Acta Numerica - Current Issue

Volume 17

Wed, 30 Apr 2008 23:00:00 GMT

Acta Numerica, Volume 17

This annual collection of review articles includes survey papers by leading researchers in numerical analysis and scientific computing. The papers present overviews of recent advances and provide state-of-the-art techniques and analysis. Covering the breadth of numerical analysis, articles are written in a style accessible to researchers at all levels and can serve as advanced teaching aids. Broad subject areas for inclusion are computational methods in linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis and nonlinear dynamical systems, as well as the application of computational techniques in science and engineering and the mathematical theory underlying numerical methods.

Linear algebra algorithms as dynamical systems

Research Articles
Moody T. Chu,
Acta Numerica, Volume 17 , pp 1-86

Abstract
Any logical procedure that is used to reason or to infer either deductively or inductively, so as to draw conclusions or make decisions, can be called, in a broad sense, a realization process. A realization process usually assumes the recursive form that one state develops into another state by following a certain specific rule. Such an action is generally formalized as a dynamical system. In mathematics, especially for existence questions, a realization process often appears in the form of an iterative procedure or a differential equation. For years researchers have taken great effort to describe, analyse, and modify realization processes for various applications.

Accurate and efficient expression evaluation and linear algebra

Research Articles
James Demmel, Ioana Dumitriu, Olga Holtz, Plamen Koev,
Acta Numerica, Volume 17 , pp 87-145

Abstract
We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: most of our results will use the so-called traditional model (TM), where the computed result of op(a, b), a binary operation like a+b, is given by op(a, b) * (1+ | 1. Here is a constant also known as machine epsilon.

Asymptotic and numerical homogenization

Research Articles
B. Engquist, P. E. Souganidis,
Acta Numerica, Volume 17 , pp 147-190

Abstract

Interior-point methods for optimization

Research Articles
Arkadi S. Nemirovski, Michael J. Todd,
Acta Numerica, Volume 17 , pp 191-234

Abstract
This article describes the current state of the art of interior-point methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twenty years.

Greedy approximation

Research Articles
V. N. Temlyakov,
Acta Numerica, Volume 17 , pp 235-409

Abstract
In this survey we discuss properties of specific methods of approximation that belong to a family of greedy approximation methods (greedy algorithms). It is now well understood that we need to study nonlinear sparse representations in order to significantly increase our ability to process (compress, denoise, etc.) large data sets. Sparse representations of a function are not only a powerful analytic tool but they are utilized in many application areas such as image/signal processing and numerical computation. The key to finding sparse representations is the concept of m-term approximation of the target function by the elements of a given system of functions (dictionary). The fundamental question is how to construct good methods (algorithms) of approximation. Recent results have established that greedy-type algorithms are suitable methods of nonlinear approximation in both m-term approximation with regard to bases, and m-term approximation with regard to redundant systems. It turns out that there is one fundamental principle that allows us to build good algorithms, both for arbitrary redundant systems and for very simple well-structured bases, such as the Haar basis. This principle is the use of a greedy step in searching for a new element to be added to a given m-term approximant.