a generalization of the optimal diagonal approximate inverse preconditioner the classical optimal in the frobenius sense diagonal preconditioner for large sparse linear systems ax b is generalized and improved the new proposed approximate inverse preconditioner n is based on the minimization of the frobenius norm of the residual matrix am i where m runs over a certain linear subspace of nxn real matrices defined by a prescribed sparsity pattern the number of nonzero entries of the nxn preconditioning matrix n is less than or equal to 2n and n of them are selected as the optimal positions in each of the n columns of matrix n all theoretical results are justified in detail in particular the comparison between the proposed preconditioner n and the optimal diagonal one is theoretically analyzed finally numerical experiments reported confirm the theory and illustrate that our generalization of the optimal diagonal preconditioner improves in general its efficiency when they do not coincide

experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

generalized weighted hardy cesaro operators and their commutators on weighted morrey spaces abstract let 0 1 0 s 0 1 r be measurable functions and be a parameter curve in r n given by t x 0 1 r n s t x s t x in this paper we study a new weighted hardy cesaro operator defined by u s f x 0 1 f s t x t d t for measurable complex valued functions f on r n under certain conditions on s t and on an absolutely homogeneous weight function we characterize the weight function such that u s is bounded on weighted morrey spaces l p and then compute the corresponding operator norm of u s we also give a necessary and sufficient condition on the function which ensures the boundedness of the commutator of the operator u s on l p with symbols in bmo

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

compressed graphs and the minimum degree algorithm the minimum degree algorithm is well known and widely used to find a permutation matrix p such that a sparse symmetric positive definite matrix a can be factored as pap t ll t to yield a lower triangular matrix l with relatively few nonzero entries in this paper we show that by compressing the graph of a when possible the run time of the minimum degree algorithm can be significantly reduced some empirical results for a collection of test matrices are presented

high performance pcg solvers for fem structural analysis the preconditioned conjugate gradient algorithm is a well known and powerful method used to solve large sparse symmetric positive definite linear systems such system are generated by the finite element discretisation in structural analysis but users of finite elements in this contest generally still rely on direct methods it is our purpose in the present work to highlight the improvement brought forward by some new preconditioning techniques and show that the preconditioned conjugate gradient method performs better than efficient direct methods

lp and bmo bounds of weighted hardy littlewood averages abstract this paper characterizes those nonnegative functions defined on 0 1 for which the weighted hardy littlewood average u f 0 1 f t t dt is bounded on either l p r n p 1 or bmo r n meanwhile the corresponding operator norms are worked out

on a class of preconditioning methods for dense linear systems from boundary elements in this paper we discuss several sparse matrix based preconditioners suitable for preconditioning dense linear systems of boundary element equations all preconditioners involve only o n nonzeros we provide a framework for constructing operator splitting based preconditioners and use it to analyze a class of sparse preconditioners for singular integral equations a more efficient preconditioner is proposed that has a band 2 structure

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

the norm inequalities for the weighted cesaro mean operators this paper gives some necessary and sufficient conditions for the weighted cesaro mean operators to be bounded on herz spaces the corresponding new operator norm inequalities are obtained

incomplete cholesky factorizations with limited memory we propose an incomplete cholesky factorization for the solution of large scale trust region subproblems and positive definite systems of linear equations this factorization depends on a parameter p that specifies the amount of additional memory in multiples of n the dimension of the problem that is available there is no need to specify a drop tolerance our numerical results show that the number of conjugate gradient iterations and the computing time are reduced dramatically for small values of p we also show that in contrast with drop tolerance strategies the new approach is more stable in terms of number of iterations and memory requirements

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

a generalization of the moore penrose inverse related to matrix subspaces of c n m a natural generalization of the classical moore penrose inverse is presented the so called s moore penrose inverse of an mxn complex matrix a denoted by a s is defined for any linear subspace s of the matrix vector space c n x m the s moore penrose inverse a s is characterized using either the singular value decomposition or for the full rank case the orthogonal complements with respect to the frobenius inner product these results are applied to the preconditioning of linear systems based on frobenius norm minimization and to the linearly constrained linear least squares problem

regular incomplete factorizations of real positive definite matrices abstract we study the classes of matrices which admit a regular incomplete factorization with respect to any graph set we extend the construction to the class of real positive definite matrices the convergence of the basic iteration associated with the splittings we have is discussed

bifurcation from stability to instability for a free boundary problem modeling tumor growth by stokes equation we consider a free boundary problem modeling tumor growth in fluid like tissue the model equations include a diffusion equation for the nutrient concentration and the stokes equation with a source which represents the proliferation of tumor cells the proliferation rate and the cell to cell adhesiveness which keeps the tumor intact are two parameters which characterize the aggressiveness of the tumor for any positive radius r there exists a unique radially symmetric stationary solution with radius r r for a sequence mn r there exist symmetry breaking bifurcation branches of solutions with free boundary r r eyn 0 o e 2 n even 2 for small e w hereyn 0 is the spherical harmonic of mode n 0 furthermore the smallest mn r s aymn r is such that n n r as r in this paper we prove that the radially symmetric stationary solution with r rs is linearly stable if n rs w heren rs mn rs and we prove that strict inequality holds if is small or if is large the biological implications of these results are discussed at the end of the paper

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

continuity of hardy type operators on rectangularly defined spaces abstract we introduce a generalization of rectangular herz spaces defined in 3 and prove boundedness of a version of the n dimensional hardy operator on them we also study the action of the hausdorff operator on some subspaces of bmo

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

robust approximate inverse preconditioning for the conjugate gradient method we present a variant of the ainv factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix the new preconditioner is breakdown free and when used in conjunction with the conjugate gradient method results in a reliable solver for highly ill conditioned linear systems we also investigate an alternative approach to a stable approximate inverse algorithm based on the idea of diagonally compensated reduction of matrix entries the results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

orderings for factorized sparse approximate inverse preconditioners the influence of reorderings on the performance of factorized sparse approximate inverse preconditioners is considered some theoretical results on the effect of orderings on the fill in and decay behavior of the inverse factors of a sparse matrix are presented it is shown experimentally that certain reorderings like minimum degree and nested dissection can be very beneficial the benefit consists of a reduction in the storage and time required for constructing the preconditioner and of faster convergence of the preconditioned iteration in many cases of practical interest

finding exact and approximate block structures for ilu preconditioning sparse matrices which arise in many applications often possess a block structure that can be exploited in iterative and direct solution methods these block matrices have as their entries small dense blocks with constant or variable dimensions block versions of incomplete lu factorizations which have been developed to take advantage of such structures give rise to a class of preconditioners that are among the most effective available this paper presents general techniques for automatically determining block structures in sparse matrices a standard graph compression algorithm used in direct sparse matrix methods is considered along with two other algorithms which are also capable of unraveling approximate block structures

multiresolution approximate inverse preconditioners we introduce a new preconditioner for elliptic pdes on unstructured meshes using a wavelet inspired basis we compress the inverse of the matrix allowing an effective sparse approximate inverse by solving the sparsity vs accuracy conflict the key issue in this compression is to use second generation wavelets which can be adapted to the unstructured mesh the true boundary conditions and even the pde coefficients we also show how this gives a new perspective on multiresolution algorithms such as multigrid interpreting the new preconditioner as a variation on node nested multigrid in particular we hope the new preconditioner will combine the best of both worlds fast convergence when multilevel methods can succeed but with robust performance for more difficult problems the rest of the paper discusses the core issues for the preconditioner ordering and construction of a factored approximate inverse in the multiresolution basis robust interpolation on unstructured meshes automatic mesh coarsening and purely algebraic alternatives some exploratory numerical experiments suggest the superiority of the new basis over the standard basis for several tough problems including discontinuous anisotropic coefficients strong convection and indefinite reaction problems on unstructured meshes with scalability like hierarchical basis methods achieved

some iterative methods related to frobenius norm minimization abstract a modification of the work in 1 is established in a way that allows to suppress the assumption of matrix symmetry similar results to the ones obtained in 1 for the spectral radius of the iteration matrix i m 1a associated to a splitting of the matrix a are obtained numerical examples show that the computational cost is equivalent besides our method requires to solve only diagonal linear systems

ilum a multi elimination ilu preconditioner for general sparse matrices standard preconditioning techniques based on incomplete lu ilu factorizations offer a limited degree of parallelism in general a few of the alternatives advocated so far consist of either using some form of polynomial preconditioning or applying the usual ilu factorization to a matrix obtained from a multicolor ordering in this paper we present an incomplete factorization technique based on independent set orderings and multicoloring we note that in order to improve robustness it is necessary to allow the preconditioner to have an arbitrarily high accuracy as is done with ilus based on threshold techniques the ilum factorization described in this paper is in this category it can be viewed as a multifrontal version of a gaussian elimination procedure with threshold dropping which has a high degree of potential parallelism the emphasis is on methods that deal specifically with general unstructured sparse matrices such as those arising from finite element methods on unstructured meshes

ordering effects on relaxation methods applied to the discrete one dimensional convection diffusion equation the authors present an analysis of relaxation methods for the one dimensional discrete convection diffusion equation based on norms of the iteration matrices in contrast to standard analytic techniques that use spectral radii these results show how the performance of iterative solvers is affected by directions of flow associated with the underlying operator and by orderings of the discrete grid points in particular for problems of size n relaxation against the flow incurs a latency of approximately n steps in which convergence is slow and red black relaxation incurs a latency of approximately n 2 steps there is no latency associated with relaxation that follows the flow these results are largely independent of the choice of discretization

quasi static motion of a capillary drop ii the three dimensional case abstract a theory is presented to analyze the nonlinear stability of a drop of incompressible viscous fluid with negligible inertia the theory is developed here on the three dimensional version of the relevant free boundary model for stokes equations within this context we show that if the free boundary initiates close to a sphere r 1 e 0 e small then there exists a global in time solution with free boundary r 1 t e 1 n 1 n t e n which approaches a sphere exponentially fast as t moreover we prove that if 0 is analytic resp c in then the velocity u x t e the pressure p x t e and the free boundary are all jointly analytic resp c in x e in an earlier paper we considered the analogous problem for a two dimensional drop although the three dimensional problem proceeds along similar lines the analysis is more complicated due to the fact that we work here with spherical harmonics and vector spherical harmonics

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

some history of the conjugate gradient and lanczos methods this paper gives some of the history of the conjugate gradient and lanczos algorithms and an annotated bibliography for the period 1948 1976

factored approximate inverse preconditioners with dynamic sparsity patterns abstract we propose two sparsity pattern selection algorithms for factored approximate inverse preconditioners to solve general sparse matrices the sparsity pattern is adaptively updated in the construction phase by using combined information of the inverse and original triangular factors of the original matrix in order to determine the sparsity pattern our first algorithm uses the norm of the inverse factors multiplied by the largest absolute value of the original factors and the second employs the norm of the inverse factors divided by the norm of the original factors experimental results show that these algorithms improve the robustness of the preconditioners to solve general sparse matrices

improving approximate inverses based on frobenius norm minimization approximate inverses based on frobenius norm minimization of real nonsingular matrices are analyzed from a purely theoretical point of view in this context this paper provides several sufficient conditions that assure us the possibility of improving in the sense of the frobenius norm some given approximate inverses moreover the optimal approximate inverses of matrix a r n x n among all matrices belonging to certain subspaces of r n x n are obtained particularly a natural generalization of the classical normal equations of the system ax b is given when searching for approximate inverses n a t such that an is symmetric and an i f

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

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

msp a class of parallel multistep successive sparse approximate inverse preconditioning strategies we develop a class of parallel multistep successive preconditioning strategies to enhance efficiency and robustness of standard sparse approximate inverse preconditioning techniques the key idea is to compute a series of simple sparse matrices to approximate the inverse of the original matrix studies are conducted to show the advantages of such an approach in terms of both improving preconditioning accuracy and reducing computational cost compared to the standard sparse approximate inverse preconditioners numerical experiments using one prototype implementation to solve a few sparse matrices on a distributed memory parallel computer are reported

preconditioners generated by minimizing norms we introduce two diagonal preconditioners one of them is a scaling of the matrix they are obtained by minimizing the norm of the iteration matrix using the frobenius and norms we prove convergence of the corresponding iterations for the class of diagonally dominant matrices for the iteration we also prove convergence for positive definite matrices both iterations compete very well with the jacobi iteration

a unifying approach to abstract matrix algebra preconditioning in earlier papers tyrtyshnikov 42 and the first author 14 considered the analysis of clustering properties of the spectra of specific toeplitz preconditioned matrices obtained by means of the best known matrix algebras here we generalize this technique to a generic banach algebra of matrices by devising general preconditioners related to convergent approximation processes 36 finally as case study we focus our attention on the tau preconditioning by showing how and why the best matrix algebra preconditioners for symmetric toeplitz systems can be constructed in this class

the p adic weighted hardy cesaro operator and an application to discrete hardy inequalities abstract this paper aims to investigate the boundedness of the p adic analog of the weighted hardy cesaro operator u s f z p f s t t d t on weighted lebesgue spaces and weighted bmo spaces in each case we obtain the corresponding operator norms u s in particular these results have a surprising relevance to discrete hardy inequalities on the real field we prove a reverse bmo hardy inequality and give a necessary condition on so that the commutator of u s is bounded on l r q p n with symbols in bmo q p n

on measures of nonnormality of matrices abstract this paper discusses several measures of nonnormality of matrices i e functions v cn n r plus where v a 0 iff a is normal besides measures already in the literature we introduce new ones and give comparisons between them at length some of these comparisons e g c14 c15 and c17 manifest well known phenomena of ill conditioned eigenproblems

hardy type theorems on fourier transforms revised abstract we obtain new conditions on periodic integrable functions so that their transformed fourier series belong to l p this improves the classical hardy and bellman results a counterpart for the fourier transforms is also established our main tool is a new extension of the hausdorff young paley inequality for fourier transforms

an efficient computation of generalized inverse of a matrix abstract we propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves 18th order of convergence by using only seven matrix multiplications per iteration loop this yields a high efficiency index for that computational task the algorithm has a relatively mild numerical instability and we stabilize it at the price of adding two extra matrix multiplications per iteration loop this implies an efficiency index that exceeds the known record for numerically stable iterations for this task which means substantial acceleration of the long standing algorithms for an important problem of numerical linear algebra our numerical tests cover a variety of examples in the category of generalized inverses such as drazin case rectangular case and preconditioning of linear systems the test results are in good accordance with our formal study

approximate inverse preconditioners via sparse sparse iterations the standard incomplete lu ilu preconditioners often fail for general sparse indefinite matrices because they give rise to unstable factors l and u in such cases it may be attractive to approximate the inverse of the matrix directly this paper focuses on approximate inverse preconditioners based on minimizing i am f where am is the preconditioned matrix an iterative descent type method is used to approximate each column of the inverse for this approach to be efficient the iteration must be done in sparse mode i e with sparse matrix by sparse vector operations numerical dropping is applied to maintain sparsity compared to previous methods this is a natural way to determine the sparsity pattern of the approximate inverse this paper describes newton global and column oriented algorithms and discusses options for initial guesses self preconditioning and dropping strategies some limited theoretical results on the properties and convergence of approximate inverses are derived numerical tests on problems from the harwell boeing collection and the fidap fluid dynamics analysis package show the strengths and limitations of approximate inverses finally some ideas and experiments with practical variations and applications are presented

pole placement preconditioning the partial pole placement problem has received considerable attention in control theory where it is applied to stabilize time invariant linear control systems this paper discusses application of the partial pole placement problem to the construction of preconditioners for linear systems of equations numerical examples show that these preconditioners can improve the rate of convergence of the restarted gmres methods significantly

orthogonal projections of the identity spectral analysis and applications to approximate inverse preconditioning many strategies for constructing different structures of sparse approximate inverse preconditioners for large linear systems have been proposed in the literature in a more general framework this paper analyzes the theoretical effectiveness of the optimal preconditioner in the frobenius norm of a linear system over an arbitrary subspace of m n left mathbb r right for this purpose the spectral analysis of the frobenius orthogonal projections of the identity matrix onto the linear subspaces of m n left mathbb r right is performed this analysis leads to a simple general criterion the effectiveness of the optimal approximate inverse preconditioners parametrized by any vectorial structure improves at the same time as the smallest singular value or the smallest eigenvalue s modulus of the corresponding preconditioned matrices increases to 1

morphological stability of an elastic tumor host interface abstract the ability of tumors to metastasize is preceded by morphological instabilities such as chains or fingers that invade the host environment therefore parameters that control the morphology of the tumor may also contribute to its invasive ability previous analyses on morphological changes were performed using surface energy of the tumor host interface in an effort to understand the role the interface stiffness plays on tumor evolution here we model the tumor host interface as an elastic membrane governed by the helfrich bending energy using an energy variation approach we derive a modified laplace young condition for the stress jump across the interface in the stokes equation we then perform a linear stability analysis and investigate how physical parameters such as viscosity bending rigidity and apoptosis affect the morphological instability results show that increased bending rigidity versus mitosis rate contributes to a more stable morphological tumor behavior on the other hand increasing tumor viscosity or apoptosis may lead to invasive fingering morphologies comparison with experimental data on glioblastoma spheroids shows good agreement especially for tumors with high adhesion and low proliferation

sparse approximate inverse preconditioners using norm minimization techniques we investigate the use of sparse approximate inverse preconditioners for the iterative solution of unsymmetric linear systems of equations we consider the approximations proposed by cosgrove diaz and griewank internat j comput math 44 1992 pp 91 110 and huckle and grote a new approach to parallel preconditioning with sparse approximate inverses tech report sccm 94 03 stanford university 1994 which are based on norm minimization techniques such methods are of particular interest because of the considerable scope for parallelization we propose a number of enhancements which may improve their performance when run in a sequential environment these methods can perform unfavorably when compared with other techniques however they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment

a linear algebraic analysis of diffusion synthetic acceleration for the boltzmann transport equation a linear algebraic formulation of discretized mono energetic steady state linear boltzmann transport equations in slab geometry is presented the discretization consists of a discrete ordinates collocation in angle and a diamond difference method in space by expressing diffusion synthetic acceleration in this formalism asymptotic results are obtained that prove the effectiveness of the associated preconditioner in various asymptotic regimes including the asymptotic diffusion limit these results hold for problems with nonconstant coefficients and nonuniform spatial zoning posed on finite domains with an incident flux and or reflection prescribed at the boundaries numerical results are also presented which demonstrate these results

robust preconditioners for linear elasticity fem analyses summary this paper deals with two forms of preconditioner which can be easily used with a conjugate gradient solver to replace a direct solution subroutine in a traditional engineering nite element package they are tested in such a package final over a range of 2 d and 3 d elasticity problems from geotechnical engineering quadratic basis functions are used a number of modications to the basic incomplete choleski ic 0 factorization preconditioner are considered an algorithm to reduce positive o diagonal entries is shown in numerical experiments to ensure stability but at the expense of slow convergence an alternative algorithm of jennings and malik is more successful and a relaxation parameter is introduced which can make a further signicant improvement in performance while maintaining stability a heuristic for determining a near optimal value of is proposed a second form of preconditioning symmetrically scaled element by element due to bartelt is also shown to perform robustly over a range of problems it does not require assembly of the global stiness matrix and has great potential for parallelization 1997 by john wiley sons ltd

hardy space estimates of hausdorff operators on the heisenberg group abstract this is a continuation of a series of works about hausdorff operator on hardy spaces in the setting of the heisenberg group in this paper we obtain the boundedness from lebesgue spaces to hardy spaces for fractional hausdorff operators and their compositions with riesz transforms we also establish the boundedness for two kinds of special hausdorff operators the hausdorff poisson operator and the hausdorff gauss operator on hardy spaces

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

polynomial preconditioners for conjugate gradient calculations dubois greenbaum and rodrigue proposed using a truncated neumann series as an approximation to the inverse of a matrix a for the purpose of preconditioning conjugate gradient iterative approximations to ax b if we assume that a has been symmetrically scaled to have unit diagonal and is thus of the form i g then the neumann series is a power series in g with unit coefficients the incomplete inverse was thought of as a replacement of the incomplete cholesky decomposition suggested by meijerink and van der vorst in the family of methods iccg n the motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors we here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in g we then show how to select the parameters to minimize the condition number of the product of the polynomial and i g theoretically the resulting algorithm

multilevel ilu decomposition in this paper the multilevel ilu mlilu decomposition is introduced during an incomplete gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels on these levels some smoothing steps are computed the mlilu decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices convergence rates independent of the number of unknowns are shown numerically for several examples many numerical experiments including unsymmetric and anisotropic problems problems with jumping coefficients as well as realistic problems are presented they indicate a very robust convergence behavior of the mlilu method

bending rigidity and higher order curvature terms for the hard sphere fluid near a curved wall in this work i derive analytic expressions for the curvature dependent fluid substrate surface tension of a hard sphere fluid on a hard curved wall in the first step the curvature thermodynamic properties are found as truncated power series in the activity in terms of the exactly known second and third order cluster integrals of the hard sphere fluid near spherical and cylindrical walls these results are then expressed as packing fraction power series and transformed to different reference regions which is equivalent to considering different positions of the dividing surface based on the truncated series it is shown that the bending rigidity of the system is non null and that higher order terms in the curvature also exist in the second step approximate analytic expressions for the surface tension the tolman length the bending rigidity and the gaussian rigidity as functions of the packing fraction are found by considering the known terms of the series expansion complemented with a simple fitting approach it is found that the obtained formulas accurately describe the curvature thermodynamic properties of the system further they are more accurate than any previously published expressions

generalized poincare embeddings and weighted hardy operator on qpα q spaces abstract the poincare embeddings w 1 n r n bmo r n and the john nirenberg inequality in bmo r n are classical results in modern analysis and partial differential equations this paper studies the generalized poincare embeddings and the john nirenberg inequality in q p q r n spaces furthermore for the weighted hardy operator u and its adjoint v this paper gives sufficient and necessary conditions on weight function to ensure the boundedness of them on q p q r n spaces

a new preconditioning technique for solving large sparse linear systems abstract a new sparse approximate triangular factorization technique for solving large sparse linear system by iterative methods is proposed the method is based on the description of the triangular factorization of a matrix as a product of elementary matrices and provides a general scheme for constructing incomplete preconditioners for the given matrix in particular the familiar incomplete choleski decomposition can be incorporated into this scheme the algorithm based on choice by value compares favorably with the incomplete choleski preconditioner and equipped with a user controlled parameter is able to tackle extremely ill conditioned problems arising in structural analysis semiconductor simulation oil reservoir modelling and other applications when applied to a positive definite symmetric matrix the algorithm produces a preconditioning matrix preserving that property

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

a family of modified regularizingcirculant preconditioners for two levels toeplitz systems customary circulant preconditioners of chan strang and inverse toeplitz type aremodified for the regularized solution of ill conditioned toeplitz systems given a symmetric two level toeplitz matrix a and two parameters t and s consider the customary preconditioner c and modify it by replacing its eigenvalues with modulus less than t with the value s the circulant matrix obtained in this way is used as a preconditioner we analyze the distribution of the eigenvalues of the preconditioned matrix and the regularization properties numerical experiments shown for the image restoration problem validate the theoretical results

normalized frobenius condition number of the orthogonal projections of the identity abstract this paper deals with the orthogonal projection in the frobenius sense a n of the identity matrix i onto the matrix subspace a s a r n n s being an arbitrary subspace of r n n lower and upper bounds on the normalized frobenius condition number of matrix a n are given furthermore for every matrix subspace s r n n a new index f a s which generalizes the normalized frobenius condition number of matrix a is defined and analyzed

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

fast approximate solution of large scale sparse linear systems abstract in this paper we apply the fast approximate inversion algorithm of frederickson to solve a large scale sparse linear system arising from the triangular finite element solution to the second order elliptic problems the main advantage of this algorithm is that the rate of convergence is independent of n for equations in the class considered several approximate inversion techniques for the algorithm are proposed and numerical results for these techniques are also provided

using approximate inverses in algebraic multilevel methods this paper deals with the iterative solution of large sparse symmetric positive definite systems we investigate preconditioning techniques of the two level type that are based on a block factorization of the system matrix whereas the basic scheme assumes an exact inversion of the submatrix related to the first block of unknowns we analyze the effect of using an approximate inverse instead we derive condition number estimates that are valid for any type of approximation of the schur complement and that do not assume the use of the hierarchical basis they show that the two level methods are stable when using approximate inverses based on modified ilu techniques or explicit inverses that meet some row sum criterion on the other hand we bring to the light that the use of standard approximate inverses based on convergent splittings can have a dramatic effect on the convergence rate these conclusions are numerically illustrated on some examples

estimates of weighted hardy littlewood averages on the p adic vector space abstract in the p adic vector space q p n we characterize those non negative functions defined on z p w q p 0 w p 1 for which the weighted hardy littlewood average u f z p f t t d t is bounded on l r q p n 1 r and on bmo q p n also in each case we find the corresponding operator norm u

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance