Local Linear Convergence Analysis of Primal-Dual Splitting Methods

In this paper, we study the local linear convergence properties of a versatile class of Primal-Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework we first show that (i) the sequences generated by Primal-Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal-Dual splitting can be specialized to cover existing ones on Forward-Backward splitting and Douglas-Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning, etc. The demonstration not only verifies the local linear convergence behaviour of Primal-Dual splitting methods, but also the insights on how to accelerate them in practice

On I-acceleration convergence of sequences of fuzzy real numbers

In this article we introduce the notion of ideal acceleration convergence of sequences of fuzzy real numbers. We have proved a decomposition theorem for ideal acceleration convergence of sequences as well as for subsequence transformations and studied different types of acceleration convergence of fuzzy real valued sequence

On Vector Sequence Transforms and Acceleration Techniques

This dissertation is devoted to the acceleration of convergence of vector sequences. This means to produce a replacement sequence from the original sequence with higher rate of convergence. It is assumed that the sequence is generated from a linear matrix iteration xi+ i = Gxi + k where G is an n x n square matrix and xI+1 , xi,and k are n x 1 vectors. Acceleration of convergence is obtained when we are able to resolve approximations to low dimension invariant subspaces of G which contain large components of the error. When this occurs, simple weighted averages of iterates x,+|, i = 1 ,2 ,... k where k \u3c n are used to produce iterates which contain approximately no error in the selfsame low dimension invariant subspaces. We begin with simple techniques based upon the resolution of a simple dominant eigenvalue/eigenvector pair and extend the notion to higher dimensional invariant spaces. Discussion is given to using various subspace iteration methods and their convergence. These ideas are again generalized by solving the eigenelement for a projection of G onto an appropriate subspace. The use of Lanzcos-type methods are discussed for establishing these projections. We produce acceleration techniques based on the process of generalized inversion. The relationship between the minimal polynomial extrapolation technique (MPE) for acceleration of convergence and conjugate gradient type methods is explored. Further acceleration techniques are formed from conjugate gradient type techniques and a generalized inverse Newton\u27s method. An exposition is given to accelerations based upon generalizations of rational interpolation and Pade approximation. Further acceleration techniques using Sherman-Woodbury-Morrison type formulas are formulated and suggested as a replacement for the E-transform. We contrast the effect of several extrapolation techniques drawn from the dissertation on a nonsymmetric linear iteration. We pick the Minimal Polynomial Extrapolation (MPE) as a representative of techniques based on orthogonal residuals, the Vector $\epsilon$-Algorithm (VEA) as a representative vector interpolation technique and a technique formulated in this dissertation based on solving a projected eigenproblem. The results show the projected eigenproblem technique to be superior for certain iterations

Abstract Fixpoint Computations with Numerical Acceleration Methods

Static analysis by abstract interpretation aims at automatically proving properties of computer programs. To do this, an over-approximation of program semantics, defined as the least fixpoint of a system of semantic equations, must be computed. To enforce the convergence of this computation, widening operator is used but it may lead to coarse results. We propose a new method to accelerate the computation of this fixpoint by using standard techniques of numerical analysis. Our goal is to automatically and dynamically adapt the widening operator in order to maintain precision

Scalar Levin-Type Sequence Transformations

Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. Transformations that depend not only on the sequence elements or partial sums $s_n$ but also on an auxiliary sequence of so-called remainder estimates $\omega_n$ are of Levin-type if they are linear in the $s_n$, and nonlinear in the $\omega_n$. Known Levin-type sequence transformations are reviewed and put into a common theoretical framework. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. As illustration, two new sequence transformations are derived. Common properties and results on convergence acceleration and stability are given. For important special cases, extensions of the general results are presented. Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given.Comment: 59 pages, LaTeX, invited review for J. Comput. Applied Math., abstract shortene

Implementation of the Combined--Nonlinear Condensation Transformation

We discuss several applications of the recently proposed combined nonlinear-condensation transformation (CNCT) for the evaluation of slowly convergent, nonalternating series. These include certain statistical distributions which are of importance in linguistics, statistical-mechanics theory, and biophysics (statistical analysis of DNA sequences). We also discuss applications of the transformation in experimental mathematics, and we briefly expand on further applications in theoretical physics. Finally, we discuss a related Mathematica program for the computation of Lerch's transcendent.Comment: 23 pages, 1 table, 1 figure (Comput. Phys. Commun., in press