Iterative Optimization in Inverse Problems

ISBN: 9781482222333 出版年：2014 页码：298 Byrne, Charles CRC Press

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