superior properties of the presb preconditioner for operators on two by two block form with square blocks matrices or operators in two by two block form with square blocks arise in numerous important applications such as in optimal control problems for pdes the problems are normally of very large scale so iterative solution methods must be used thereby the choice of an efficient and robust preconditioner is of crucial importance since some time a very efficient preconditioner the preconditioned square block presb method has been used by the authors and coauthors in various applications in particular for optimal control problems for pdes it has been shown to have excellent properties such as a very fast and robust rate of convergence that outperforms other methods in this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs under certain conditions the condition number of the preconditioned matrix is bounded by 2 or even smaller furthermore under certain assumptions the rate of convergence is superlinear

preconditioning and boundary conditions without h 2 estimates l 2 condition numbers and the distribution of the singular values this work deals with the behavior in the l sub 2 norm of the condition number and distribution of the l sub 2 singular values of the preconditioned operators b sub h sup minus 1 a sub h and a sub h b sub h sup minus 1 sub h where a sub h and b sub h are finite element discretizations of second order elliptic operators a and b in an earlier work manteuffel and parter proved that b sub h sup minus 1 a sub h a sub h b sub h sup minus 1 have a uniformly bounded l sub 2 condition number if and only if a sup and b sup a and b have the same boundary conditions this earlier work used the h sub 2 regularity of a and b as well as optimal l sub 2 error estimates and a quasi uniform grid for the finite element spaces in the present paper the authors first extend these condition number results to the case in which neither h sub 2 regularity and hence optimal l sub 2 error estimates nor the quasi uniformity assumption need to be satisfied instead it is assumed that the principle part of the preconditioning operator b is a scalar multiple 1 mu of the principle part of a

inverse eigenvalue problems for extended hessenberg and extended tridiagonal matrices in inverse eigenvalue problems one tries to reconstruct a matrix satisfying some constraints given some spectral information here two inverse eigenvalue problems are solved first given the eigenvalues and the first components of the associated eigenvectors called the weight vector an extended hessenberg matrix with prescribed poles is computed possessing these eigenvalues and satisfying the eigenvector constraints the extended hessenberg matrix is retrieved by executing particularly designed unitary similarity transformations on the diagonal matrix containing the eigenvalues this inverse problem closely links to orthogonal rational functions the extended hessenberg matrix contains the recurrence coefficients given the nodes eigenvalues poles poles of the extended hessenberg matrix and a weight vector first eigenvector components determining the discrete inner product moreover it is also sort of the inverse of the rational arnoldi algorithm instead of using the rational arnoldi method to compute a krylov basis to approximate the spectrum we will reconstruct the orthogonal krylov basis given the spectral info in the second inverse eigenvalue problem we do the same but refrain from unitarity as a result we execute possibly non unitary similarity transformations on the diagonal matrix of eigenvalues to retrieve a non symmetric extended tridiagonal matrix the algorithm will be less stable but it will be faster as the extended tridiagonal matrix admits a low cost factorization of o n n equals the number of eigenvalues whereas the extended hessenberg matrix does not again there is a close link with orthogonal function theory the extended tridiagonal matrix captures the recurrence coefficients of bi orthogonal rational functions moreover it is again sort of inverse of the nonsymmetric lanczos algorithm given spectral properties we reconstruct the two basis krylov matrices linked to the nonsymmetric lanczos algorithm

orthogonal hessenberg reduction and orthogonal krylov subspace bases we study necessary and sufficient conditions that a nonsingular matrix a can be b orthogonally reduced to upper hessenberg form with small bandwidth by this we mean the existence of a decomposition av vh where h is upper hessenberg with few nonzero bands and the columns of v are orthogonal in an inner product generated by a hermitian positive definite matrix b the classical example for such a decomposition is the matrix tridiagonalization performed by the hermitian lanczos algorithm also called the orthogonal reduction to tridiagonal form does there exist such a decomposition when a is nonhermitian in this paper we completely answer this question the related but not equivalent question of necessary and sufficient conditions on a for the existence of short term recurrences for computing b orthogonal krylov subspace bases was completely answered by the fundamental theorem of faber and manteuffel siam j numer anal 21 1984 pp 352 362 we give a detailed analysis of b normality the central condition in both the faber manteuffel theorem and our main theorem and show how the two theorems are related our approach uses only elementary linear algebra tools we thereby provide new insights into the principles behind krylov subspace methods that are not provided when more sophisticated tools are employed

superlinearly convergent cg methods via equivalent preconditioning for nonsymmetric elliptic operators the convergence of the conjugate gradient method is studied for preconditioned linear operator equations with nonsymmetric normal operators with focus on elliptic convection diffusion operators in sobolev space superlinear convergence is proved first for equations whose preconditioned form is a compact perturbation of the identity in a hilbert space then the same convergence result is verified for elliptic convection diffusion equations using different preconditioning operators the convergence factor involves the eigenvalues of the corresponding operators for which an estimate is also given the above results enable us to verify the mesh independence of the superlinear convergence estimates for suitable finite element discretizations of the convection diffusion problems

krylov subspace acceleration of waveform relaxation in this paper we describe and analyze krylov subspace techniques for accelerating the convergence of waveform relaxation for solving time dependent problems a new class of accelerated waveform methods convolution krylov subspace methods is presented in particular we give convolution variants of the cg algorithm and the gmres algorithm and analyze their convergence behavior we prove that the convolution krylov subspace algorithms for initial value problems have the same convergence bounds as their linear algebra counterparts analytical examples are given to illustrate the operation of convolution krylov subspace methods experimental results are presented which show the convergence behavior of traditional and convolution waveform methods applied to solving a linear initial value problem as well as the convergence behavior of static krylov subspace methods applied to solving the associated linear algebraic equation

preconditioning cubic spline collocation discretizations of elliptic equations this work considers the uniformly elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 and partial u over partial nu alpha u on gamma 1 and its discretization based on hermite cubic spline spaces and collocation at the gauss points using an interpolatory basis with support on the gauss points one obtains the matrix hat a n h 1 over n we discuss the h 1 condition numbers and the distribution of tilde beta n singular values of the preconditioned matrices tilde beta n 1 hat a n where tilde beta n is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator b given by b v delta v b 0 v in omega with boundary conditions v 0 on gamma 0 partial v over partial nu beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by gauss points or the space of continuous functions which are linear on the triangles of the triangulation of omega using the gauss points when a b we obtain results on the eigenvalues of tilde beta n 1 hat a n in the general case we obtain bounds and clustering results on the tilde beta n singular values of tilde beta n 1 hat a n these results are related to the results of manteuffel and parter mp parter and wong pw and wong w for finite element discretizations as well as the results of parter and rothman pr for discretizations based on legendre spectral collocation

diffraction problems for quasilinear reaction diffusion systems abstract the aim of this paper is to investigate existence uniqueness and qualitative properties of the weak and piecewise classical solutions of the diffraction problem for weakly coupled reaction diffusion systems monotony and stability with respect to the time variable of the weak solutions is proved as an application is considered a model reaction diffusion system describing an enzyme sensor for concentration of phenol in the chemical industry

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

efficient dynamic finite element method for 3d objects with uniform cross section an efficient implicit dynamic finite element method fem for elastic 3d objects with uniform cross sections was developed in this method the finite element mesh is generated in such a way that the object to be analysed is at first sliced into layers with the same thickness along its generatrix and then each layer is discretized into finite elements of the same pattern this way of discretization makes the mass viscosity and stiffness matrices into the repetitive block tridiagonal matrices the repetitive block tridiagonal matrix has the characteristic that the sequence of matrices which appears in the gaussian elimination for the repetitive block tridiagonal matrix is a rapid convergent sequence the process of the gaussian elimination can be terminated when the sequence converges the rest of the sequence is not necessary to be stored the present method can save the computational time and memory by utilising this characteristic of the repetitive block tridiagonal matrix a few examples of analyses including whole hopkinson bar analysis were performed to demonstrate the effectiveness of the present method the present method is applicable not only to the elasto dynamics but also to many other problems such as thermal problems electrical problems and plastic problems without geometric non linearity copyright 2005 john wiley sons ltd

bifurcation of steady state solutions in predator prey and competition systems we discuss steady state solutions of systems of semilinear reaction diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region it is easy to find solutions to the systems such that either u or v is identically zero such solutions correspond to the case where one of the species is extinct by using decoupling and global bifurcation theory techniques we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co exist

a preconditioned semi implicit method for magnetohydrodynamics equations the paper describes the development of a semi implicit solver designed for the need to follow slow evolution flows encountered in nonlinear resistive computational plasma dynamics cpd spatial discretization uses a second order finite difference approximation while temporal advance is achieved by a second order semi implicit predictor corrector scheme that reduces the severe time step constraints imposed by the fast magnetosonic and alfven waves on standard explicit schemes an efficient preconditioner adapted to magnetohydrodynamics mhd problems and associated with conjugate gradient like linear solvers considerably increases the cpu saving when compared with an explicit advance this scheme enables the use of a large time step as well as the necessary high spatial resolution an application of this scheme to a class of astrophysical nonlinear mhd problems allows one to perform numerical experiments relevant to the slow mhd evolution of the magnetic field dominating the outer atmosphere of the sun that leads to small scales formation

exponential time differencing for mimetic multilayer ocean models abstract a framework for exponential time discretization of the multilayer rotating shallow water equations is developed in combination with a mimetic discretization in space the method is based on a combination of existing exponential time differencing etd methods and a careful choice of approximate jacobians the discrete hamiltonian structure and conservation properties of the model are taken into account in order to ensure stability of the method for large time steps and simulation horizons in the case of many layers further efficiency can be gained by a layer reduction which is based on the vertical structure of fast and slow modes numerical experiments on the example of a mid latitude regional ocean model confirm long term stability for time steps increased by an order of magnitude over the explicit cfl while maintaining accuracy for key statistical quantities

solving hermitian positive definite systems using indefinite incomplete factorizations incomplete ldl factorizations sometimes produce an indefinite preconditioner even when the input matrix is hermitian positive definite the two most popular iterative solvers for symmetric systems cg and minres cannot use such preconditioners they require a positive definite preconditioner one approach that has been extensively studied to address this problem is to force positive definiteness by modifying the factorization process we explore a different approach use the incomplete factorization with a krylov method that can accept an indefinite preconditioner the conventional wisdom has been that long recurrence methods like gmres or alternatively non optimal short recurrence methods like symmetric qmr and bicgstab must be used if the preconditioner is indefinite we explore the performance of these methods when used with an incomplete factorization but also explore a less known krylov method called pcg odir that is both optimal and uses a short recurrence and can use an indefinite preconditioner furthermore we propose another optimal short recurrence method called ip minres that can use an indefinite preconditioner and a variant of pcg odir which we call ip cg that is more numerically stable and usually requires fewer iterations

fast direct solution of the helmholtz equation with a perfectly matched layer or an absorbing boundary condition we consider the efficient numerical solution of the helmholtz equation in a rectangular domain with a perfectly matched layer pml or an absorbing boundary condition abc standard bilinear trilinear finite element discretization on an orthogonal mesh leads to a separable system of linear equations for which we describe a cyclic reduction type fast direct solver we present numerical studies to estimate the reflection of waves caused by an absorbing boundary and a pml and we optimize certain parameters of the layer to minimize the reflection copyright 2003 john wiley sons ltd

a new taxonomy of conjugate gradient methods abstract in this paper a framework is proposed for conjugate gradient and conjugate direction algorithms based on an indefinite quadratic form by the choice of the hessian g of this form and of another arbitrary matrix k no fewer than nineteen distinct cg algorithms may be succinctly defined four of these are believed to be new the choice of certain other vectors leads either to a two term or a three term recurrence formula and it is shown that to every two term formula there corresponds a three term formula although the converse is not generally true the two matrices g and k also determine the stability characteristics of the two term methods

mesh independent superlinear pcg rates via compact equivalent operators the subject of the paper is the mesh independent convergence of the preconditioned conjugate gradient pcg method for nonsymmetric elliptic problems the approach of equivalent operators is involved in which one uses the discretization of another suitable elliptic operator as preconditioning matrix by introducing the notion of compact equivalent operators it is proved that for a wide class of elliptic problems the superlinear convergence of the obtained pcg method is mesh independent under finite element discretizations that is the rate of superlinear convergence is given in the form of a sequence which is mesh independent and is determined only by the elliptic operators

sign definite solutions in some linear elliptic systems on considere un ensemble faiblement couple de deux equations aux derivees partielles ou la matrice de couplage a des elements variables et la partie principale de chaque equation est le meme operateur uniformement elliptique on obtient des conditions necessaires pour le decouplage on determine le signe algebrique des composantes de solution

restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems the restrictively preconditioned conjugate gradient rpcg method for solving large sparse system of linear equations of a symmetric positive definite and block two by two coefficient matrix is further studied in fact this rpcg method is essentially the classical preconditioned conjugate gradient pcg method with a specially structured preconditioner within this setting we present algorithmic descriptions of two restrictive preconditioners that respectively employ the block jacobi and the block symmetric gauss seidel matrix splitting matrices as approximations to certain matrices involved in them and give convergence analyses of the correspondingly induced two pcg methods numerical results show that these restrictive preconditioners can lead to practical and effective pcg methods for solving large sparse systems of linear equations of symmetric positive definite and block two by two coefficient matrices

a study on q appell polynomials from determinantal point of view this research is aimed to give a determinantal definition for the q appell polynomials and show some classical properties as well as find some interesting properties of the mentioned polynomials in the light of the new definition

preconditioning legendre spectral collocation approximations to elliptic problems this work deals with the h 1 condition numbers and the distribution of the tilde beta n m singular values of the preconditioned operators tilde beta n m 1 w n m hat a n m hat a n m is the matrix representation of the legendre spectral collocation discretization of the elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 frac partial u partial nu a alpha u on gamma 1 tilde beta n m is the stiffness matrix associated with the finite element discretization of the positive definite elliptic operator b defined by bv delta v b 0 v in omega with boundary conditions v 0 on gamma 0 frac partial v partial nu b beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by the legendre gauss lobatto lgl points or the s

a generalized conjugate gradient least square method a generalizeds term truncated conjugate gradient method of least square type proposed in 1a b is extended to a form more suitable for proving when the truncated version is identical to the full term version advantages with keeping a control term in the truncated version is pointed out a computationally efficient new algorithm based on a special inner product with a small demand of storage is also presented we also give simplified and slightly extended proofs of termination of the iterative sequence and of existence of ans term recursion identical to the full term version important earlier results on this latter topic are found in 15 16 8 and 11

a determinantal approach to appell polynomials a new definition by means of a determinantal form for appell 1880 1 polynomials is given general properties some of them new are proved by using elementary linear algebra tools finally classic and non classic examples are considered and the coefficients calculated by an ad hoc mathematica code for particular sequences of appell polynomials are given

finite time balanced truncation for linear systems via shifted legendre polynomials abstract in this paper we present a finite time model order reduction method for linear systems via shifted legendre polynomials the main idea of the approach is to use finite time empirical gramians which are constructed from impulse responses by solving block tridiagonal linear systems to generate approximate balanced system for the large scale system the balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted legendre polynomials without individual reduction of the gramians and a separate eigenvector solve then the reduced order model is constructed by truncating the states corresponding to the small approximate hankel singular values hsvs the stability preservation of the reduced model is briefly discussed and in combination with the dominant subspace projection method we modify the reduction procedure to alleviate the shortcomings of the above method which may unexpectedly lead to unstable systems even though the original one is stable furthermore the properties of the resulting reduced models are considered finally numerical experiments are given to demonstrate the effectiveness of the proposed methods

qmr smoothing for lanczos type product methods based on three term rrecurrences for the solution of large sparse non hermitian linear systems lanczos type product methods that are based on the lanczos three term recurrence are derived in a systematic way these methods either square the lanczos process or combine it with a local minimization of the residual for them a quasi minimal residual qmr smoothing is proposed that can also be implemented by short term recurrences the practical performance of these methods and the qmr smoothing is demonstrated in a number of numerical examples

domain decomposition iterative procedures for solving scalar waves in the frequency domain the propagation of dispersive waves can be modeled relevantly in the frequency domain a wave problem in the frequency domain is difficult to solve numerically in addition to having a complex valued solution the problem is neither hermitian symmetric nor coercive in a wide range of applications in geophysics or quantum mechanics in this paper we consider a parallel domain decomposition iterative procedure for solving the problem by finite differences or conforming finite element methods the analysis includes the decomposition of the domain into either the individual elements or larger subdomains blocks of finite elements to accelerate the speed of convergence we introduce relaxation parameters on the subdomain interfaces and an artificial damping iteration the convergence rate of the resulting algorithm turns out to be independent on the mesh size and the wave number numerical results carried out on an ncube2 parallel computer are presented to show the effectiveness of the method

equivalent operator preconditioning for elliptic problems with nonhomogeneous mixed boundary conditions the numerical solution of linear elliptic partial differential equations often involves finite element discretization where the discretized system is usually solved by some conjugate gradient method the crucial point in the solution of the obtained discretized system is a reliable preconditioning that is to keep the condition number of the systems under control no matter how the mesh parameter is chosen the pcg method is applied to solving convection diffusion equations with nonhomogeneous mixed boundary conditions using the approach of equivalent and compact equivalent operators in hilbert space it is shown that for a wide class of elliptic problems the superlinear convergence of the obtained preconditioned cgm is mesh independent under fem discretization

a parallel fictitious domain method for the three dimensional helmholtz equation the application of a fictitious domain domain embedding method to the three dimensional helmholtz equation with absorbing boundary conditions is considered the finite element discretization is performed by using locally fitted meshes and an algebraic fictitious domain method with a separable preconditioner is used in the iterative solution of the resultant linear systems such a method is based on embedding the original domain into a larger one with a simple geometry with this approach it is possible to realize the gmres iterations in a low dimensional subspace and use the partial solution method to solve the linear systems with the preconditioner an efficient parallel implementation of the iterative algorithm is introduced results of numerical experiments demonstrate good scalability properties on distributed memory parallel computers and the ability to solve high frequency acoustic scattering problems

study on semi conjugate direction methods for non symmetric systems some theoretical problems and implementation problems are studied here for the semi conjugate direction method established by yuan golub plemmons and cecilio 2002 the existence of semi conjugate directions is proved for almost all matrices except skew symmetric matrices a new technique is proposed to overcome the breakdown problem appeared in the semi conjugate direction method in the implementation of the semi conjugate direction method the generation of the semi conjugate direction is very important and necessary but very expensive the technique of limited memory is introduced to economize the cost of the generation of the semi conjugate direction in the yuan golub plemmons cecilio algorithm finally some numerical experiments are given to confirm our theoretical results our results illustrate that the semi conjugate direction method is very nice alternative for solving non symmetric systems and the limited memory left conjugate direction method is a good improvement of the left conjugate direction method copyright 2004 john wiley sons ltd

newton ll method for the second order semi linear elliptic partial differential equations newton s method with first order system least squares fosls finite element method has been widely used to approximately solve a system of nonlinear partial differential equations adler et al 2010 9 codd et al 2003 10 manteuffel et al 2006 11 in this paper we propose to use the first order system ll method to find a correction in each newton s iteration which provides an l2 approximation of the second order semi linear elliptic partial differential equations while the typical newton fosls method providesh1 approximations the numerical tests have been conducted to validate the theory

preconditioning a class of fourth order problems by operator splitting we develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space we consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two discrete linear second order elliptic operators and exploit this property in designing the preconditioners the underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available we propose symmetric and non symmetric preconditioners along with theory and numerical experiments they both document crucial properties of the preconditioners as well as their practical performance it is important to note that we neither need h s regularity s 1 of the continuous problem nor quasi uniform grids

a domain decomposition solver for acoustic scattering by elastic objects in layered media a finite element solution procedure is presented for accurately computing time harmonic acoustic scattering by elastic targets buried in sediment an improved finite element discretization based on trilinear basis functions leading to fourth order phase accuracy is described for sufficiently accurate discretizations 100 million to 1 billion unknowns are required the resulting systems of linear equations are solved iteratively using the gmres method with a domain decomposition preconditioner employing a fast direct solver due to the construction of the discretization and preconditioner iterations can be reduced onto a sparse subspace associated with the interfaces numerical experiments demonstrate capability to evaluate the scattered field with hundreds of wavelengths

a parallel fast direct solver for block tridiagonal systems with separable matrices of arbitrary dimension a parallel fast direct solution method for linear systems with separable block tridiagonal matrices is considered such systems appear for example when discretizing the poisson equation in a rectangular domain using the five point finite difference scheme or the piecewise linear finite elements on a triangulated possibly nonuniform rectangular mesh the method under consideration has the arithmetical complexity mathcal o n log n and it is closely related to the cyclic reduction method but instead of using the matrix polynomial factorization the so called partial solution technique is employed hence in this paper the method is called the partial solution variant of the cyclic reduction method pscr method the method is presented and analyzed in a general radix q framework and based on this analysis the radix 4 variant is chosen for parallel implementation using the mpi standard the generalization of the method to the case of arbitrary block dimension is described the numerical experiments show the sequential efficiency and numerical stability of the pscr method compared to the well known blktri implementation of the generalized cyclic reduction method the good scalability properties of the parallel pscr method are demonstrated in a distributed memory cray t3e 750 computer

a mesh independent superlinear algorithm for some nonlinear nonsymmetric elliptic systems the numerical solution of nonlinear elliptic transport systems is considered an outer inner damped inexact newton plus pcg type iteration is proposed for the finite element discretization of the problem and mesh independent superlinear convergence is proved for both the outer and inner iterations numerical experiments are enclosed

two conjugate gradient type methods for unsymmetric linear equations we propose two new conjugate gradient type methods for the solution of sparse unsymmetric linear systems we present a new tridiagonalization process for unsymmetric matrices that is closely related to the lanczos process we use orthogonal factorizations of the tridiagonal matrix to derive the new algorithms usymlq and usymqr in the same fashion as symmlq and minres c c paige and m a saunders solution of sparse indefinite systems of linear equations siam j numer anal 12 1975 pp 617 629 for symmetric matrices some numerical results for the new methods and comparisons with other methods are presented

on eigenfunctions with sign definite components in weakly coupled linear elliptic systems on considere le systeme lu m 11 u m 12 v lv m 21 v m 22 v dans u v 0 sur ou r n est un domaine borne a frontiere lisse l est un operateur uniformement fortement elliptique d ordre 2

preconditioning nonconforming finite element methods for treating dirichlet boundary conditions ii this work deals with theh 1 condition numbers and the distribution of theb h singular values of the preconditioned operators b h 1 a h 0 h 1 wherea h andb h are finite element discretizations of second order elliptic operators a andb respectively b is also assumed to be self adjoint and positive definite for conforming finite elements parter and wong have shown that the singular values cluster in a positive finite interval goldstein also has derived results on the spectral distribution ofb h 1 a h using a different approach as a generalization of the results of parter and wong the current work includes nonconforming finite element methods which deal with dirichlet boundary conditions it will be shown that in this more general setting the singular values also cluster in a positive finite interval in particular if the leading part ofb is the same as the leading part ofa then the singular values cluster about the point 1 two specific methods are given as applications of this theory they are the penalty method of babuska and the method of nearly zero boundary conditions of nitsche finally it will be shown that the same results can be proven by an approach generalized from the work of goldstein

block tridiagonal reduction of perturbed normal and rank structured matrices abstract it is well known that if a matrix a c n n solves the matrix equation f a a h 0 where f x y is a linear bivariate polynomial then a is normal a and a h can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and moreover the spectrum of a is located on a straight line in the complex plane in this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed hermitian and unitary matrices

exponential convergence through linear finite element discretization of stratified subdomains motivated by problems where the response is needed at select localized regions in a large computational domain we devise a novel finite element discretization that results in exponential convergence at pre selected points the key features of the discretization are a use of midpoint integration to evaluate the contribution matrices and b an unconventional mapping of the mesh into complex space named complex length finite element method cfem the technique is linked to pade approximants that provide exponential convergence of the dirichlet to neumann maps and thus the solution at specified points in the domain exponential convergence facilitates drastic reduction in the number of elements this combined with sparse computation associated with linear finite elements results in significant reduction in the computational cost the paper presents the basic ideas of the method as well as illustration of its effectiveness for a variety of problems involving laplace helmholtz and elastodynamics equations

comparison principle for non cooperative elliptic systems this paper presents some sufficient conditions for the validity of the comparison principle for weak and classical solutions of non cooperative weakly coupled systems of elliptic second order pde

idr s a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations we present idr s a new family of efficient short recurrence methods for large nonsymmetric systems of linear equations the new methods are based on the induced dimension reduction idr method proposed by sonneveld in 1980 idr s generates residuals that are forced to be in a sequence of nested subspaces although idr s behaves like an iterative method in exact arithmetic it computes the true solution using at most n n s matrix vector products with n the problem size and s the codimension of a fixed subspace we describe the algorithm and the underlying theory and present numerical experiments to illustrate the theoretical properties of the method and its performance for systems arising from different applications our experiments show that idr s is competitive with or superior to most bi cg based methods and outperforms bi cgstab when s 1

preconditioning nonconforming finite element methods this paper is concerned with the h sup 1 condition number and singular value distribution of preconditioned nonself adjoint discrete elliptic operators in particular a wide class of nonconforming finite element methods are investigated the main results give a precise interval on which the singular values cluster results of this kind have an important bearing on the convergence behavior of the conjugate gradient method applied to the normal equations

preconditioning operators and sobolevgradients for nonlinear elliptic problems a preconditioning framework is presented for the iterative solution of nonlinear elliptic problems based on the preconditioning operator approach various fixed preconditioning operators are used in the iteration which can be interpreted as a weighted sobolev gradient method

preconditioningp1 nonconforming finite elements condition numbers and singular value distributions this work deals with the condition numbers and the distribution of theb h singular values of the preconditioned operators b h 1 a h 0 h 1 wherea h andb h are discretizations of second order elliptic operatorsa andb usingp 1 nonconforming finite elements of crouzeix and raviart b is also assumed to be self adjoint and positive definite for conforming finite elements parter and wong have shown that the singular values cluster in a positive finite interval these reults are being extended to the aforementioned nonconforming finite elements it will be shown that for quasiuniform grids theb h singular values are bounded above and below by positive constants which are independent of the grid sizeh moreover they also cluster in a smaller but usually estimable interval issues of implementation are also discussed

preconditioning isoparametric finite element methods taking into account numerical integration this paper is concerned with the h 1 condition numbers and the b h singular value distribution of preconditioned operators b h 1a h 0 h 1 where a h and b h are isoparametric finite element discretizations of second order elliptic operators a and b by taking into account numerical integration respectively where a is nonselfad joint and possibly indefinite and b is selfadjoint and positive definite it is proved that the h 1 condition numbers of operators b h 1a h 0 h 1 are uniformly bounded and the b h singular values cluster in a positive finite interval finally the implementation of b h 1 is discussed

sobolev gradient preconditioning for the electrostatic potential equation sobolev gradient type preconditioning is proposed for the numerical solution of the electrostatic potential equation a constructive representation of the gradients leads to efficient laplacian preconditioners in the iteration thanks to the available fast poisson solvers convergence is then verified for the corresponding sequence in sobolev space implying mesh independent convergence results for the discretized problems a particular study is devoted to the case of a ball due to the radial symmetry of this domain a direct realization without discretization is feasible the simplicity of the algorithm and the fast linear convergence are finally illustrated in a numerical test example

superlinearly convergent pcg algorithms for some nonsymmetric elliptic systems a preconditioned conjugate gradient method is applied to finite element discretizations of some nonsymmetric elliptic systems mesh independent superlinear convergence is proved which is an extension of a similar earlier result from a single equation to systems the proposed preconditioning method involves decoupled preconditioners which yields small and parallelizable auxiliary problems

an analysis of equivalent operator preconditioning for equation free newton krylov methods we consider the computation of a fixed point of a time stepper using newton krylov methods and propose and analyze equivalent operator preconditioning for the resulting linear systems for a linear scalar advection reaction diffusion equation we investigate in detail how the convergence rate depends upon the choice of preconditioner parameters and upon the time discretization the results are especially valuable when computing fixed points of a coarse time stepper in the equation free multiscale framework in which one simulates an unavailable coarse scale model by wrapping a set of computational routines around appropriately initialized fine scale simulations both analytical results and numerical experiments are presented showing that one can speed up the convergence of iterative methods significantly for a wide range of parameter values in the preconditioner

necessary and sufficient conditions for the existence of a conjugate gradient method we characterize the class cg s of matrices a for which the linear system a bf x bf b can be solved by an s term conjugate gradient method we show that except for a few anomalies the class cg s consists of matrices a for which conjugate gradient methods are already known these matrices are the hermitian matrices a a and the matrices of the form ae i theta di b with b b

an efficient algorithm for a model with a bidiagonal coefficient matrix bidiagonal matrices arise directly in the discretization of models using one step methods and indirectly through the factoring of tridiagonal systems with special characteristics we first look at the solution of a bidiagonal system and then relate it to an experimental method that has been used to determine the overall heat transfer coefficient in a concentric tube parallel flow heat exchanger the discretization of the governing partial differential equation with its boundary conditions gives rise to a near toeplitz bidiagonal system here we consider a method for the parallel processing of the resulting systems to provide measurements at fixed spatial and time increments

iterative k step methods for computing possibly repulsive fixed points in banach spaces abstract we discuss iterative methods of the form xn 0 xx 1 1xn 1 kxn k n k k 1 for computing a fixed point x of a frechet differentiable self mapping of a sub set in a banach space by suitably choosing the coefficients 0 k the local convergence is established under certain assumptions on the spectrum of x this spectrum need not be contained in the unit disk however if it is convergence can often be speeded up considerably compared to picard iteration the methods are generalizations of the methods of v n kublanovskaya and w niethammer for linear systems of equations a more general type of iteration with nonstationary coefficients is considered also for the proof of their local convergence a generalization of the convergence theorems of perron ostrowski and kitchen is presented the connection with other methods in particular mann s iterative processes is also discussed

superlinear convergence of krylov subspace methods for self adjoint problems in hilbert space the conjugate gradient and minimum residual methods for self adjoint problems in hilbert space are considered linear and superlinear convergence results with respect to both q and r rates are reviewed new results on ell step q superlinear and r superlinear convergence for the minimum residual method are provided and examples are considered to underscore the relevance of a hilbert space theory

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

the aitken like acceleration of the schwarz method on nonuniform cartesian grids in this paper we present a family of domain decomposition based on an aitken like acceleration of the schwarz method seen as an iterative procedure with a linear rate of convergence this paper is a generalization of the method first introduced at the 12th international conference on domain decomposition that was restricted to regular cartesian grids the potential of this method to provide scalable parallel computing on a geographically broad grid of parallel computers was demonstrated for some linear and nonlinear elliptic problems discretized by finite differences on a cartesian mesh the main purpose of this paper is to present a generalization of the method to nonuniform cartesian meshes the salient feature of the method consists of accelerating the sequence of traces on the artificial interfaces generated by the schwarz procedure using a good approximation of the main eigenvectors of the trace transfer operator for linear separable elliptic operators our solver is a direct solver for nonlinear operators we use an approximation of the eigenvectors of the jacobian of the trace transfer operator the acceleration is then applied to the sequence generated by the schwarz algorithm applied directly to the nonlinear operator

numerical solution of the nonlinear helmholtz equation using nonorthogonal expansions in j comput phys 171 2001 632 677 we developed a fourth order numerical method for solving the nonlinear helmholtz equation which governs the propagation of time harmonic laser beams in media with a kerr type nonlinearity a key element of the algorithm was a new nonlocal two way artificial boundary condition abc set in the direction of beam propagation this two way abc provided for reflectionless propagation of the outgoing waves while also fully transmitting the given incoming beam at the boundaries of the computational domain altogether the algorithm of j comput phys 171 2001 632 677 has allowed for a direct simulation of nonlinear self focusing without neglecting nonparaxial effects and backscattering to the best of our knowledge this capacity has never been achieved previously in nonlinear optics in the current paper we propose an improved version of the algorithm the principal innovation is that instead of using the dirichlet boundary conditions in the direction orthogonal to that of the laser beam propagation we now introduce sommerfeld type local radiation boundary conditions which are constructed directly in the discrete framework numerically implementation of the sommerfeld conditions requires evaluation of eigenvalues and eigenvectors for a non hermitian matrix subsequently the separation of variables which is a key building block of the aforementioned nonlocal abc is implemented through an expansion with respect to the nonorthogonal basis of the eigenvectors numerical simulations show that the new algorithm offers a considerable improvement in its numerical performance as well as in the range of physical phenomena that it is capable of simulating

optimal equivalent preconditioners preconditioning strategies for elliptic partial differential operators and corresponding difference operators are compared by way of a model problem the technique considered here makes use of a selfadjoint positive definite operator b as the preconditioner for the nonselfadjoint convection diffusion operator a preconditioning by a s leading term plus a positive zeroth order term sigma i and optimizing over sigma is proposed the analysis of a model problem indicates that when a has large first order terms both the spectral and norm condition of b 1 a are minimized by sigma gg 0

acceleration of the schwarz method for elliptic problems in this paper we present a family of domain decomposition methods based on aitken like acceleration of the schwarz method which is an iterative procedure with linear rate of convergence this paper is a generalization of our method first introduced in 2000 that was restricted to cartesian grids we consider the finite volume fv approximation on general quadrangle meshes introduced by faille in 1992 we first present the so called aitken schwarz procedure for a linear differential operator this solver is a direct solver but its computational cost in the general case might be prohibitive second we introduce the steffensen schwarz variant which is an iterative domain decomposition solver that can be applied to linear and nonlinear problems we show that this iterative solver has reasonable numerical efficiency and can be applied successfully to several classes of linear and nonlinear elliptic problems however the salient feature of our method is that our algorithm has high tolerance for slow networks in the context of distributed parallel computing and is attractive to use with computer architectures for which performance is limited by the memory bandwidth rather than the flop performance of the cpu this is currently the case for most parallel computers using risc processor architecture such as beowulf systems

iterative solution methods for modeling multiphase flow in porous media fully implicitly we discuss several fully implicit techniques for solving the nonlinear algebraic system arising in an expanded mixed finite element or cell centered finite difference discretization of two and three phase porous media flow every outer nonlinear newton iteration requires solution of a nonsymmetric jacobian linear system two major types of preconditioners supercoarsening multigrid scmg and two stage are developed for the gmres iteration applied to the solution of the jacobian system the scmg reduces the three dimensional system to two dimensions using a vertical aggregation followed by a two dimensional multigrid the two stage preconditioners are based on decoupling the system into a pressure and concentration equations several pressure preconditioners of different types are described extensive numerical results are presented using the integrated parallel reservoir simulator ipars and indicate that these methods have low arithmetical complexity per iteration and good convergence rates

difference equations of q appell polynomials in this paper we study some properties of the q appell polynomials including the recurrence relations and the q difference equations which extend some known classical q 1 results we also provide the recurrence relations and the q difference equations for q bernoulli polynomials q euler polynomials q genocchi polynomials and for q hermite polynomials as special cases of q appell polynomials