Fixed-Point Algorithms for Inverse Problems in Science and Engineering presents some of the most recent work from leading researchers in variational and numerical analysis. The contributions in this collection provide state-of-the-art theory and practice in first-order fixed-point algorithms, identify emerging problems driven by applications, and discuss new approaches for solving these problems.

This book is a compendium of topics explored at the Banff International Research Station "Interdisciplinary Workshop on Fixed-Point Algorithms for Inverse Problems in Science and Engineering" in November of 2009. The workshop included a broad range of research including variational analysis, numerical linear algebra, biotechnology, materials science, computational solid-state physics, and chemistry.

Key topics and features of this book include:

· Theory of Fixed-point algorithms: variational analysis, convex analysis, convex and nonconvex 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

· Applications: Image and signal processing, antenna optimization, location problems

The wide scope of applications presented in this volume easily serve as a basis for new and innovative research and collaboration.

Zusammenfassung

"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.

Inhalt

-1. Chebyshev Sets, Klee Sets, and Chebyshev Centers with respect to Bregman Distances: Recent Results and Open Problems (H. Bauschke, M. Macklem, S.X. Wang). -2. Self-dual Smooth Approximations of Convex Functions via the Proximal Average (H. Bauschke, S. Moffat, S.X. Wang). -3. A Linearly Convergent Algorithm for Solving a Class of Nonconvex/Affine Feasibility Problems (A. Beck, M. Teboulle). -4. The Newton Bracketing Method for Convex Minimization: Convergence Analysis (A. Ben-Israel, Y. Levin). -5. Entropic regularization of the 0 function (J. Borwein, D. Luke). -6. The Douglas-Rachford algorithm in the absence of convexity (J. Borwein, B. Sims). -7. A comparison of some recent regularity conditions for Fenchel duality (R. Bo, E. Czetnek). -8. Non-Local Functionals for Imaging (J. Boulanger, P. Elbau, C. Pontow, O. Scherzer). -9. Opial-Type Theorems and the Common Fixed Point Problem (A. Cegielski, Y. Censor). -10. Proximal Splitting Methods in Signal Processing (P. Combettes, J. Pesquet). -11. Arbitrarily Slow Convergence of Sequences of Linear Operators: A Survey (F. Deutsch, H. Hundal). -12. Graph-Matrix Calculus for Computational Convex Analysis (B. Gardiner, Y. Lucet). -13. Identifying Active Manifolds in Regularization Problems (W. Hare). -14. Approximation methods for nonexpansive type mappings in Hadamard manifolds (G. López, V. Martín-Márquez). -15. Existence and Approximation of Fixed Points of Bregman Firmly Nonexpansive Mappings in Reflexive Banach Spaces (S. Reich, S. Sabach). -16. Regularization procedure for monotone operators: recent advances (J. Revalski). -17. Minimizing the Moreau Envelope of Nonsmooth Convex Functions over the Fixed Point Set of Certain Quasi-Nonexpansive Mappings (I. Yamada, M. Yukawa, M. Yamagishi). -18. The Brézis-Browder Theorem revisted and properties of Fitzpatrick functions of order n (L. Yao).