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Radon series on computational and applied mathematics.
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Annotation
This book is thesecond volume of three volume series recording the ""Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment"" taking place in Linz, Austria, October 3-7, 2011. The volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications.
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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
- 1 Bayesian statistics
- 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
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