This book is about computational methods based on operator-splitting. The three editors, while all being applied mathematicians whose expertise include splitting methods, work in different application areas including computational mechanics, image processing, wireless communication, nonlinear optics (optical fibers), and so on. Therefore, the book will present very versatile aspects of splitting methods and their applications, motivating the cross-fertilization of ideas.
Introduction.- Some Facts about Operator-Splitting and Alternating Direction Methods.- Operator Splitting.- Convergence Rate Analysis of Several Splitting Schemes.- Self Equivalence of the Alternating Direction Method of Multipliers.- Application of the Strictly Contractive Peaceman-Rachford Splitting Method to Multi-block Separable Convex Programming.- Nonconvex Sparse Regularization and Splitting Algorithms.- ADMM and Non-convex Variational Problems.- Operator Splitting Methods in Compressive Sensing and Sparse Approximation.- First Order Algorithms in Variational Image Processing.- A Parameter Free ADI-like Method for the Numerical Solution of Large Scale Lyapunov Equations.- Splitting Enables Overcoming the Curse of Dimensionality.- ADMM Algorithmic Regularization Paths for Sparse Statistical Machine Learning.- Decentralized Learning for Wireless Communications and Networking.- Splitting Methods for SPDEs: From Robustness to Financial Engineering, Optimal Control and Nonlinear
Filtering.- Application of Operator Splitting Methods in Finance.- A Numerical Method to Solve Multi-marginal Optimal Transport Problems with Coulomb Cost.- Robust Split-step Fourier Methods for Simulating the Propagation of Ultra-short Pulses in Single- and Two-mode Optical Communication Fibers.- Operator Splitting Methods with Error Estimator and Adaptive Time-stepping: Application to the Simulation of Combustion Phenomena.- Operator Splitting Algorithms for Free Surface Flows: Application to Extrusion Processes.- An Operator Splitting Approach to the Solution of Fluid-structure Interaction Problems with Hemodynamics.- On Circular cluster Formation in a Rotating Suspension of Non-Brownian Settling Particles in a Fully Filled Circular Cylinder: An Operator Splitting Approach to the Numerical Simulation.
Introduction.- Some Facts about Operator-Splitting and Alternating
Direction Methods.- Operator Splitting.- Convergence Rate Analysis of Several
Splitting Schemes.- Self Equivalence of the Alternating Direction Method of
Multipliers.- Application of the Strictly Contractive Peaceman-Rachford
Splitting Method to Multi-block Separable Convex Programming.- Nonconvex
Sparse Regularization and Splitting Algorithms.- ADMM and Non-convex
Variational Problems.- Operator Splitting Methods in Compressive Sensing and
Sparse Approximation.- First Order Algorithms in Variational Image
Processing.- A Parameter Free ADI-like Method for the Numerical Solution of
Large Scale Lyapunov Equations.- Splitting Enables Overcoming the Curse of
Dimensionality.- ADMM Algorithmic Regularization Paths for Sparse Statistical
Machine Learning.- Decentralized Learning for Wireless Communications and
Networking.- Splitting Methods for SPDEs: From Robustness to Financial
Engineering, Optimal Control and Nonlinear Filtering.- Application of
Operator Splitting Methods in Finance.- A Numerical Method to Solve
Multi-marginal Optimal Transport Problems with Coulomb Cost.- Robust
Split-step Fourier Methods for Simulating the Propagation of Ultra-short
Pulses in Single- and Two-mode Optical Communication Fibers.- Operator
Splitting Methods with Error Estimator and Adaptive Time-stepping:
Application to the Simulation of Combustion Phenomena.- Operator Splitting
Algorithms for Free Surface Flows: Application to Extrusion Processes.- An
Operator Splitting Approach to the Solution of Fluid-structure Interaction
Problems with Hemodynamics.- On Circular cluster Formation in a Rotating
Suspension of Non-Brownian Settling Particles in a Fully Filled Circular
Cylinder: An Operator Splitting Approach to the Numerical Simulation.
Roland Glowinski works in Computational Mechanics and Physics and more generally in areas involving the numerical solution of partial differential equations and inequalities. Stanley Osher is a Professor of Mathematics, Computer Science and Electrical Engineering at UCLA, and is an Associate Director of the NSF funded Institute for Pure and Applied Mathematics. Wotao Yin works in optimization theory, develops many fast algorithms for compressive sensing, image processing, medical imaging, wireless networking, etc.
