Iterative optimization in inverse problems

Content: Background  Overview  An Urns Model for Remote Sensing  Hidden Markov Models  Measuring the Fourier Transform Transmission Tomography Emission Tomography  A Unifying Framework        Sequential Optimization  Overview  Examples of SUM  Auxiliary-Function Methods  The SUMMA Class of AF Methods        Barrier-Function and Penalty-Function Methods  Barrier Functions  Examples of Barrier Functions  Penalty Functions  Examples of Penalty Functions Basic Facts         Proximal Minimization  The Basic Problem  Proximal Minimization Algorithms  Some Obstacles  All PMA Are SUMMA  Convergence of the PMA  The Non-Differentiable Case The IPA  Projected Gradient Descent  Relaxed Gradient Descent  Regularized Gradient Descent  The Projected Landweber Algorithm  The Simultaneous MART  A Convergence Theorem  Another Job for the PMA  The Goldstein-Osher Algorithm  A Question         The Forward-Backward Splitting Algorithm Moreau\'s Proximity Operators  The FBS Algorithm  Convergence of the FBS Algorithm  Some Examples Minimizing f2 over a Linear Manifold  Feasible-Point Algorithms        Operators  Overview  Operators  Contraction Operators Convex Sets in RJ  Orthogonal Projection Operators  Firmly Nonexpansive Gradients  Exercises        Averaged and Paracontractive Operators  Averaged Operators  Gradient Operators  Two Useful Identities  The Krasnosel\'skii-Mann-Opial Theorem  Affine Linear Operators  Paracontractive Operators  Exercises         Convex Feasibility and Related Problems  Convex Constraint Sets Using Orthogonal Projections The ART  Regularization  Avoiding the Limit Cycle Exercises         Eigenvalue Bounds  Introduction and Notation  Overview  Cimmino\'s Algorithm  The Landweber Algorithms  Some Upper Bounds for L  Simultaneous Iterative Algorithms  Block-Iterative Algorithms  Exercises        Jacobi and Gauss-Seidel Methods  The Jacobi and Gauss-Seidel Methods: An Example  Splitting Methods  Some Examples of Splitting Methods  Jacobi\'s Algorithm and JOR  The Gauss-Seidel Algorithm and SOR Summary         The SMART and EMML Algorithms  The SMART Iteration  The EMML Iteration  The EMML and the SMART as AM  The SMART as SUMMA  The SMART as PMA  Using KL Projections  The MART and EMART Algorithms  Extensions of MART and EMART  Convergence of the SMART and EMML  Regularization  Modifying the KL Distance  The ABMART Algorithm  The ABEMML Algorithm        Alternating Minimization  Alternating Minimization Exercises        The EM Algorithm  Overview  A Non-Stochastic Formulation of EM  The Stochastic EM Algorithm  The Discrete Case  Missing Data  The Continuous Case  EM and the KL Distance  Finite Mixture Problems         Geometric Programming and the MART  Overview  An Example of a GP Problem  The Generalized AGM Inequality  Posynomials and the GP Problem  The Dual GP Problem  Solving the GP Problem  Solving the DGP Problem Constrained Geometric Programming  Exercises        Variational Inequality Problems and Algorithms  Monotone Functions  The Split-Feasibility Problem  The Variational Inequality Problem  Korpelevich\'s Method for the VIP  On Some Algorithms of Noor  Split Variational Inequality Problems  Saddle Points  Exercises         Set-Valued Functions in Optimization  Overview  Notation and Definitions  Basic Facts  Monotone Set-Valued Functions  Resolvents  Split Monotone Variational Inclusion  Solving the SMVIP  Special Cases of the SMVIP  The Split Common Null-Point Problem  Exercises   Fenchel Duality  The Legendre-Fenchel Transformation  Fenchel\'s Duality Theorem  An Application to Game Theory  Exercises         Compressed Sensing Compressed Sensing  Sparse Solutions  Minimum One-Norm Solutions  Why Sparseness?  Compressed Sampling         Appendix: Bregman-Legendre Functions  Essential Smoothness and Essential Strict Convexity  Bregman Projections onto Closed Convex Sets  Bregman-Legendre Functions         Bibliography     Index