harbourne schenck and seceleanu s conjecture abstract in 2 conjecture 5 5 2 harbourne schenck and seceleanu conjectured that for r 6 and all r 8 the artinian ideal i l 1 2 l r 1 2 k x 1 x r generated by the square of r 1 general linear forms l i fails the weak lefschetz property this paper is entirely devoted to prove this conjecture it is worthwhile to point out that half of the conjecture namely the case when the number of variables r is even was already proved in 5 theorem 6 1

a weakly overlapping domain decomposition preconditioner for the finite element solution of elliptic partial differential equations we present a new two level additive schwarz domain decomposition preconditioner which is appropriate for use in the parallel finite element solution of elliptic partial differential equations pdes as with most parallel domain decomposition methods each processor may be assigned one or more subdomains and the preconditioner is such that the processors are able to solve their own subproblem s concurrently the novel feature of the technique proposed here is that it requires just a single layer of overlap in the elements which make up each subdomain at each level of refinement and it is shown that this amount of overlap is sufficient to yield an optimal preconditioner some numerical experiments posed in both two and three space dimensions are included to confirm that the condition number when using the new preconditioner is indeed independent of the level of mesh refinement on the test problems considered

on the simplification of generalized conjugate gradient methods for nonsymmetrizable linear systems abstract the conjugate gradient cc method developed by hestenes and stiefel in 1952 can be effectively used to solve the linear system au b when a is symmetrixable in the sense that za and z are symmetric and positive definite spd for some z a number of generalizations of the cg method have been proposed by the authors and by others for handling the nonsymmetrizable case for many problems the amount of computer memory and computational effort required may be so large as to make the procedures not feasible truncated schemes are often used but in some cases the truncated methods may not converge even though the nontruncated schemes converge however it is well known that if a is symmetric the generalized cg schemes can be greatly simplified even though a is not spd so that the truncated schemes are equivalent to the nontruncated schemes in the present paper it is shown that such a simplification can occur if a nonsingular matrix h is available such that ha ath of course if a at then h can be taken to be the identity matrix it is also shown that such an h always exists however it may not be practical to compute h these results are used to derive three variations of the lanczos method for solving nonsymmetrizable systems two of the forms are well known but the third appears to be new an argument is given for choosing the third form over the other two

on optimal short recurrences for generating orthogonal krylov subspace bases we analyze necessary and sufficient conditions on a nonsingular matrix a such that for any initial vector r 0 an orthogonal basis of the krylov subspaces cal k n a r 0 is generated by a short recurrence orthogonality here is meant with respect to some unspecified positive definite inner product this question is closely related to the question of existence of optimal krylov subspace solvers for linear algebraic systems where optimal means the smallest possible error in the norm induced by the given inner product the conditions on a we deal with were first derived and characterized more than 20 years ago by faber and manteuffel siam j numer anal 21 1984 pp 352 362 their main theorem is often quoted and appears to be widely known its details and underlying concepts however are quite intricate with some subtleties not covered in the literature we are aware of our paper aims to present and clarify the existing important results in the context of the faber manteuffel theorem furthermore we review attempts to find an easier proof of the theorem and explain what remains to be done in order to complete that task

preconditioned mixed spectral element methods for elasticity and stokes problems this paper continues our study of efficient preconditioners for the indefinite systems obtained by discretizing the linear elasticity and stokes problems with mixed spectral methods in three dimensions our study is extended here in two new directions to include i more general multidomain mixed spectral element methods and ii a new class of block triangular preconditioners the stiffness matrices resulting from stokes and mixed elasticity problems have the structure of saddle point problems with a penalty term which is associated with the poisson ratio for elasticity problems or with stabilization techniques for stokes problems the main result of this paper shows that the convergence rate of the proposed preconditioned iterative methods is independent of the penalty parameter nu the number of spectral elements n and mildly dependent on the spectral degree n via the inf sup constant the number of iterations required by the block triangular preconditioner is much smaller than the one required by the block diagonal preconditioner on the other hand the block triangular preconditioner is no longer symmetric and has a small extra cost these results and several numerical experiments presented in the final section show that these algorithms are a practical and efficient strategy for the iterative solution of the indefinite systems arising from mixed spectral element discretizations of elliptic systems

minimal resolution of relatively compressed level algebras a relatively compressed algebra with given socle degrees is an artinian quotient a of a given graded algebra r c whose hilbert function is maximal among such quotients with the given socle degrees for us c is usually a general complete intersection and we usually require that a be level the precise value of the hilbert function of a relatively compressed algebra is open and we show that finding this value is equivalent to the froberg conjecture we then turn to the minimal free resolution of a level algebra relatively compressed with respect to a general complete intersection when the algebra is gorenstein of even socle degree we give the precise graded betti numbers when it is of odd socle degree we give good bounds on the graded betti numbers we also relate this case to the minimal resolution conjecture of musta for points on a projective variety finding the graded betti numbers is essentially equivalent to determining to what extent there can be redundant summands i e ghost terms in the minimal free resolution i e when copies of the same r t can occur in two consecutive free modules this is easy to arrange using koszul syzygies we show that it can also occur in more surprising situations that are not koszul using the equivalence to the froberg conjecture we show that in a polynomial ring where that conjecture holds e g in three variables the possible non koszul ghost terms are extremely limited finally we use the connection to the froberg conjecture as well as the calculation of the minimal free resolution for relatively compressed gorenstein algebras to find the minimal free resolution of general artinian almost complete intersections in many new cases this greatly extends previous work of the first two authors

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

an iterative solver for p version finite elements in three dimensions abstract a new iterative method for systems arising from p version finite elements is presented and computational experience reported for conforming solid serendipity elements of order up to p 8 and real world structures with 5000 elements and over 1 million degrees of freedom the method uses a domain decomposition technique with each element taken to be a subdomain and a coarse space consisting of p 1 elements to overcome the i o bottleneck only access to a fast matrix vector multiplication instead of stiffness data is needed during iterations in terms of wall clock time the method outperforms on ibm rs 6000 an earlier method running on cray xmp

conjugate gradient method for dual dual mixed formulations we deal with the iterative solution of linear systems arising from so called dual dual mixed finite element formulations the linear systems are of a two fold saddle point structure they are indefinite and ill conditioned we define a special inner product that makes matrices of the two fold saddle point structure after a specific transformation symmetric and positive definite therefore the conjugate gradient method with this special inner product can be used as iterative solver for a model problem we propose a preconditioner which leads to a bounded number of cg iterations numerical experiments for our model problem confirming the theoretical results are also reported

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

on the stability of viscous free surface flow supported by a rotating cylinder using an adaptive finite element fe scheme developed recently by the authors we shed new light on the long standing fundamental problem of the unsteady free surface stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field for supportable loads we observe that the steadystate is more readily attained for near maximal fluid loads on the cylinder than for significantly sub maximal loads for the latter we investigate large time dynamics by means of a finite difference approximation to the thin film equations which is also used to validate the adaptive fe simulations applied to the full stokes equations for these significantly sub maximal loads conversely by comparing results of the two methods we assess the validity of the thin film approximation as either the load is increased or the rotation rate of the cylinder is decreased results are presented on the independent effects of gravity surface tension and initial film thickness on the decay to steady state finally new numerical simulations of load shedding are presented

the weak lefschetz property of gorenstein algebras of codimension three associated to the apery sets abstract it has been conjectured that all graded artinian gorenstein algebras of codimension three have the weak lefschetz property over a field of characteristic zero in this paper we study the weak lefschetz property of associated graded algebras a of the apery set of m pure symmetric numerical semigroups generated by four natural numbers in 2010 bryant proved that these algebras are graded artinian gorenstein algebras of codimension three in a recent article guerrieri showed that if a is not a complete intersection then a is of form a r i with r k x y z and i x a y b x b z z c x a b y b y b z c where 1 b 1 max 1 b a 1 min b 1 c 1 and a c 2 we prove that a has the weak lefschetz property in the following cases max 1 b a c 1 b 1 and a b c 2 2 a 2 b c and a b c 1 b 1 one of a b c is at most five

a note on the generalized conjugate gradient method each iterate generated by the generalized conjugate gradient method of concus and golub lecture notes in economics and mathematical systems 134 springer verlag new york 1976 and widlund siam j numer anal 15 1978 pp 801 812 is shown to be the best approximation to the solution from a certain affine subspace although not from the natural afline krylov subspace this property is used to improve the error bounds given by widlund and hageman luk and young siam j numer anal 17 1980 pp 852 873

on nonsymmetric saddle point matrices that allow conjugate gradient iterations linear systems in saddle point form are usually highly indefinite which often slows down iterative solvers such as krylov subspace methods it has been noted by several authors that negating the second block row of a symmetric indefinite saddle point matrix leads to a nonsymmetric matrix mathcal a whose spectrum is entirely contained in the right half plane in this paper we study conditions so that mathcal a is diagonalizable with a real and positive spectrum these conditions are based on necessary and sufficient conditions for positive definiteness of a certain bilinear form with respect to which mathcal a is symmetric in case the latter conditions are satisfied there exists a well defined conjugate gradient cg method for solving linear systems with mathcal a we give an efficient implementation of this method discuss practical issues such as error bounds and present numerical experiments

balancing domain decomposition for mixed finite elements the rate of convergence of the balancing domain decomposition method applied to the mixed finite element discretization of second order elliptic equations is analyzed the balancing domain decomposition method introduced by mandel in 24 is a substructuring method that involves at each iteration the solution of a local problem with dirichlet data a local problem with neumann data and a coarse grid problem to propagate information globally and to insure the consistency of the neumann problems it is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size in both two and three dimensions and for elements of arbitrary order the bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains the key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the h 1 2 norm of an interplant of the boundary data computational results from a message passing parallel implementation on an intel delta machine demonstrate the scalability properties of the method and show almost optimal linear observed speed up for up to 64 processors

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

a deflated version of the conjugate gradient algorithm we present a deflated version of the conjugate gradient algorithm for solving linear systems the new algorithm can be useful in cases when a small number of eigenvalues of the iteration matrix are very close to the origin it can also be useful when solving linear systems with multiple right hand sides since the eigenvalue information gathered from solving one linear system can be recycled for solving the next systems and then updated

the method of conjugate residuals for solving the galerkin equations associated with symmetric positive semidefinite ill posed problems for the numerical solution of the galerkin equations associated with linear ill posed problems that are symmetric and positive semidefinite the method of conjugate residuals is considered an a posteriori stopping rule is introduced and associated estimates for the approximations are provided which are order optimal with respect to noise in the right hand side and with respect to the discretization error

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

optimal left and right additive schwarz preconditioning for minimal residual methods with euclidean and energy norms for the solution of non symmetric or indefinite linear systems arising from discretizations of elliptic problems two level additive schwarz preconditioners are known to be optimal in the sense that convergence bounds for the preconditioned problem are independent of the mesh and the number of subdomains these bounds are based on some kind of energy norm however in practice iterative methods which minimize the euclidean norm of the residual are used despite the fact that the usual bounds are non optimal i e the quantities appearing in the bounds may depend on the mesh size see x c cai j zou some observations on the l 2 convergence of the additive schwarz preconditioned gmres method numer linear algebra appl 9 2002 379 397 in this paper iterative methods are presented which minimize the same energy norm in which the optimal schwarz bounds are derived thus maintaining the schwarz optimality as a consequence bounds for the euclidean norm minimization are also derived thus providing a theoretical justification for the practical use of euclidean norm minimization methods preconditioned with additive schwarz both left and right preconditioners are considered and relations between them are derived numerical experiments illustrate the theoretical developments 2006 elsevier b v all rights reserved

preserving symmetry in preconditioned krylov subspace methods we consider the problem of solving a linear system a x b when a is nearly symmetric and when the system is preconditioned by a symmetric positive definite matrix m in the symmetric case we can recover symmetry by using m inner products in the conjugate gradient cg algorithm this idea can also be used in the nonsymmetric case and near symmetry can be preserved similarly like cg the new algorithms are mathematically equivalent to split preconditioning but do not require m to be factored better robustness in a specific sense can also be observed when combined with truncated versions of iterative methods tests show that this is more effective than the common practice of forfeiting near symmetry altogether

orthogonal error methods in this paper we discuss the conjugate gradient method generalized to nonsymmetric matrices constructing a class of algorithms that includes both the conjugate gradient algorithms as described in 12 and the orthogonal residual algorithms cf elman 10 we call this class orthogonal error algorithms and characterize the class of matrices for which finite term orthogonal error algorithms exist

on the weak lefschetz property for almost complete intersections generated by uniform powers of general linear forms abstract in 10 conjecture 6 6 migliore the first author and nagel conjectured that for all n 4 the artinian ideal i l 0 d l 2 n 1 d r k x 0 x 2 n generated by the d th powers of 2 n 2 general linear forms fails to have the weak lefschetz property if and only if d 1 this paper is entirely devoted to prove partially this conjecture more precisely we prove that r i fails to have the weak lefschetz property provided 4 n 8 d 4 or d 2 r 1 r 8 4 n 2 r r 2 1

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

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

on the choice of preconditioner for minimum residual methods for non hermitian matrices abstract we consider the solution of left preconditioned linear systems p 1 c x p 1 c where p c c n n are non hermitian c c n and c p and p 1 c are diagonalisable with spectra symmetric about the real line we prove that when p and c are self adjoint with respect to the same hermitian sesquilinear form the convergence of a minimum residual method in a particular nonstandard inner product applied to the preconditioned linear system is bounded by a term that depends only on the spectrum of p 1 c the inner product is related to the spectral decomposition of p when p is self adjoint with respect to a nearby hermitian sesquilinear form to c the convergence of a minimum residual method in this nonstandard inner product applied to the preconditioned linear system is bounded by a term involving the eigenvalues of p 1 c and a constant factor the size of this factor is related to the nearness of the hermitian sesquilinear forms numerical experiments indicate that for certain matrices eigenvalue dependent convergence is observed both for the nonstandard method and for standard gmres

investigation of a preconditioner for the p2 p1 finite element solution of stokes problem abstract discretizing the classical stokes problem in the p 2 p 1 finite element method gives rise to a symmetric indefinite stiffness matrix in this paper a suitable preconditioner matrix is considered to be used in the preconditioned conjugate residual and it is shown that it can produce very reasonable condition number

necessary and sufficient conditions for the simplification of generalized conjugate gradient algorithms recent papers by faber and manteuffel characterize the cases where orthodir a generalized conjugate gradient algorithm for solving complex non hermitian linear systems can be defined by a short recurrence formula and thus simplified in the present paper similar results are obtained for two other generalized conjugate gradient algorithms namely orthomin and orthores necessary and sufficient conditions on the coefficient matrix a are given so that simplification is possible illustrative examples are given

on the orthogonality of residual polynomials of minimax polynomial preconditioning this paper studies polynomials used in polynomial preconditioning for solving linear systems of equations optimum preconditioning polynomials are obtained by solving some constrained minimax approximation problems the resulting residual polynomials are referred to as the de boor rice and grcar polynomials it will be shown in this paper that the de boor rice and grcar polynomials are orthogonal polynomials over several intervals more specifically each de boor rice or grcar polynomial belongs to an orthogonal family but the orthogonal family varies with the polynomial this orthogonality property is important because it enables one to generate the minimax preconditioning polynomials by three term recursive relations some results on the convergence properties of certain preconditioning polynomials are also presented

conjugate gradient like algorithms for solving nonsymmetric linear systems this paper presents a unified formulation of a class of the conjugate gradient like algorithms for solving nonsymmetric linear systems the common framework is the petrov galerkin method on krylov subspaces we discuss some practical points concerning the methods and point out some of the interrelations between them 1 introduction in the recent few years a large number of generalizations of the conjugate gradient and conjugate residual methods which are very successful in solving symmetric positive definite linear systems have been proposed for solving nonsymmetric linear systems 3 6 5 7 12 in this paper we present an abstract framework which includes most of these methods and many new ones our goal is to understand the relationships among the methods and to synthesize consider the general linear system

leapfrog variants of iterative methods for linear algebraic equations abstract two iterative methods are considered richardson s method and a general second order method for both methods a variant of the method is derived for which only even numbered iterants are computed the variant is called a leapfrog method comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large in the case of richardson s method it is possible to express the final iterate in terms of only the initial approximation a variant of the iteration called the grand leap method in the case of the grand leap variant a set of parameters is required an algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for gaussian quadrature general algorithms to implement the leapfrog and grand leap methods are presented algorithms for the important special case of the chebyshev method are also given

abstract perturbed krylov methods abstract we introduce the framework of abstract perturbed krylov methods this is a new and unifying point of view on krylov subspace methods based solely on the matrix equation aq k f k q k 1 c k q k c k q k 1 c k 1 k e k t and the assumption that the matrix ck is unreduced hessenberg we give polynomial expressions relating the ritz vectors quasi orthogonal residual iterates and quasi minimal residual iterates to the starting vector q1 and the perturbation term fk the properties of these polynomials and similarities between them are analyzed in some detail the results suggest the interpretation of abstract perturbed krylov methods as additive overlay of several abstract exact krylov methods

preconditioned conjugate residual methods for mixed spectral discretizations of elasticity and stokes problems abstract mixed spectral methods applied to elasticity and stokes problems in three dimensions produce large symmetric indefinite systems of equations these systems are sparse and more ill conditioned than those produced by h version finite element methods these stiffness matrices have the structure of saddle point problems with a penalty term in this paper preconditioned iterative methods of conjugate residual type are introduced for these systems the convergence rates of the resulting algorithms are proven to be independent of the penalty parameter and mildly dependent on the spectral degree n several numerical experiments on three dimensional model problems confirm the results obtained

balancing domain decomposition for problems with large jumps in coefficients the balancing domain decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent the abstract theory introduced recently by the first named author is used to develop condition number bounds for conforming linear elements in two and three dimensions the bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains and grow only as the squared logarithm of the mesh size h computational experiments for two and three dimensional problems confirm the theory

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

a hybrid iterative method for symmetric indefinite linear systems hybrid iterative methods that combine a conjugate direction method with a simpler iteration scheme such as chebyshev or richardson iteration were first proposed in the 1950s the ease with which chebyshev and richardson iteration can be implemented efficiently on a large variety of computer architectures has in recent years lead to renewed interest in iterative methods that use chebyshev or richardson iteration this paper presents a new hybrid iterative method for the solution of linear systems of equations with a symmetric indefinite matrix our method combines the conjugate residual method with richardson iteration special attention is paid to the determination of two real intervals one on each side of the origin that contain most of the eigenvalues of the matrix these intervals are used to compute suitable iteration parameters for richardson iteration we also discuss when to switch between the methods the hybrid scheme typically uses the richardson method for most iterations and this reduces the number of arithmetic vector operations significantly compared with the number of arithmetic vector operations required when only the conjugate residual method is used computed examples illustrate the competitiveness of the hybrid scheme

generalized conjugate directions abstract this paper presents a simple unifying framework for a wide class of conjugate directions algorithms whose iterates minimize some quadratic functional over a subspace our approach is motivated by its advantages for nonlinear minimization but the purpose of this paper is to present the greatly simplified convergence analysis that results for the linear case

on improving linear solver performance a block variant of gmres the increasing gap between processor performance and memory access time warrants the re examination of data movement in iterative linear solver algorithms for this reason we explore and establish the feasibility of modifying a standard iterative linear solver algorithm in a manner that reduces the movement of data through memory in particular we present an alternative to the restarted gmres algorithm for solving a single right hand side linear system ax b based on solving the block linear system ax b algorithm performance i e time to solution is improved by using the matrix a in operations on groups of vectors experimental results demonstrate the importance of implementation choices on data movement as well as the effectiveness of the new method on a variety of problems from different application areas

schwarz methods to symmetrize or not to symmetrize a preconditioning theory is presented which establishes sufficient conditions for multiplicative and additive schwarz algorithms to yield self adjoint positive definite preconditioners it allows for the analysis and use of nonvariational and nonconvergent linear methods as preconditioners for conjugate gradient methods and it is applied to domain decomposition and multigrid it is illustrated why symmetrizing may be a bad idea for linear methods it is conjectured that enforcing minimal symmetry achieves the best results when combined with conjugate gradient acceleration also it is shown that the absence of symmetry in the linear preconditioner is advantageous when the linear method is accelerated by using the bi cgstab method numerical examples are presented for two test problems which illustrate the theory and conjectures

combinatorics and geometry of power ideals we investigate ideals in a polynomial ring which are generated by powers of linear forms such ideals are closely related to the theories of fat point ideals cox rings and box splines we pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement a we prove that their hilbert series are determined by the combinatorics of a and can be computed from its tutte polynomial we also obtain formulas for the hilbert series of certain closely related fat point ideals and zonotopal cox rings our work unifies and generalizes results due to dahmen micchelli holtz ron postnikov shapiro shapiro and sturmfels xu among others it also settles a conjecture of holtz ron on the spline interpolation of functions on the lattice points of a zonotope

the weak and strong lefschetz properties for artinian k algebras abstract let a i 0ai be a standard graded artinian k algebra where chark 0 then a has the weak lefschetz property if there is an element l of degree 1 such that the multiplication l a i a i 1 has maximal rank for every i and a has the strong lefschetz property if l d a i a i d has maximal rank for every i and d the main results obtained in this paper are the following 1 every height three complete intersection has the weak lefschetz property our method surprisingly uses rank two vector bundles on p 2 and the grauert mulich theorem 2 we give a complete characterization including a concrete construction of the hilbert functions that can occur for k algebras with the weak or strong lefschetz property and the characterization is the same one 3 we give a sharp bound on the graded betti numbers achieved by our construction of artinian k algebras with the weak or strong lefschetz property and fixed hilbert function this bound is again the same for both properties some hilbert functions in fact force the algebra to have the maximal betti numbers 4 every artinian ideal in k x y possesses the strong lefschetz property this is false in higher codimension

the second generation feti methods and their application to the parallel solution of large scale linear and geometrically non linear structural analysis problems the feti algorithms are a family of numerically scalable domain decomposition methods they have been designed in the early 1990s for solving iteratively and on parallel machines large scale systems of equations arising from the finite element discretization of solid mechanics structural engineering structural dynamics and acoustic scattering problems and for analyzing complex structures obtained from the assembly of substructures with incompatible discrete interfaces in this paper we present the second generation of these methods that operate more efficiently on large numbers of subdomains offer greater robustness better performance and more flexibility for implementation on a wider variety of computational platforms we also report on the application and performance of these methods for the solution of geometrically non linear structural analysis problems we discuss key aspects of their implementation on shared and distributed memory parallel processors benchmark them against optimized direct sparse solvers and highlight their potential with the solution of large scale structural mechanics problems with several million degrees of freedom

two level domain decomposition methods with lagrange multipliers for the fast iterative solution of acoustic scattering problems we present two different but related lagrange multiplier based domain decomposition dd methods for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed methods are essentially two distinct extensions of the regularized finite element tearing and interconnecting feti method to indefinite or complex problems the first method employs a single lagrange multiplier field to glue the local solutions at the subdomain interface boundaries the second method employs two lagrange multiplier fields for that purpose the key ingredients of both of these feti methods are the regularization of each subdomain matrix by a complex lumped mass matrix defined on the subdomain interface boundary and the preconditioning of the global interface problem by a coarse second level problem constructed with planar waves we show numerically that both methods are scalable with respect to the mesh size the subdomain size and the wavenumber but that the feti method with a single lagrange multiplier field labeled feti h h for helmholtz in this paper delivers superior computational performances we apply the feti h method to the parallel solution on a 24 processor origin 2000 of an acoustic scattering problem with a submarine shaped obstacle and report performance results that highlight the unique efficiency of this dd method for the solution of high frequency acoustic scattering problems

on demazure polynomials abstract the dimension of the demazure module e w for an affine kac moody algebra is a polynomial in called a demazure polynomial in this article we prove that it is harmonic for the algebra sl n we define a subset e of w of non null density we calculate explicitly all the demazure polynomials for this subset

a preconditioned conjugate gradient method for nonselfadjoint or indefinite orthogonal spline collocation problems we study the computation of the orthogonal spline collocation solution of a linear dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form lu sum a ij x u x ix j sum b i x u x i c x u we apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator and prove that the resulting algorithm has a convergence rate independent of the partition step size we solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm on a uniform n n partition the cost of the algorithm for computing the collocation solution within a tolerance epsilon is o n 2 ln n ln epsilon

numerical experiments of some krylov subspace methods for black oil model this paper discusses preconditioned krylov subspace methods for solving large scale linear systems that originate from oil reservoir numerical simulations two types of preconditioners one being based on an incomplete lu decomposition and the other being based on iterative algorithms are used together in a combination strategy in order to achieve an adaptive and efficient preconditioner numerical tests show that different krylov subspace methods combining with appropriate preconditioners are able to achieve optimal performance

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

an inexact newton algorithm for solving the tokamak edge plasma fluid equations on a multiply connected domain newton s method is combined with a preconditioned conjugate gradient like algorithm and finite volume discretization to solve the steady state two dimensional tokamak edge plasma fluid equations a numerical evaluation of the jacobian is employed mesh sequencing pseudo transient continuation and adaptive damping are used to increase the radius of convergence the computations are performed on a multiply connected curvilinear geometry in a fully coupled manner the preconditioned conjugate gradient like algorithm is shown to have a significant storage advantage over the previously used banded gaussian elimination while maintaining the excellent convergence characteristics of the overall algorithm simulations of a high recycling divertor and a gaseous divertor on the diii d tokamak geometry are used to demonstrate algorithm performance

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

changing the norm in conjugate gradient type algorithms a conjugate gradient type method for solving ax b or more precisely an orthogonal error method operatorname oe b c a is determined by a formal inner product matrix b and a left preconditioning matrix c relations are pointed out between quantities such as iterates residuals direction vectors and recurrence coefficients of orthogonal error methods with different but in a particular way related matrices b namely for operatorname oe b ca k c a k 0 1 ldots and for operatorname oe a h c h k b c a k 0 1 ldots the relations for the first sequence of methods have to do with rutishauser s lr algorithm those for the second one are based on a generalization of the schonauer weiss smoothing algorithm relevant for practice are the cases k 0 1 for b ca and b ca h

preconditioning saddle point systems with applications in optimization saddle point systems arise in many applications areas in fact in any situation where an extremum principle arises with constraints the stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from pdes and in the area of optimization such problems are ubiquitous in this paper we present a framework into which many well known methods for solving saddle point systems fit based on this description we show how new approaches for the solution of saddle point systems arising in optimization can be derived from the bramble pasciak conjugate gradient approach widely used in pdes and more recent generalizations thereof in particular we derive a class of new solution methods based on the use of preconditioned conjugate gradients in nonstandard inner products and demonstrate how these can be understood through more standard machinery we show connections to constraint preconditioning and give the results of numerical computations on a number of standard optimization test examples

numerical relativity as a tool for computational astrophysics abstract the astrophysics of compact objects which requires einstein s theory of general relativity for understanding phenomena such as black holes and neutron stars is attracting increasing attention in general relativity gravity is governed by an extremely complex set of coupled nonlinear hyperbolic elliptic partial differential equations the largest parallel supercomputers are finally approaching the speed and memory required to solve the complete set of einstein s equations for the first time since they were written over 80 years ago allowing one to attempt full 3d simulations of such exciting events as colliding black holes and neutron stars in this paper we review the computational effort in this direction and discuss a new 3d multi purpose parallel code called cactus for general relativistic astrophysics directions for further work are indicated where appropriate

polytopes associated to demazure modules of symmetrizable kac moody algebras of rank two let 1 2 be the two fundamental weights of a symmetrizable kac moody algebra g of rank two hence necessarily affine or finite and an element of the weyl group in this paper we construct polytopes p 1 p 2 rl and a linear map rl h such that for any dominant weight k1 1 k2 2 we have char e e e x where the sum is over all the integral points x of the polytope k1p 1 k2p 2 furthermore we show that there exists a flat deformation of the schubert variety s into the toric variety defined by p 1 p 2

on the eigenvalues of a class of saddle point matrices we study spectral properties of a class of block 2 2 matrices that arise in the solution of saddle point problems these matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system we give conditions for having a positive real spectrum and for ensuring diagonalizability of the matrix in particular we show that these properties hold for the discrete stokes operator and we discuss the implications of our characterization for augmented lagrangian formulations for krylov subspace solvers and for certain types of preconditioners

bicr type methods for families of shifted linear systems the shifted linear systems with non hermitian matrices often arise from the numerical solutions for time dependent pdes computing the large scale eigenvalue problems control theory and so on in the present paper we develop two shifted variants of bicr type methods for solving such linear systems these variants of bicr type methods take advantage of the shifted structure so that the number of matrix vector multiplications and the number of inner products are the same as a single linear system finally extensive numerical examples are reported to illustrate the performance and effectiveness of the proposed methods

on conjugate gradient type methods and polynomial preconditioners for a class of complex non hermitian matrices we consider conjugate gradient type methods for the solution of large linear systemsa x b with complex coefficient matrices of the typea t i i wheret is hermitian and a real scalar three different conjugate gradient type approaches with iterates defined by a minimal residual property a galerkin type condition and an euclidean error minimization respectively are investigated in particular we propose numerically stable implementations based on the ideas behind paige and saunder s symmlq and minres for real symmetric matrices and derive error bounds for all three methods it is shown how the special shift structure ofa can be preserved by using polynomial preconditioning and results on the optimal choice of the polynomial preconditioner are given also we report on some numerical experiments for matrices arising from finite difference approximations to the complex helmholtz equation

a taxonomy for conjugate gradient methods the conjugate gradient method of hestenes and stiefel is an effective method for solving large sparse hermitian positive definite hpd systems of linear equations ax b generalizations to non hpd matrices have long been sought the recent theory of faber and manteuffel gives necessary and sufficient conditions for the existence of a cg method this paper uses these conditions to develop and organize such methods it is shown that any cg method for ax b is characterized by an hpd inner product matrix b and a left preconditioning matrix c at each step the method minimizes the b norm of the error over a krylov subspace this characterization is then used to classify known and new methods finally it is shown how eigenvalue estimates may be obtained from the iteration parameters generalizing the well known connection between cg and lanczos such estimates allow implementation of a stopping criterion based more nearly on the true error

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

performance of certain iterative methods in solving implicit difference schemes for 2 d navier stokes equations the performance of three iterative methods local sor preconditioned generalized conjugate gradient methods such as orthomin 1 and the gcg ls method with preconditioning matrices derived by incomplete modified and unmodified pointwise factorizations of the matrix corresponding to implicit finite difference schemes for the 2 d navier stokes equations is compared the relative merits of these methods in terms of cpu time average number of iterations with respect to the reynolds number and discretization parameters are emphasized

an expanded mixed finite element approach via a dual dual formulation and the minimum residual method abstract we apply an expanded mixed finite element method which introduces the gradient as a third explicit unknown to solve a linear second order elliptic equation in divergence form instead of using the standard dual form we show that the corresponding variational formulation can be written as a dual dual operator equation we establish existence and uniqueness of solution for the continuous and discrete formulations and provide the corresponding error analysis by using raviart thomas elements in addition we show that the corresponding dual dual linear system can be efficiently solved by a preconditioned minimum residual method some numerical results illustrating this fact and the rate of convergence of the mixed finite element method are also provided

fast iterative solution of stabilised stokes systems part i using simple diagonal preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems the use of stabilisation methods for convenient but unstable mixed elements introduces stabilisation parameters we show how these can be chosen to obtain rapid iterative convergence we propose a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems describe the effects of stabilisation on the algebraic structure of the discrete stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends here we discuss the simple case of diagonal preconditioning our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable we demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete laplacian are approachable for the stokes problem