On OBV Methods

COO-2280-3879-10-NAM-01EY-76-S-02-2880info:eu-repo/semantics/publishe

Forgivable variational crimes

It is common use to call variatonal crimes those applications of the variational method or more generally, of the Galerkin method, where not all the assumptions needed to validate the method are exactly satisfied. This covers a.o. the use of trial functions for second order elliptic problems that are only approximately continuous along element boundaries or the use of quadrature formulas to only approximately compute the entries of the stiffness matrix or the replacement of the exact domain boundary by an approximate one, to cite the main examples. Classical techniques based on bounding the so-called consistency error terms have been used to analyse and most often absolve such crimes. In the present contribution, we propose an alternate approach for a restricted class of variational crimes such that there is no approximation on the representation of the RHS of the exact equation and such that a generalized variational principle can be introduced in such way that both the exact and the approximate problems appear as Galerkin approximations of this generalized problem. This reduces their analysis to successive applications of the variational method itself and produces essentially the same error bounds as perfectly legal applications of the variational method, whence our suggestion to consider such crimes as forgivable. Examples of applications including and generalizing the PCD method presented elsewhere in this conference are considered by way of illustration. © Springer-Verlag Berlin Heidelberg 2003.SCOPUS: ar.kinfo:eu-repo/semantics/publishe

Implementation strategies for block recursive factorizations

Recent advances have shown that block recursive approximate factorizations provide among the best preconditioners for solving elasticity equations on large 3D meshes such as arising in geomechanical applications. However, to get full efficiency of these methods, a wide range of implementation strategies need be put into work. In this contribution, we attempt to survey these strategies and explain their purpose. This includes level orderings and fill-in strategies, diagonal and offdiagonal relaxation, graph perturbations, reduction techniques, W-cycles and smoothing steps. © 1999 IMACS/Elsevier Science B.V. All rights reserved.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Approximate factorizations with S/P consistently ordered M-factors

The behaviour of PCG methods for solving a finite difference or finite element positive definite linear system Ax=b with a (pre)conditioning matrix B=UTP-1U (where U is upper triangular and P=diag(U)) obtained from a modified incomplete factorization, is unpredictable in the present status of knowledge whenever the upper triangular factor is not strictly diagonally dominant and 2 P -D, where D=diag(A), is not symmetric positive definite. The origin of this rather surprising shortcoming of the theory is that all upper bounds on the associated spectral condition number κ(B-1A) obtained so far require either the strict diagonal dominance of the upper triangular factor or the strict positive definiteness of 2 P -D. It is our purpose here to improve the theory in this respect by showing that, when the triangular factors are "S/P consistently ordered"M-matrices, nonstrict diagonal dominance is generally a sufficient requirement, without additional condition on 2 P -D. As a consequence, the new analysis does not require diagonal perturbations (otherwise needed to keep control of the diagonal dominance of U or of the positive definiteness of 2 P -D). Further, the bounds obtained here on κ(B-1A) are independent of the lower spectral bound of D-1A meaning that quasi-singular problems can be solved at the same speed as regular ones, an unexpected result. © 1989 BIT Foundations.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Iterative solution methods

This presentation is intended to review the state-of-the-art of iterative methods for solving large sparse linear systems such as arising in finite difference and finite element approximations of boundary value problems. However, in order to keep this review within reasonable bounds, we only review those methods for which an algebraic analysis has been achieved. We first review the basic principles and components of iterative solution methods and describe in more detail the main devices used to design preconditioners, showing how the present day complex preconditioners are built through additive and/or multiplicative composition of simpler ones. We also note that acceleration methods may sometimes be viewed, and thus used, as preconditioners. Next, using approximate factorizations as basic framework, we show how their development led to the study of so-called modified methods and why attention then shifted to specific orderings, of multilevel type. Finally we show how the successful development of multigrid and hierarchical basis methods prompted the introduction of equivalent algebraic techniques: besides recursive orderings, an additional step called stabilization by polynomial preconditioning that plays here the role of the W-cycles of the multigrid method and an algebraic version of V-cycles with smoothing. © 2004 IMACS. Published by Elsevier B.V. All rights reserved.SCOPUS: cp.jinfo:eu-repo/semantics/publishe

Etude du transport des neutrons en milieu hétérogène à une vitesse par la méthode des sources superficielles

Doctorat en sciences appliquéesinfo:eu-repo/semantics/nonPublishe

On a characterization theorem for M-operators

Published in Warsaw, 1977INR 1693/CYFRONET/PM/Ainfo:eu-repo/semantics/publishe

Upper eigenvalue bounds for pencils of matrices

Eigenvalue bounds are obtained for pencils of matrices A - vB where A is a Stieltjes matrix and B is positive definite, under assumptions suitable for the estimation of asymptotic convergence rates of factorization iterative methods, where B represents the approximate factorization of A. The upper bounds obtained depend on the "connectivity" structure of the matrices involved, which enters through matrix graph considerations; in addition, a more classical argument is used to obtain a lower bound. Potential applications of these results include a partial confirmation of Gustafsson's conjecture concerning the nonnecessity of Axelsson's perturbations. © 1984.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Semistrict diagonal dominance

The notion of semistrict diagonal dominance is introduced and shown to provide a means for separating the properties of diagonally dominant matrices which depend on irreducibility from those which do not. Refinements of well-known monotonicity criteria are obtained as applications.Copyright © 1976 Society for Industrial and Applied Mathematicsinfo:eu-repo/semantics/publishe

Conditioning analysis of positive definite matrices by approximate factorizations

The conditioning analysis of positive definite matrices by approximate LU factorizations is usually reduced to that of Stieltjes matrices (or even to more specific classes of matrices) by means of perturbation arguments like spectral equivalence. We show in the present work that a wider class, which we call "almost Stieltjes" matrices, can be used as reference class and that it has decisive advantages for the conditioning analysis of finite element approximations of large multidimensional steady-state diffusion problems. © 1989.SCOPUS: ar.jinfo:eu-repo/semantics/publishe