analytical bounds for block approximate factorization methods abstract we derive simple analytical upper bounds on the spectral condition number associated with the preconditioning of large sparse linear systems by the so called modified block incomplete factorizations the results prove to handle a large class of discrete second order elliptic partial differential equations and support the conclusion that sparse approximate block matrix factorizations provide powerful preconditioners

a non oscillation theorem for the emden fowler equation ground states for semilinear elliptic equations with critical exponents abstract let be a ball centered at the origin in r n n 2 its boundary and 0 0 a given function this article is concerned with radial solutions of the boundary value problem u x u p 0 u 0 in u 0 on 1 radial solutions of 1 are functions u of the variable r x that satisfy the ordinary differential equation u n 1 r u r u p 0 2 it is shown that if p p where p is the critical sobolev exponent p n 2 n 2 then 2 is non oscillatory at the origin if there exists a 0 such that the function r r log 1 r is non decreasing near the origin the result is a corollary of a similar result for the emden fowler equation x t 2 1 g t x 1 2 0 3 where g is a given positive valued function and a positive constant the relation between the solutions of 2 and 3 is established via the transformations x t u r g t r where t n 2 r n 2 it is shown that 3 is nonoscillatory at infinity if there exists a 0 such that the function t g t log t is non increasing for all sufficiently large t the proof of this result constitutes the core of the article the result is of independent interest in the theory of the emden fowler equation and improves earlier oscillation results of nehari and chiou

some results on sparse block factorization iterative methods abstract we consider the general sparse block factorization iterative methods as beauwens and ben bouzid did we develop some existence and convergence theorems for general nonsingular h matrices

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

block ilu factorization preconditioners for a block tridiagonal h matrix abstract we propose new parallelizable block ilu incomplete lu factorization preconditioners for a nonsymmetric block tridiagonal h matrix theoretical properties of these block ilu preconditioners are compared with those of block ilu preconditioners for the corresponding comparison matrix numerical results of the bicgstab using these block preconditioners are compared with those of the bicgstab using a standard ilu factorization preconditioner to see the effectiveness of the block ilu preconditioners lastly parallel computations of the block ilu preconditioners are carried out and evaluated on the cray c90

incomplete blockwise factorizations of block h matrices the use of incomplete blockwise factorizations as preconditioners in conjugate gradient like methods has become more and more popular in recent years most of the theory concerning existence and applicability of these factorizations has been limited to m matrices so far here we introduce a more general definition of block h matrices robert 8 and we extend the theory to this class of matrices

a general incomplete block matrix factorization method abstract a general method for incomplete factorization of m matrices in block matrix form is presented the method generalizes previous methods for pointwise incomplete factorization methods of varga 1960 of wo nicki 1973 and of meijerink and van der vorst 1977 as well as a method for tridiagonal block matrices also we present a new and more general result on the existence of so called modified incomplete factorizations which satisfy a generalized rowsum criterion

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

self similarity in a cut and paste model of coarsening we study certain redistribution processes for a resource distributed among a large number of resource holders clusters in the processes we consider the smallest clusters are cut up into pieces according to a given statistical law and the pieces are randomly redistributed among the remaining clusters we derive an evolution equation for the cluster size distribution show that self similar solutions exist and characterize their structure in a limiting case when the pieces are small we show that in general solutions approach a singular self similar form

on some versions of incomplete block matrix factorization iterative methods abstract various forms of preconditioning matrices for iterative acceleration methods are discussed the preconditioning is based on two versions of incomplete block matrix factorization

semilinear elliptic equations with near critical growth rates on etudie les solutions symetriques de l equation elliptique semi lineaire u u u u p 1 0 u b 0 ou b est la boule unite dans r 3 on prend p proche de 5 on obtient une description asymptotique des solutions avec une grande norme

on the conditioning analysis of block approximate factorization methods abstract the paper is devoted to the conditioning analysis of modified block incomplete factorizations of a given stieltjes matrix we obtain new results improve other theories and compare all existing upper bounds through numerical experiments applied to discrete elliptic pdes our results show that an o h 1 spectral bound can be achieved for a large class of problems

comparison theorems for radial solutions of semilinear elliptic equations on obtient des resultats sur les solutions radiales des equations elliptiques de la forme u f u 0

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

on the multi grid technique for solving three dimensional boundary value engineering problems the paper proposes an iterative technique with a similar multi grid operator for solving complex spatial bounctary value problems the description of the method and efficient vectorial algorithms based on a simple ctata structure applicable to solving three dimensional engineering problems are presented the discussion of the method s convergence includes theoretical grounds and practical aspects we give a general proof of convergence and consider the model cases to estimate the rate of convergence we also address the histories of convergence derived from the numerical tests and consider two examples of practical problems

incomplete block matrix factorization preconditioning methods the ultimate answer abstract a considerable interest has been devoted to block matrix incomplete factorization preconditioning methods in recent time see 12 2 3 4 6 7 8 10 and 21 the purpose of this paper is to survey some of these results and to present new results showing how much more efficient and robust such methods can be as compared to other admittingly also efficient methods such as pointwise incomplete factorization preconditioning methods and multilevel methods with hierarchical basis functions both on scalar as well as on parallel or vectorizable computers

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

conditioning analysis of modified block incomplete factorizations abstract we improve the conditioning analysis of modified block incomplete factorizations of stieltjes matrices letting n denote the number of diagonal blocks our results show that the spectral condition number is bounded by n for a large class of two dimensional pdes

semilinear elliptic equations and supercritical growth where p p the aim of the present paper is to understand for the case p pc the behaviour of solutions to 1 i with large supremum norm to this end 0 is restricted to be the unit ball f3c r3 as in this case the behaviour of solutions to 1 1 can be described in some detail according to the results proved in gidas ni and nirenberg s all positive solutions of 1 1 must be radial when q b with the scaling iid ilip in the case n 3 problem 1 1 reduces to the following ordinary differential equation problem

a comparison of direct and preconditioned iterative techniques for sparse unsymmetric systems of linear equations in this paper we compare direct and preconditioned iterative methods for the solution of nonsymmetric sparse systems of linear algebraic equations these problems occur in finite difference and finite element simulations of semiconductor devices and fluid flow problems we consider five iterative methods that appear to be the most promising for this class of problems the biconjugate gradient method the conjugate gradient squared method the generalized minimal residual method the generalized conjugate residual method and the method of orthogonal minimization each of these methods was tested using similar preconditioning incomplete lu factorization on a set of large sparse matrices arising from finite element simulation of semiconductor devices results are shown where we compare the computation time and memory requirements for each of these methods against one another as well as against a direct method that uses lu factorization to solve these problems the results of our numerical experiments show that preconditioned iterative methods are a practical alternative to direct methods in the solution of large sparse systems of equations and can offer significant savings in storage and cpu time

a convergence analysis of block accelerated over relaxation iterative methods for weak block h matrices to partition π abstract the aim of this paper is to establish the convergence of the block iteration methods such as the block successively accelerated over relaxation method baor and the symmetric block successively accelerated over relaxation method bsaor let a c n m m be a weak block h matrix to partition then for 0 r 2 1 b j a b l r 1 b j a b s r 1 b j a 2 and exact convergence and divergence domains for the block sor and block ssor iterative methods are obtained as it has been obtained to h matrices based on these results the main results in bai parallel computing 25 1999 and cvetkovic appl numer math 41 2002 can be improved

analysis of incomplete matrix factorizations as multigrid smoothers for vector and parallel computers the cost of smoothing is usually a major expense in multigrid codes efficient vectorizable and parallelizable versions of incomplete blockmatrix factorization methods used as smoothers for multigrid methods are discussed in this paper the methods are particularly interesting for computers with parallel processors with pipelines because both multitasking with little overhead and vectorization can be achieved

lower eigenvalue bounds for pencils of matrices abstract 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

vectorizable preconditioners for elliptic difference equations in three space dimensions abstract until now research in vectorizable preconditioners has mainly focused on problems in two space dimensions employing all degrees of freedom we here present preconditioners for 3d matrices based on point line and plane factorizations both in vectorizable and for reasons of comparisons in recursive versions by a limit analysis and accompanying numerical tests of elliptic difference equations with anisotropy we find that methods based on line block factorizations are to be preferred in general among the vectorizable methods a line method where euler expansion of the factors of the pivot blocks is used is the most robust with respect to the problem parameters furthermore this method turns out to approach the best of the recursive methods in efficiency in terms of operation counts it has the added bonus of almost maximal vectorlength