Large Scale Inverse Problems: Computational Methods and Applications in the Earth Sciences

✍ Scribed by Mike Cullen (editor); Melina A Freitag (editor); Stefan Kindermann (editor); Robert Scheichl (editor)

• Publisher: De Gruyter
• Year: 2013
• Tongue: English
• Leaves: 216
• Series: Radon Series on Computational and Applied Mathematics; 13
• Category: Library

✦ Synopsis

This book is the second volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications.

The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary.

This collection of survey articles focusses on the large inverse problems commonly arising in simulation and forecasting in the earth sciences. For example, operational weather forecasting models have between 10⁷ and 10⁸ degrees of freedom. Even so, these degrees of freedom represent grossly space-time averaged properties of the atmosphere. Accurate forecasts require accurate initial conditions. With recent developments in satellite data, there are between 10⁶ and 10⁷ observations each day. However, while these also represent space-time averaged properties, the averaging implicit in the measurements is quite different from that used in the models. In atmosphere and ocean applications, there is a physically-based model available which can be used to regularise the problem. We assume that there is a set of observations with known error characteristics available over a period of time. The basic deterministic technique is to fit a model trajectory to the observations over a period of time to within the observation error. Since the model is not perfect the model trajectory has to be corrected, which defines the data assimilation problem. The stochastic view can be expressed by using an ensemble of model trajectories, and calculating corrections to both the mean value and the spread which allow the observations to be fitted by each ensemble member. In other areas of earth science, only the structure of the model formulation itself is known and the aim is to use the past observation history to determine the unknown model parameters.

The book records the achievements of Workshop 2 "Large-Scale Inverse Problems and Applications in the Earth Sciences". It involves experts in the theory of inverse problems together with experts working on both theoretical and practical aspects of the techniques by which large inverse problems arise in the earth sciences.

✦ Table of Contents

Preface
Synergy of inverse problems and data assimilation techniques
1 Introduction
2 Regularization theory
3 Cycling, Tikhonov regularization and 3DVar
4 Error analysis
5 Bayesian approach to inverse problems
6 4DVar
7 Kalman filter and Kalman smoother
8 Ensemble methods
9 Numerical examples
9.1 Data assimilation for an advection-diffusion system
9.2 Data assimilation for the Lorenz-95 system
10 Concluding remarks
Variational data assimilation for very large environmental problems
1 Introduction
2 Theory of variational data assimilation
2.1 Incremental variational data assimilation
3 Practical implementation
3.1 Model development
3.2 Background error covariances
3.3 Observation errors
3.4 Optimization methods
3.5 Reduced order approaches
3.6 Issues for nested models
3.7 Weak-constraint variational assimilation
4 Summary and future perspectives
Ensemble filter techniques for intermittent data assimilation
1 Bayesian statistics
1.1 Preliminaries
1.2 Bayesian inference
1.3 Coupling of random variables
1.4 Monte Carlo methods
2 Stochastic processes
2.1 Discrete time Markov processes
2.2 Stochastic difference and differential equations
2.3 Ensemble prediction and sampling methods
3 Data assimilation and filtering
3.1 Preliminaries
3.2 SequentialMonte Carlo method
3.3 Ensemble Kalman filter (EnKF)
3.4 Ensemble transform Kalman–Bucy filter
3.5 Guided sequential Monte Carlo methods
3.6 Continuous ensemble transform filter formulations
4 Concluding remarks
Inverse problems in imaging
1 Mathematicalmodels for images
2 Examples of imaging devices
2.1 Optical imaging
2.2 Transmission tomography
2.3 Emission tomography
2.4 MR imaging
2.5 Acoustic imaging
2.6 Electromagnetic imaging
3 Basic image reconstruction
3.1 Deblurring and point spread functions
3.2 Noise
3.3 Reconstruction methods
4 Missing data and prior information
4.1 Prior information
4.2 Undersampling and superresolution
4.3 Inpainting
4.4 Surface imaging
5 Calibration problems
5.1 Blind deconvolution
5.2 Nonlinear MR imaging
5.3 Attenuation correction in SPECT
5.4 Blind spectral unmixing
6 Model-based dynamic imaging
6.1 Kinetic models
6.2 Parameter identification
6.3 Basis pursuit
6.4 Motion and deformation models
6.5 Advanced PDE models
The lost honor of ℓ2-based regularization
1 Introduction
2 ℓ1-based regularization
3 Poor data
4 Large, highly ill-conditioned problems
4.1 Inverse potential problem
4.2 The effect of ill-conditioning on L1 regularization
4.3 Nonlinear, highly ill-posed examples
5 Summary
List of contributors