Fixed-Point Algorithms for Inverse Problems in Science and Engineering

This book presents recent work from top-notch researchers studying projection and other first-order fixed-point algorithms in several areas of mathematics and the applied sciences. It provides a survey of the theory and practice in fixed-point algorithms.

"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.

International group of expert editors and contributors. Presents all new material in the areas of projection and fixed point algorithms for mathematics and the applied sciences. Basis for innovative research from a broad range of topics such as variational analysis, numerical linear algebra, biotechnology, materials science, computational solid state physics, and chemistry. Areas of application include 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). Includes supplementary material: sn.pub/extras

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).