On the real convergence rate of the conjugate gradient method

AbstractWe present a parametrized class of matrices for which the rate of convergence of the conjugate gradient method varies greatly with the parameter and does not appreciably depend on the algorithm implementation. A small change in the eigenvalue distribution can lead to a large change in the sensitivity of CG to rounding errors. A theorem is proved which gives a necessary and sufficient condition for ordering exact arithmetic CG processes for systems with different spectra according to the energy norm of the error. Theorems 4.1 and 4.2 continue Paige's and Greenbaum's work

The Czech Republic, 27. 11. -9

Abstract: Our goal is to show on several examples the great progress made in numerical analysis in the past decades together with the principal problems and relations to other disciplines. We restrict ourselves to numerical linear algebra, or, more specifically, to solving Ax = b where A is a real nonsingular n by n matrix and b a real n−dimensional vector, and to computing eigenvalues of a sparse matrix A. We discuss recent developments in both sparse direct and iterative solvers, as well as fundamental problems in computing eigenvalues. The effects of parallel architectures to the choice of the method and to the implementation of codes are stressed throughout the contribution

A posteriori error estimates including algebraic error: computable upper bounds and stopping criteria for iterative solvers

International audienceFor the finite volume discretization of a second-order elliptic model problem, we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system. We show that the algebraic error can be bounded by constructing an equilibrated Raviart--Thomas--Nédélec discrete vector field whose divergence is given by a proper weighting of the residual vector. Next, claiming that the discretization error and the algebraic one should be in balance, we construct stopping criteria for iterative algebraic solvers. An attention is paid, in particular, to the conjugate gradient method which minimizes the energy norm of the algebraic error. Using this convenient balance, we also prove the efficiency of our a posteriori estimates, i.e., we show that they also represent a lower bound, up to a generic constant, for the overall energy error. A local version of this result is also stated. This makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error. Numerical experiments illustrate the proposed estimates and construction of efficient stopping criteria for algebraic iterative solvers

5th Workshop of the ERCIM Working Group on Matrix Computations and Statistics Numerical Methods for Statistics Aims and Scopes

Similarly to the previous ERCIM WG workshops we plan several plenary lectures and specialize