solvers for mathcal o n electronic structure in the strong scaling limit we present a hybrid openmp charm tt framework for solving the mathcal o n self consistent field eigenvalue problem with parallelism in the strong scaling regime p gg n where p is the number of cores and n is a measure of system size i e the number of matrix rows columns basis functions atoms molecules etc this result is achieved with a nested approach to spectral projection and the sparse approximate matrix multiply bock and challacombe siam j sci comput 35 2013 pp c72 c98 and involves a recursive task parallel algorithm often employed by generalized n body solvers to occlusion and culling of negligible products in the case of matrices with decay employing classic technologies associated with generalized n body solvers including overdecomposition recursive task parallelism orderings that preserve locality and persistence based load balancing we obtain scaling beyond hundreds of cores per molecule for small water clusters h 2 o n n in 30

ordering anisotropy and factored sparse approximate inverses we consider ordering techniques to improve the performance of factored sparse approximate inverse preconditioners concentrating on the ainv technique of m benzi and m t r u ma several practical existing unweighted orderings are considered along with a new algorithm minimum inverse penalty mip that we propose we show how good orderings such as these can improve the speed of preconditioner computation dramatically and also demonstrate a fast and fairly reliable way of testing how good an ordering is in this respect our test results also show that these orderings generally improve convergence of krylov subspace solvers but may have difficulties particularly for anisotropic problems we then argue that weighted orderings which take into account the numerical values in the matrix will be necessary for such systems after developing a simple heuristic for dealing with anisotropy we propose several practical algorithms to implement it while these show promise we conclude a better heuristic is required for robustness

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

subvarieties of suc 2 and 2theta divisors in the jacobian let suc 2 l denote the projective moduli variety of semistable rank 2 vector bundles with determinant l pic c on a smooth curve c of genus g 2 and suppose that deg l is even it is well known that on the one hand the singular locus of suc 2 l is isomorphic to the kummer variety of the jacobian and on the other hand that when c is nonhyperelliptic suc 2 o has an injective morphism into the linear series 2 on the jacobian jg 1 c which restricts to the kummer embedding a 7 a a on the singular locus dually suc 2 k injects into the linear series l on j c where l o 2 for any theta characteristic and again this map restricts to the kummer map jg 1 c 2 l on the singular locus this map to projective space the two cases are of course isomorphic comes from the complete series on the ample generator of the picard group and at least for a generic curve is an embedding of the moduli space moreover its image contains much of the geometry studied in connection with the schottky problem notably the configuration of prym kummer varieties in this paper we explore a little of the interplay via this embedding between the geometry of vector bundles and the geometry of 2 divisors on the vector bundle side we are principally concerned with the brill noether loci wr suc 2 k defined by the condition h e r on stable bundes e these are analogous to the very classical varieties w r g 1 jg 1 c unlike the line bundle theory however general results connectedness dimension smoothness and so on are not known for the varieties wr see 6 on the 2 side we shall consider the fay trisecants of the 2 embedded kummer variety and the subseries p 00 l consisting of divisors having multiplicity 4 at the origin this subseries is known to be important in the study of principally polarised abelian varieties 10 in the jacobian of a curve its base locus is the surface c c j c plus a pair of isolated points

a robust preconditioner with low memory requirements for large sparse least squares problems this paper describes a technique for constructing robust preconditioners for the cgls method applied to the solution of large and sparse least squares problems the algorithm computes an incomplete ldlt factorization of the normal equations matrix without the need to form the normal matrix itself the preconditioner is reliable pivot breakdowns cannot occur and has low intermediate storage requirements numerical experiments illustrating the performance of the preconditioner are presented a comparison with incomplete qr preconditioners is also included

global simpler gmres for nonsymmetric systems with multiple right hand sides a new version of the simpler gmres for solving nonsymmetric linear system of equations with multiple right hand sides is presented the main advantage over the simpler block gmres is that the new method requires less storage and computational work per cycle preconditioning techniques can also be used to further accelerate the convergence of the new method finally numerical examples are given to illustrate the effectiveness of the proposed method and its some preconditioned versions

a quasi minimal residual variant of idrstab using the residual smoothing technique abstract quasi minimal residual qmr type methods have been developed for improving the oscillations in the residual norms of the bi conjugate gradient bi cg method and the hybrid bi cg methods the idrstab method which combines the induced dimension reduction idr s method with higher order stabilizing polynomials is often more effective than idr s and the hybrid bi cg methods but the residual norms oscillate in this paper we design a qmr variant of idrstab by using a residual smoothing technique in order to obtain smoother convergence behavior numerical experiments show the effectiveness of our qmr variant of idrstab

eigenvalue estimates of an indefinite block triangular preconditioner for saddle point problems in this paper an indefinite block triangular preconditioner for symmetric saddle point problems is discussed the bounds for all the real eigenvalues of the preconditioned matrix are provided and the bounds for the real and imaginary parts of all the complex eigenvalues of the preconditioned matrix are also provided the corresponding theoretical results of paper by bergamaschi 2012 are improved numerical experiments are reported to confirm our theoretical results

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

decay rates of inverses of banded m matrices that are near to toeplitz matrices the decaying behavior of inverses of positive definite band matrices is analysed for m matrices that are in some sense close to toeplitz matrices estimates based on the factorization are derived that are better than existing ones in particular for nonsymmetric matrices some examples are given

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

preconditioning sparse nonsymmetric linear systems with the sherman morrison formula let ax b be a large sparse nonsymmetric system of linear equations a new sparse approximate inverse preconditioning technique for such a class of systems is proposed we show how the matrix a0 1 a 1 where a0 is a nonsingular matrix whose inverse is known or easy to compute can be factorized in the form u omega v t using the sherman morrison formula when this factorization process is done incompletely an approximate factorization may be obtained and used as a preconditioner for krylov iterative methods for a0 sin where in is the identity matrix and s is a positive scalar the existence of the preconditioner for m matrices is proved in addition some numerical experiments obtained for a representative set of matrices are presented results show that our approach is comparable with other existing approximate inverse techniques

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

comparison criteria for parallel orderings in ilu preconditioning this paper introduces block red black ordering in a general three dimensional form for parallel incomplete lu ilu preconditioning this parallel ordering method is designed to attain fast convergence with reduced synchronization among processors in parallelized forward and backward substitutions in this method the grid nodes are divided into blocks and red black ordering is applied to them since blocks with identical colors never have a data dependency the blocks in each color can be processed in parallel moreover in order to compare parallel orderings we propose a new tool for investigating orderings on convergence the analytic index for convergence which is based on the remainder matrix is easily computed and has a unique value for a fixed ordering nodes are classified into seven groups and the effect of each group on convergence is estimated numerical tests using a three dimensional problem confirm the validity of both proposed parallel ordering and analytic convergence comparison methods

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

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

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

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

a block version of left looking ainv preconditioner with one by one or two by two block pivots abstract in this paper we present a block format of left looking ainv preconditioner for a nonsymmetric matrix this preconditioner has block 1 1 or 2 2 pivot entries it is introduced based on a block format of gaussian elimination process which has been studied in 14 we have applied the multilevel nested dissection reordering as the preprocessing and have compared this block preconditioner by the plain left looking ainv preconditioner if we mix the multilevel nested dissection by the maximum weighted matching process then the numerical experiments indicate that the number of 2 2 pivot entries in the block preconditioner will grow up in this case the block preconditioner makes gmres method convergent in a smaller number of iterations in the numerical section we have also compared the ilut and block left looking ainv preconditioners

robust preconditioners for optimal control with time periodic parabolic equation abstract based on a new approximation of a schur complement matrix and some preconditioning techniques considered in earlier papers three robust preconditioners are presented for solving the kkt systems arising from discretizing optimal control problem with time periodic parabolic equation the eigenvalue bounds of the preconditioned matrices and some new theoretical results are obtained moreover selection of quasi optimal parameter is studied and an explicit expression of the quasi optimal parameter is given numerical examples are given to illustrate the effectiveness of the three proposed optimized preconditioners

fourier analysis of frequency filtering decomposition preconditioners in this paper frequency filtering decomposition ffd preconditioner is analyzed by the approach of fourier analysis the condition number estimation of a preconditioned 2 d model problem is presented analysis reveals that condition number of the preconditioned matrix grows like o h 1 with h be the mesh size by using the framework of ffd a stabilized frequency filtering decomposition sffd method is proposed and analyzed by fourier method results show that sffd preconditioner is superior to ffd preconditioner in the sense that k m s f f d 1a k m f f d 1a numerical tests are performed to illustrate the theoretical results and the superiority of sffd preconditioner

contraction and optimality properties of an adaptive legendre galerkin method the multi dimensional case we analyze the theoretical properties of an adaptive legendre galerkin method in the multidimensional case after the recent investigations for fourier galerkin methods in a periodic box and for legendre galerkin methods in the one dimensional setting the present study represents a further step towards a mathematically rigorous understanding of adaptive spectral hp hp discretizations of elliptic boundary value problems the main contribution of the paper is a careful construction of a multidimensional riesz basis in h 1 h1 based on a quasi orthonormalization procedure this allows us to design an adaptive algorithm to prove its convergence by a contraction argument and to discuss its optimality properties in the sense of non linear approximation theory in certain sparsity classes of gevrey type

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

overlapped multicolor milu preconditioning milu preconditioned iterative methods are useful for solving large sparse linear systems which arise from the finite difference approximations of three dimensional second order elliptic partial differential equations in these schemes the milu preconditioning which accelerates the convergence is the most difficult part to vectorize on vector supercomputers currently the reordering techniques like the multicolor ordering strategy are commonly used to obtain sufficiently long vector lengths however these reordering techniques deteriorate the convergence compared with the standard milu preconditioner and gustafsson s acceleration does not work well for them in this paper a new preconditioning is proposed and its advantages compared with the multicolor and hyperplane orderings are shown through experiments on nec vector supercomputer sx 3

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

singular sturmian comparison theorems for linear hamiltonian systems abstract in this paper we prove singular comparison theorems on unbounded intervals for two nonoscillatory linear hamiltonian systems satisfying the sturmian majorant condition and the legendre condition at the same time we do not impose any controllability condition the results are phrased in terms of the comparative index and the numbers of proper focal points of the minimal principal solutions of these systems at both endpoints of the considered interval the main idea is based on an application of new transformation theorems for principal and antiprincipal solutions at infinity and on new limit properties of the comparative index involving these solutions this work generalizes the recently obtained sturmian separation theorems on unbounded intervals for one system by the authors 2019 as well as the sturmian comparison theorems and transformation theorems on compact intervals by j elyseeva 2016 and 2018 we note that all the results are new even in the completely controllable case

refining an approximate inverse direct methods have made remarkable progress in the computational efficiency of factorization algorithms during the last three decades the advances in graph theoretic algorithms have not received enough attention from the iterative methods community for example we demonstrate how symbolic factorization algorithms from direct methods can accelerate the computation of a factored approximate inverse preconditioner for very sparse preconditioners however a reformulation of the algorithm with outer products can exploit even more zeros to good advantage we also explore the possibilities of improving cache efficiency in the application of the preconditioner through reorderings the article ends by proposing a block version of the algorithm for further gains in efficiency and robustness

a matrix free preconditioner for sparse symmetric positive definite systems and least squares problems we analyze and discuss matrix free and limited memory preconditioners for sparse symmetric positive definite systems and normal equations of large and sparse least squares problems the preconditioners are based on a partial cholesky factorization and can be coupled with a deflation strategy the construction of the preconditioners requires only matrix vector products is breakdown free and does not need to form the matrix the memory requirements of the preconditioners are defined in advance and they do not depend on the number of nonzero entries in the matrix when eigenvalue deflation is used the preconditioners turn out to be suitable for solving sequences of slowly changing systems or linear systems with different right hand sides numerical results are provided including the case of sequences arising in nonnegative linear least squares problems solved by interior point methods

a two level ilu preconditioner for electromagnetic applications computational electromagnetics based on the solution of the integral form of maxwell s equations with boundary element methods require the solution of large and dense linear systems for large scale problems the solution is obtained by using iterative krylov type methods provided that a fast method for performing matrix vector products is available in addition for ill conditioned problems some kind of preconditioning technique must be applied to the linear system in order to accelerate the convergence of the iterative method and improve its performance for many applications it has been reported that incomplete factorizations often suffer from numerical instability due to the indefiniteness of the coefficient matrix in this context approximate inverse preconditioners based on frobenius norm minimization have emerged as a robust and highly parallel alternative in this work we propose a two level ilu preconditioner for the preconditioned gmres method the computation and application of the preconditioner is based on graph partitioning techniques numerical experiments are presented for different problems and show that with this technique it is possible to obtain robust ilu preconditioners that perform competitively compared with frobenius norm minimization preconditioners

fast computational methods for locating fold points for the power flow equations voltage collapse in a power system can occur following a progressive decline in voltage magnitude at the system buses and mathematically this phenomenon is associated with a fold bifurcation point occurring in the nonlinear algebraic equations used to model the power system in this paper we first discuss some of the methods used to speed up the process of detecting a fold bifurcation focussing on designing test function methods to predict a performance index we then discuss some iterative methods that can be used to improve the newton iterations in particular we present new and efficient preconditioners of the two level type for the jacobian matrix numerical results are given using standard ieee test bus systems

orderings for incomplete factorization preconditioning of nonsymmetric problems numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned krylov subspace methods for the solution of nonsymmetric linear systems is shown the preconditioners used in this study are different variants of incomplete factorizations it is shown that certain reorderings for direct methods such as reverse cuthill mckee can be very beneficial the benefit can be seen in the reduction of the number of iterations and also in measuring the deviation of the preconditioned operator from the identity

incomplete hyperbolic gram schmidt based preconditioners for the solution of large indefinite least squares problems abstract we propose to precondition the cgils method based on the incomplete hyperbolic gram schmidt methods for the solution of the large and sparse indefinite least squares ils problem which requires minimization of an indefinite quadratic form numerical experiments show that the incomplete hyperbolic classical modified gram schmidt ihcgs ihmgs preconditioners can generally greatly reduce the number of iterations and ihmgs has a better quality for ill conditioned problems but may be expensive in computation cost

parallelizable approximate solvers for recursions arising in preconditioning abstract for the recursions used in the modified incomplete lu preconditioner namely the construction of incomplete decomposition the forward elimination and the back substitution parallelizable approximate solvers are introduced the analysis shows that the solutions of the recursions depend only weakly on their initial conditions which may be interpreted to indicate that the inexact solution is close in some sense to the exact one the method is based on a domain decomposition approach suitable for parallel implementations on message passing architectures it requires a small number of communication steps per preconditioned iteration independent of the number of subdomains and the size of the problem the overlapping subdomains are either squares suitable for mesh connected arrays of processors or defined by the data flow rule of the recursions suitable for line connected arrays of simd or vector processors numerical examples show that in both cases the overhead in the number of preconditioned iterations required for convergence is small in comparison with the speedup gained

sturmian comparison theorems for completely controllable linear hamiltonian systems in singular case abstract in this paper we consider two linear hamiltonian differential systems on an open unbounded interval we assume that the systems are completely controllable and satisfy the sturmian majorant condition and the legendre condition we derive a singular sturmian comparison theorem for the case when the minorant system has a solution which is principal at both endpoints of the considered interval the main result is new even for the second order differential equations and it generalizes the singular comparison theorem obtained by aharonov and elias 2010 we illustrate our new theory by several examples

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

factorized approximate inverses with adaptive dropping this paper presents a new approach to constructing factorized approximate inverses for a symmetric and positive definite matrix a the proposed strategy is based on adaptive dropping that reflects the quality of preserving the relation uz i between the direct factor u and the inverse factor z satisfying a u tu and a 1 zz t an important part of the approach is column pivoting used to minimize the growth of the condition number of leading principal submatrices of u that occurs explicitly in the dropping criterion numerical experiments demonstrate that the resulting approximate inverse factorization is robust as a preconditioner for solving large and sparse systems of linear equations

an approach to making spai and psai preconditioning effective for large irregular sparse linear systems we investigate the adaptive sparse approximate inverse spai and adaptive power sparse approximate inverse psai preconditioning procedures and shed light on two important features of them i for the large linear system ax b with a irregular sparse i e with a having s relatively dense columns spai may be very costly to implement and the resulting sparse approximate inverses may be ineffective for preconditioning psai can be effective for preconditioning but may require excessive storage and be unacceptably time consuming ii the situation is improved drastically when a is regular sparse that is all its columns are sparse in this case both spai and psai are efficient moreover spai and especially psai are more likely to construct effective preconditioners motivated by these features we propose an approach to making spai and psai more practical for ax b with a irregular sparse we first split a into a regular sparse tilde a and a matrix of low rank s then exploitin

on the decay of the inverse of matrices that are sum of kronecker products abstract decay patterns of matrix inverses have recently attracted considerable interest due to their relevance in numerical analysis and in applications requiring matrix function approximations in this paper we analyze the decay pattern of the inverse of banded matrices of the form s m i n i n m where m is tridiagonal symmetric and positive definite i n is the identity matrix and stands for the kronecker product it is well known that the inverses of banded matrices exhibit an exponential decay pattern away from the main diagonal however the entries in s 1 show a non monotonic decay which is not caught by classical bounds by using an alternative expression for s 1 we derive computable upper bounds that closely capture the actual behavior of its entries we also show that similar estimates can be obtained when m has a larger bandwidth or when the sum of kronecker products involves two different matrices numerical experiments illustrating the new bounds are also reported

multilevel ilu with reorderings for diagonal dominance this paper presents a preconditioning method based on combining two sided permutations with a multilevel approach the nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix the method can be regarded as a complete pivoting version of the incomplete lu factorization this leads to an effective incomplete factorization preconditioner for general nonsymmetric irregularly structured sparse linear systems

an iteration method for solving the linear system ax b abstract in this paper based on a convergence splitting of the matrix a we present an inner outer iteration method for solving the linear system a x b we analyze the overall convergence of this method without any other restriction on its parameters moreover we give the accelerated inner outer iteration method and discuss how to apply the inner outer iterations as a preconditioner for the krylov subspace methods the inner outer iteration method can also be used for the solution of a x b c finally several numerical examples are given to validate the performance of our proposed algorithms

an efficient preconditioned variant of the pss preconditioner for generalized saddle point problems abstract for the generalized saddle point problems from the stokes equations we propose an efficient preconditioned variant of the positive semidefinite and skew hermitian pvpss preconditioner the new preconditioner is established by adopting matrix preconditioning strategy and relaxation technique for the pss one derived by pan et al 2006 compared with the pss one the pvpss preconditioner is much closer to the coefficient matrix and easier to be implemented if proper preconditioning matrices are adopted we prove the convergence of the pvpss iteration method under some restrictions and discuss the spectral properties of the pvpss preconditioned matrix meanwhile the implementation and a practical way to choose the parameter of the pvpss preconditioner are discussed comparisons between the pvpss preconditioner and some existing ones are also given numerical experiments are carried out to illustrate that the proposed preconditioner is effective for the generalized saddle point problems and outperforms several other commonly used ones

an inexact relaxed dpss preconditioner for saddle point problem based on the relaxed deteriorated positive definite and skew hermitian splitting dpss preconditioner in this paper we proposed a class of relaxed deteriorated positive definite and skew hermitian splitting rdpss preconditioner for solving the saddle point problem the proposed rdpss preconditioner is a technical modification of the deteriorated positive definite and skew hermitian splitting dpss preconditioner 36 the pss preconditioner is a straightforward application of the positive definite and skew hermitian splitting pss iteration method for solving non hermitian positive definite linear systems initially established by bai et al 37 numerical results have shown that the proposed rdpss preconditioner is advantageous over the existing dpss preconditioner

a variant of the hss preconditioner for complex symmetric indefinite linear systems abstract using the equivalent block two by two real linear systems we establish a new variant of the hermitian and skew hermitian splitting hss preconditioner for a class of complex symmetric indefinite linear systems the new preconditioner is not only a better approximation to the block two by two real coefficient matrix than the well known hss preconditioner but also resulting in an unconditional convergent fixed point iteration the quasi optimal parameter which minimizes an upper bound of the spectral radius of the iteration matrix is analyzed eigen properties and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed finally two numerical examples are provided to show the efficiency of the new preconditioner

analysis on block diagonal and triangular preconditioners for a pml system of an electromagnetic scattering problem abstract we shall propose several block triangular preconditioners for a pml system of an electromagnetic wave scattering problem and analyze the spectral behavior of the preconditioned systems when the pml system is discretized by edge element methods it results in a discrete system with its stiffness matrix being complex symmetric but indefinite which can be formulated into a real symmetric but indefinite saddle point system in order to preserve the symmetry of the coefficient matrix we present block triangular preconditioners with two sided preconditioning for the discrete pml system we will estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively on the other hand one may also like to apply some iteration methods for nonsymmetric linear systems in applications although the discrete systems are symmetric to this end we propose a block triangular preconditioner to precondition the systems only from one side and analyze the spectrum of the preconditioned systems in addition we have also established a spectral estimate of the preconditioned system by an effective preconditioner that was recently developed in literature numerical experiments are presented to demonstrate the effectiveness and robustness of these new preconditioners and our theoretical predictions on the spectral bounds of the preconditioned systems

singular sturm comparison theorems sturm s comparison theorem is discussed for differential equations whose coefficients are continuous in an open finite or infinite interval but the coefficients cannot necessarily be extended continuously to the boundary points of the interval

sharp error bounds of some krylov subspace methods for non hermitian linear systems1project 19601036 supported by the national natural science foundation of china 1 the convergence of the krylov subspace methods e g full orthogonal method fom and generalized minimal residual method gmres etc for solving large non hermitian linear systems is studied in a unified and detailed way when the coefficient matrix is defective in particular when its spectrum lies in the open right left half plane or is on the real axis related theoretical error bounds are established which reveal some intrinsic relationships between the convergence properties and the eigen characteristics of the coefficient matrix these results not only generalize all the known ones for the diagonalizable matrices in the literature but also sharp the corresponding estimates in jia acta mathematica sinica new series 14 1998 507 518

preconditioning techniques for nonsymmetric and indefinite linear systems abstract the standard preconditioning techniques for conjugate gradient methods often fail for matrices that are indefinite and or strongly nonsymmetric the most common alterative considered for these cases are either to use expensive direct solvers or to resort to one of many techniques based on the normal equations this paper examines several such alternatives and compares them in particular an incomplete lq factorization is proposed and some of its implementation details are described a number of experiments are reported to compare these methods

fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations abstract the discretizations of two and three dimensional spatial fractional diffusion equations with the shifted finite difference formulas of the grunwald letnikov type can result in discrete linear systems whose coefficient matrices are of the form d t where d is a nonnegative diagonal matrix and t is a block toeplitz with toeplitz block matrix or a block toeplitz with each block being block toeplitz with toeplitz block matrix for these discrete spatial fractional diffusion matrices we construct diagonal and block circulant with circulant block splitting preconditioner for the two dimensional case and diagonal and block circulant with each block being block circulant with circulant block splitting preconditioner for the three dimensional case to further accelerate the convergence rates of krylov subspace iteration methods and we analyze the eigenvalue distributions for the corresponding preconditioned matrices theoretical results show that except for a small number of outliners the eigenvalues of the preconditioned matrices are located within a complex disk centered at 1 with the radius being exactly less than 1 and numerical experiments demonstrate that these structured preconditioners can significantly improve the convergence behavior of the krylov subspace iteration methods moreover this approach is superior to the geometric multigrid method and the preconditioned conjugate gradient methods incorporated with the approximate inverse circulant plus diagonal preconditioners in both iteration counts and computing times

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

stopping criteria for iterations in finite element methods this work extends the results of arioli 1 2 on stopping criteria for iterative solution methods for linear finite element problems to the case of nonsymmetric positive definite problems we show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to convergence in a finite element context we then use krylov solvers to provide alternative ways of calculating or estimating this quantity and present numerical experiments which validate our criteria

a class of efficient parameterized shift splitting preconditioners for block two by two linear systems abstract based on the effective constructing an equivalent block two by two linear systems we establish a class of efficient parameterized shift splitting eps preconditioners for solving block two by two linear systems theoretical analysis shows that the eps iteration method is unconditionally convergent and then we study the spectral properties of the corresponding preconditioned matrix by removing one of the shift term of the eps preconditioners two classes of local eps preconditioners are presented and the eigenvalues distribution of the preconditioned matrices are also studied numerical experiments are provided to demonstrate the feasibility and effectiveness of the proposed preconditioners

a flexible and adaptive simpler block gmres with deflated restarting for linear systems with multiple right hand sides block gmres is one of the most popular algorithms for solving large non hermitian linear systems with multiple right hand sides the simpler block gmres algorithm is a variation of block gmres which avoids the factorization of a block upper hessenberg matrix and it is much simpler to program and requires less work than block gmres however theoretical results given in this paper indicate that it is less stable due to the ill conditioning of the basis used in order to overcome this difficulty we propose an adaptive simpler block gmres algorithm for large linear systems with multiple right hand sides theoretical analysis is made to show the reason why the new algorithm is superior to its original counterpart moreover we consider how to restart and precondition the adaptive simpler block gmres algorithm and propose a flexible and adaptive simpler block gmres algorithm with deflated restarting numerical experiments show the numerical behavior of our new algorithm

the generalized modified shift splitting preconditioners for nonsymmetric saddle point problems for a nonsymmetric saddle point problem the modified shift splitting mss preconditioner has been proposed by zhou et al by replacing the parameter in 2 2 block in the mss preconditioner by another parameter a generalized mss gmss preconditioner is established in this paper which results in a fixed point iteration called the gmss iteration method we provide the convergent and semi convergent analysis of the gmss iteration method which show that this method is convergence and semi convergence if the related parameters satisfy suitable restrictions meanwhile the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analyzed in detail finally numerical examples show that the gmss method is more feasibility and robustness than the mss uzawa hss and pu sts methods as a solver and the gmss preconditioner outperforms the gsor uzawa hss mss and lmss preconditioners for the gmres method for solving both the nonsingular and the singular saddle point problems with nonsymmetric positive definite and symmetric dominant 1 1 parts

pre symmetric approach and applications to indefinite non symmetric problems a class of pre symmetric minimal residual method to solve large scale system of linear algebraic equations with indefinite symmetrizable coefficient matrix is presented in this paper briefly called as psminres method the psminres method is established by first transforming the non symmetric system of equations into symmetric system of equations by the pre symmetric technique and then utilizing the minimal residual minres method to solve the symmetric system of equations and getting a new approximation to the original system of linear equations theoretical analysis and numerical computations show that the pre symmetric technique is feasible and behaviors of the psminres are superior to some krylov subspace methods which are usual adopted for solving general non symmetric linear systems such as cgs gmres etc

motivations and realizations of krylov subspace methods for large sparse linear systems we briefly introduce typical and important direct and iterative methods for solving systems of linear equations concretely describe their fundamental characteristics in viewpoints of both theory and applications and clearly clarify the substantial differences among these methods in particular the motivations of searching the solution of a linear system in a krylov subspace are described and the algorithmic realizations of the generalized minimal residual gmres method are shown and several classes of state of the art algebraic preconditioners are briefly reviewed all this is useful for correctly deeply and completely understanding the application scopes theoretical properties and numerical behaviors of these methods and is also helpful in designing new methods for solving systems of linear equations

guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems abstract this paper presents incomplete decompositions for various types of matrices as they occur in the implicit discretisation of practical problems a review is given of methods for the usual five point discretisation of a self adjoint elliptic second order partial differential equation in two dimensions on a square the matrices which occur in this type of problem are symmetric m matrices of very regular structure the convergence behaviour of the different decompositions for this case is demonstrated by numerical experiments the paper also gives decompositions for the following type of matrices i symmetric m matrices of a different structure ii symmetric positive definite matrices iii non symmetric matrices

a note on parameterized block triangular preconditioners for generalized saddle point problems in this paper based on the results presented by jiang et al m q jiang y cao l q yao on parameterized block triangular preconditioners for generalized saddle point problems appl math comput 216 2010 1777 1789 we consider an indefinite parameterized block triangular preconditioner for symmetric saddle point problems the eigenvalue bound of the preconditioned matrix is discussed some presented results are improved numerical experiments are given to confirm our theoretical results

parameterized generalized shift splitting preconditioners for nonsymmetric saddle point problems abstract to solve nonsymmetric saddle point problems the parameterized generalized shift splitting pgss preconditioner is presented and analyzed the corresponding pgss iteration method can be applied not only to the nonsingular saddle point problems but also to the singular ones the convergence and semi convergence of the pgss iteration method are discussed carefully meanwhile the spectral properties of the preconditioned matrix and the strategy of the choices of the parameters are given numerical experiments further demonstrate that the pgss iteration method and the pgss preconditioner are efficient and have better performance than some existing iteration methods and newly proposed preconditioners respectively for solving both the nonsingular and singular nonsymmetric saddle point problems

extensions of certain graph based algorithms for preconditioning the original tpablo algorithms are a collection of algorithms which compute a symmetric permutation of a linear system such that the permuted system has a relatively full block diagonal with relatively large nonzero entries this block diagonal can then be used as a preconditioner we propose and analyze three extensions of this approach we incorporate a nonsymmetric permutation to obtain a large diagonal we use a more general parametrization for tpablo and we use a block gauss seidel preconditioner which can be implemented to have the same execution time as the corresponding block jacobi preconditioner experiments are presented showing that for certain classes of matrices the block gauss seidel preconditioner used with the system permuted with the new algorithm can outperform the best ilut preconditioners in a large set of experiments

cimgs an incomplete orthogonal factorization preconditioner a new preconditioner for symmetric positive definite systems is proposed analyzed and tested the preconditioner compressed incomplete modified gram schmidt cimgs is based on an incomplete orthogonal factorization cimgs is robust both theoretically and empirically existing in exact arithmetic for any full rank matrix numerically it is more robust than an incomplete cholesky factorization preconditioner ic and a complete cholesky factorization of the normal equations theoretical results show that the cimgs factorization has better backward error properties than complete cholesky factorization for symmetric positive definite m matrices cimgs induces a regular splitting and better estimates the complete cholesky factor as the set of dropped positions gets smaller cimgs lies between complete cholesky factorization and incomplete cholesky factorization in its approximation properties these theoretical properties usually hold numerically even when the matrix is not an m matrix when the drop set satisfies a mild and easily verified or enforced property the upper triangular factor cimgs generates is the same as that generated by incomplete cholesky factorization this allows the existence of the ic factorization to be guaranteed based solely on the target sparsity pattern

analysis on inexact block diagonal preconditioners for elliptic pde constrained optimization problems abstract by using the galerkin finite element method the distributed control problems can be discretized into a saddle point problem with a coefficient matrix of block three by three which can be reduced to a linear system with lower order we first introduce a class of inexact block diagonal preconditioners and estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively based on the cholesky decomposition of the known matrices we also analyze a lower triangular preconditioner to accelerate the minimal residual method for the reduced linear system and discuss its real and complex eigenvalues respectively moreover these bounds do not rely on the regularization parameter and the eigenvalues of the matrices in the discrete system numerical experiments are also presented to demonstrate the effectiveness and robustness of the two new preconditioners

numerical study on incomplete orthogonal factorization preconditioners we design analyse and test a class of incomplete orthogonal factorization preconditioners constructed from givens rotations incorporating some dropping strategies and updating tricks for the solution of large sparse systems of linear equations comprehensive accounts about how the preconditioners are coded what storage is required and how the computation is executed for a given accuracy are presented a number of numerical experiments show that these preconditioners are competitive with standard incomplete triangular factorization preconditioners when they are applied to accelerate krylov subspace iteration methods such as gmres and bicgstab

polynomial preconditioners based on factorized sparse approximate inverses let ax b be a linear system where a is a symmetric positive definite matrix an additive polynomial preconditioner for the conjugate gradient method based on multisplittings is proposed the multisplittings are obtained by computing some factorized sparse approximate inverses of the coefficient matrix namely splittings of the form a email protected email protected t 1 n email protected email protected t a 1 induced by the ainv and fsai factorized approximate inverse preconditioners applied to diagonal blocks of a are used the applicability of this preconditioner is studied moreover the results of the numerical experiments obtained on a cray t3e for a representative set of matrices are presented specifically structural analysis and transport diffusion problems are considered the effect of the reverse cuthill mckee rcm and multiple minimum degree mmd orderings is evaluated

on preconditioned iterative methods for unsteady incompressible navier stokes equations when the implicit finite difference scheme and linearization technique in conjunction with the artificial compressibility method are employed to discretize the unsteady incompressible navier stokes equations in each pseudo time step a structured system of linear equations is obtained and needs to be solved numerically in this paper preconditioning techniques are applied to solve such linear systems numerical example is given to illustrate the effectiveness of the proposed method we also find that a combination of the linearization technique with the preconditioned gmres method is quite efficient for the implicit finite difference discretization of the incompressible navier stokes equations

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

singular sturmian separation theorems on unbounded intervals for linear hamiltonian systems abstract in this paper we develop new fundamental results in the sturmian theory for nonoscillatory linear hamiltonian systems on an unbounded interval we introduce a new concept of a multiplicity of a focal point at infinity for conjoined bases and based on this notion we prove singular sturmian separation theorems on an unbounded interval the main results are formulated in terms of the minimal principal solutions at both endpoints of the considered interval and include exact formulas as well as optimal estimates for the numbers of proper focal points of one or two conjoined bases as a natural tool we use the comparative index which was recently implemented into the theory of linear hamiltonian systems by the authors and independently by j elyseeva throughout the paper we do not assume any controllability condition on the system our results turn out to be new even in the completely controllable case

efficient preconditioning of sequences of nonsymmetric linear systems we present a new approach for approximate updates of factorized nonsymmetric preconditioners for solving sequences of linear algebraic systems this approach is algebraic and it is theoretically motivated it generalizes diagonal updates introduced by benzi and bertaccini bit 43 2003 pp 231 244 and bertaccini electron trans numer anal 18 2004 pp 49 64 it is shown experimentally that this approach can be very beneficial for example it is successful in significantly decreasing the number of iterations of a preconditioned iterative method for solving subsequent systems of a sequence when compared with freezing the preconditioner from the first system of the sequence in some cases the updated preconditioners offer a rate of convergence similar to or even higher than the rate obtained when preconditioning with recomputed preconditioners since the updates are typically cheap and straightforward their use is of practical interest they can replace recomputing preconditioners which is often expensive especially in parallel and matrix free environments