5 Frequentist performance of Bayes estimators for a BNM 2.6 Summary Bibliography 3 Subjective Knowledge or Objective Belief? An Oblique Look to Bayesian Methods 3.1 Introduction 3.2 Belief, information and probability 3.3 Bayes' formula and updating probabilities 3.4 Computed examples involving hypermodels 3.5 Dynamic updating of beliefs 3.6 Discussion Bibliography 4 Bayesian and Geostatistical Approaches to Inverse Problems 4.1 Introduction 4.2 The Bayesian and Frequentist Approaches 4.3 Prior Distribution 4.4 A Geostatistical Approach 4.5 Concluding Bibliography 5 Using the Bayesian Framework to Combine Simulations and Physical Observations for Statistical Inference 5.1 Introduction 5.2 Bayesian Model Formulation  5.3 Application: Cosmic Microwave Background 5.4 Discussion Bibliography 6 Bayesian Partition Models for Subsurface Characterization 6.1 Introduction 6.2 Model equations and problem setting 6.3 Approximation of the response surface using the Bayesian Partition Model and two-stage MCMC 6.4 Numerical results 6.5 Conclusions Bibliography 7 Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems 7.1 Introduction 7.2 Reducing the computational cost of solving statistical inverse problems 7.3 General formulation 7.4 Model reduction 7.5 Stochastic spectral methods 7.6 Illustrative example 7.7 Conclusions Bibliography 8 Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs; Application to real-time Bayesian parameter estimation 8.1 Introduction 8.2 Linear Parabolic Equations 8.3 Bayesian Parameter Estimation 8.4 Concluding Remarks Bibliography 9 Calibration and Uncertainty Analysis for Computer Simulations with Multivariate Output 9.1 Introduction 9.2 Gaussian Process Models 9.3 Bayesian Model Calibration 9.4 Case Study: Thermal Simulation of Decomposing Foam 9.5 Conclusions Bibliography 10 Bayesian Calibration of Expensive Multivariate Computer Experiments 10.1 Calibration of computer experiments 10.2 Principal component emulation  10.3 Multivariate calibration 10.4 Summary Bibliography 11 The Ensemble Kalman Filter and Related Filters 11.1 Introduction 11.2 Model Assumptions 11.3 The Traditional Kalman Filter (KF) 11.4 The Ensemble Kalman Filter (EnKF) 11.5 The Randomized Maximum Likelihood Filter (RMLF) 11.6 The Particle Filter (PF) 11.7 Closing Remarks 11.8 Appendix A: Properties of the EnKF Algorithm 11.9 Appendix B: Properties of the RMLF Algorithm Bibliography 12 Using the ensemble Kalman Filter for history matching and uncertainty quantification of complex reservoir models 12.1 Introduction 12.2 Formulation and solution of the inverse problem 12.3 EnKF history matching workflow 12.4 Field Case 12.5 Conclusion Bibliography 13 Optimal Experimental Design for the Large-Scale Nonlinear Ill-posed Problem of Impedance Imaging 13.1 Introduction 13.2 Impedance Tomography 13.3 Optimal Experimental Design - Background 13.4 Optimal Experimental Design for Nonlinear Ill-Posed Problems 13.5 Optimization Framework 13.6 Numerical Results 13.7 Discussion and Conclusions Bibliography 14 Solving Stochastic Inverse Problems: A Sparse Grid Collocation Approach 14.1 Introduction 14.2 Mathematical developments 14.3 Numerical Examples 14.4 Summary Bibliography 15 Uncertainty analysis for seismic inverse problems: two practical examples 15.1 Introduction 15.2 Traveltime inversion for velocity determination. 15.3 Prestack stratigraphic inversion 15.4 Conclusions Bibliography 16 Solution of inverse problems using discrete ODE adjoints 16.1 Introduction 16.2 Runge-Kutta Methods 16.3 Adaptive Steps 16.4 Linear Multistep Methods 16.5 Numerical Results 16.6 Application to Data Assimilation 16.7 Conclusions Bibliography TBD