Analysis of Function Component Complexity for Hypercube Homotopy Algorithms

Probability-one homotopy algorithms are a class of methods for solving nonlinear systems of equations that globally convergent from an arbitrary starting point with probability one. The essence of these homotopy algorithms is the construction of a homotopy map p-sub a and the subsequent tracking of a smooth curve y in the zero set p-sub a to the -1 (0) of p-sub a. Tracking the zero curve y requires repeated evaluation of the map p-sub a, its n x (v + 1) Jacobian matrix Dp-sub a and numerical linear algebra for calculating the kernel of Dp-sub a. This paper analyzes parallel homotopy algorithms on a hypercube, considering the numerical algebra, several communications topologies and problem decomposition strategies, functions component complexity, problem size, and the effect of different component complexity distributions. These parameters interact in complicated ways, but some general principles can be inferred based on empirical results

Numerical Analysis

Acknowledgements: This article will appear in the forthcoming Princeton Companion to Mathematics, edited by Timothy Gowers with June Barrow-Green, to be published by Princeton University Press.\ud \ud In preparing this essay I have benefitted from the advice of many colleagues who corrected a number of errors of fact and emphasis. I have not always followed their advice, however, preferring as one friend put it, to "put my head above the parapet". So I must take full responsibility for errors and omissions here.\ud \ud With thanks to: Aurelio Arranz, Alexander Barnett, Carl de Boor, David Bindel, Jean-Marc Blanc, Mike Bochev, Folkmar Bornemann, Richard Brent, Martin Campbell-Kelly, Sam Clark, Tim Davis, Iain Duff, Stan Eisenstat, Don Estep, Janice Giudice, Gene Golub, Nick Gould, Tim Gowers, Anne Greenbaum, Leslie Greengard, Martin Gutknecht, Raphael Hauser, Des Higham, Nick Higham, Ilse Ipsen, Arieh Iserles, David Kincaid, Louis Komzsik, David Knezevic, Dirk Laurie, Randy LeVeque, Bill Morton, John C Nash, Michael Overton, Yoshio Oyanagi, Beresford Parlett, Linda Petzold, Bill Phillips, Mike Powell, Alex Prideaux, Siegfried Rump, Thomas Schmelzer, Thomas Sonar, Hans Stetter, Gil Strang, Endre Süli, Defeng Sun, Mike Sussman, Daniel Szyld, Garry Tee, Dmitry Vasilyev, Andy Wathen, Margaret Wright and Steve Wright

A literature survey of low-rank tensor approximation techniques

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors