Large-Scale Inverse Problems and Quantification of Uncertainty

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.

• 1 Introduction1.1 Introduction1.2 Statistical Methods1.3 Approximation Methods1.4 Kalman Filtering1.5 Optimization 2 A Primer of Frequentist and Bayesian Inference in Inverse Problems2.1 Introduction2.2 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 BNM2.6 SummaryBibliography 3 Subjective Knowledge or Objective Belief? An Oblique Look to Bayesian Methods3.1 Introduction3.2 Belief, information and probability3.3 Bayes' formula and updating probabilities3.4 Computed examples involving hypermodels3.5 Dynamic updating of beliefs3.6 DiscussionBibliography 4 Bayesian and Geostatistical Approaches to Inverse Problems4.1 Introduction4.2 The Bayesian and Frequentist Approaches4.3 Prior Distribution4.4 A Geostatistical Approach4.5 ConcludingBibliography 5 Using the Bayesian Framework to Combine Simulations and Physical Observationsfor Statistical Inference5.1 Introduction5.2 Bayesian Model Formulation 5.3 Application: Cosmic Microwave Background5.4 DiscussionBibliography 6 Bayesian Partition Models for Subsurface Characterization6.1 Introduction6.2 Model equations and problem setting6.3 Approximation of the response surface using the Bayesian Partition Model and two-stageMCMC6.4 Numerical results6.5 ConclusionsBibliography 7 Surrogate and reduced-order modeling: a comparison of approaches for large-scalestatistical inverse problems7.1 Introduction7.2 Reducing the computational cost of solving statistical inverse problems7.3 General formulation7.4 Model reduction7.5 Stochastic spectral methods7.6 Illustrative example7.7 ConclusionsBibliography8 Reduced basis approximation and a posteriori error estimation for parametrizedparabolic PDEs; Application to real-time Bayesian parameter estimation8.1 Introduction8.2 Linear Parabolic Equations8.3 Bayesian Parameter Estimation8.4 Concluding RemarksBibliography 9 Calibration and Uncertainty Analysis for Computer Simulations with MultivariateOutput9.1 Introduction9.2 Gaussian Process Models9.3 Bayesian Model Calibration9.4 Case Study: Thermal Simulation of Decomposing Foam9.5 ConclusionsBibliography 10 Bayesian Calibration of Expensive Multivariate Computer Experiments10.1 Calibration of computer experiments10.2 Principal component emulation 10.3 Multivariate calibration10.4 SummaryBibliography 11 The Ensemble Kalman Filter and Related Filters11.1 Introduction11.2 Model Assumptions11.3 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 Remarks11.8 Appendix A: Properties of the EnKF Algorithm11.9 Appendix B: Properties of the RMLF AlgorithmBibliography 12 Using the ensemble Kalman Filter for history matching and uncertainty quantificationof complex reservoir models12.1 Introduction12.2 Formulation and solution of the inverse problem12.3 EnKF history matching workflow12.4 Field Case12.5 ConclusionBibliography13 Optimal Experimental Design for the Large-Scale Nonlinear Ill-posed Problem ofImpedance Imaging13.1 Introduction13.2 Impedance Tomography13.3 Optimal Experimental Design - Background13.4 Optimal Experimental Design for Nonlinear Ill-Posed Problems13.5 Optimization Framework13.6 Numerical Results13.7 Discussion and ConclusionsBibliography 14 Solving Stochastic Inverse Problems: A Sparse Grid Collocation Approach14.1 Introduction14.2 Mathematical developments14.3 Numerical Examples14.4 SummaryBibliography 15 Uncertainty analysis for seismic inverse problems: two practical examples15.1 Introduction15.2 Traveltime inversion for velocity determination.15.3 Prestack stratigraphic inversion15.4 Conclusions Bibliography16 Solution of inverse problems using discrete ODE adjoints16.1 Introduction16.2 Runge-Kutta Methods16.3 Adaptive Steps16.4 Linear Multistep Methods16.5 Numerical Results16.6 Application to Data Assimilation16.7 ConclusionsBibliographyTBD