Large-scale Inverse Problems and Quantification of Uncertainty

by Biegler, Lorenz; Biros, George; Ghattas, Omar; Heinkenschloss, Matthias; Keyes, David; Mallick , Bani; Tenorio, Luis; van Bloemen Waanders, Bart; Willcox, Karen; Marzouk, Youssef

• ISBN13: 9780470697436
• ISBN10: 0470697431
• Edition: 1st
• Format: Hardcover
• Copyright: 2010-11-15
• Publisher: Wiley

Summary

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 Biography

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.

Table of Contents

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

Table of Contents provided by Publisher. All Rights Reserved.