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.

• 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