Fixed-Point Algorithms for Inverse Problems in Science and Engineering

by Bauschke, Heinz H.; Burachik, Regina S.; Combettes, Patrick L.; Elser, Veit; Luke, D. Russell

• ISBN13: 9781441995681
• ISBN10: 1441995684
• Format: Hardcover
• Copyright: 2011-06-01
• Publisher: Springer Verlag

Summary

"Fixed-Point Algorithms for Inverse Problems in Science and Engineering" presents some of the most recent work from top-notch researchers studying projection and other first-order fixed-point algorithms in several areas of mathematics and the applied sciences. The material presented provides a survey of the state-of-the-art theory and practice in fixed-point algorithms, identifying emerging problems driven by applications, and discussing new approaches for solving these problems. This book incorporates diverse perspectives from broad-ranging areas of research including, variational analysis, numerical linear algebra, biotechnology, materials science, computational solid-state physics, and chemistry. Topics presented include: Theory of Fixed-point algorithms: convex analysis, convex optimization, subdifferential calculus, nonsmooth analysis, proximal point methods, projection methods, resolvent and related fixed-point theoretic methods, and monotone operator theory. Numerical analysis of fixed-point algorithms: choice of step lengths, of weights, of blocks for block-iterative and parallel methods, and of relaxation parameters; regularization of ill-posed problems; numerical comparison of various methods. Areas of Applications: engineering (image and signal reconstruction and decompression problems), computer tomography and radiation treatment planning (convex feasibility problems), astronomy (adaptive optics), crystallography (molecular structure reconstruction), computational chemistry (molecular structure simulation) and other areas. Because of the variety of applications presented, this book can easily serve as a basis for new and innovated research and collaboration.

Table of Contents

Chebyshev Sets, Klee Sets, and Chebyshev Centers with Respect to Bregman Distances: Recent Results and Open Problems	p. 1
Self-Dual Smooth Approximations of Convex Functions via the Proximal Average	p. 23
A Linearly Convergent Algorithm for Solving a Class of Nonconvex/Affine Feasibility Problems	p. 33
The Newton Bracketing Method for Convex Minimization: Convergence Analysis	p. 49
Entropic Regularization of the l0 Function	p. 65
The Douglas-Rachford Algorithm in the Absence of Convexity	p. 93
A Comparison of Some Recent Regularity Conditions for Fenchel Duality	p. 111
Non-Local Functionals for Imaging	p. 131
Opial-Type Theorems and the Common Fixed Point Problem	p. 155
Proximal Splitting Methods in Signal Processing	p. 185
Arbitrarily Slow Convergence of Sequences of Linear Operators: A Survey	p. 213
Graph-Matrix Calculus for Computational Convex Analysis	p. 243
Identifying Active Manifolds in Regularization Problems	p. 261
Approximation Methods for Nonexpansive Type Mappings in Hadamard Manifolds	p. 273
Existence and Approximation of Fixed Points of Bregman Firmly Nonexpansive Mappings in Reflexive Banach Spaces	p. 301
Regularization Procedures for Monotone Operators: Recent Advances	p. 317
Minimizing the Moreau Envelope of Nonsmooth Convex Functions over the Fixed Point Set of Certain Quasi-Nonexpansive Mappings	p. 345
The Brézis-Browder Theorem Revisited and Properties of Fitzpatrick Functions of Order n	p. 391
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