additive and multiplicative two level spectral preconditioning for general linear systems in this paper we introduce new preconditioning techniques for the solution of general symmetric and unsymmetric linear systems a x b these approaches borrow some ideas of the multigrid philosophy designed for the solution of linear systems arising from the discretization of elliptic partial differential equations we attempt to improve the convergence rate of a prescribed preconditioner m 1 in a two grid framework this preconditioner is viewed as a smoother and the coarse space is spanned by the eigenvectors associated with the smallest eigenvalues of m 1a we derive both additive and multiplicative variants of the resulting iterated two level preconditioners for unsymmetric linear systems that can also be adapted for hermitian positive definite problems we show that these two level preconditioners shift the smallest eigenvalues to one and tend to better cluster around one those eigenvalues that m 1 already succeeded in moving into the neighborhood of one we illustrate the behavior of our method through extensive numerical experiments on a set of general linear systems finally we show the effectiveness of these approaches on two challenging real applications the first comes from a nonoverlapping domain decomposition method in semiconductor device modeling the second from industrial electromagnetism applications

factored sparse approximate inverse of block tridiagonal and block pentadiagonal matricies this paper is concerned with approaches to compute a factored sparse approximate inverse for block tridiagonal and block pentadiagonal matrices recurrence formulas are developed for computing sparse approximate inverse factors of these matrices using bordering technique resulting factored sparse approximate inverse is used as a preconditioner for the conjugate gradient method pcg as an application these formulas are simplified for computing the preconditioner for solving lyapanuv matrix equations by pcg method numerical experiments on linear system arising from discretization of partial differential equations are presented

dynamic instabilities in scalar neural field equations with space dependent delays in this paper we consider a class of scalar integral equations with a form of space dependent delay these non local models arise naturally when modelling neural tissue with active axons and passive dendrites such systems are known to support a dynamic oscillatory turing instability of the homogeneous steady state in this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability the appropriate amplitude equations are found to be the coupled mean field ginzburg landau equations describing a turing hopf bifurcation with modulation group velocity of o 1 importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio temporal kernels defining the neural field equation of interest indeed our results cover not only models with axonal or dendritic delays but those which are described by a more general distribution of delayed spatio temporal interactions we illustrate the predictive power of this form of analysis with comparison against direct numerical simulations paying particular attention to the competition between standing and travelling waves and the onset of benjamin feir instabilities

spectral preconditioners for the efficient numerical solution of a continuous branched transport model abstract we consider the efficient solution of sequences of linear systems arising in the numerical solution of a branched transport model whose long time solution for specific parameter settings is equivalent to the solution of the monge kantorovich equations of optimal transport galerkin finite element discretization combined with explicit euler time stepping yield a linear system to be solved at each time step characterized by a large sparse very ill conditioned symmetric positive definite spd matrix a extreme cases even prevent the convergence of preconditioned conjugate gradient pcg with standard preconditioners such as an incomplete cholesky ic factorization of a which cannot always be computed we investigate several preconditioning strategies that incorporate partial approximated spectral information we present numerical evidence that the proposed techniques are efficient in reducing the condition number of the preconditioned systems thus decreasing the number of pcg iterations and the overall cpu time

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

a class of spectral two level preconditioners it is well known that the convergence of krylov methods for solving the linear system often depends to a large extent on the eigenvalue distribution in many cases it is observed that removing the smallest eigenvalues can greatly improve the convergence several techniques have been proposed in the past few years that attempt to tackle this problem the proposed approaches can be split into two main families depending on whether the scheme enlarges the generated krylov space or adaptively updates the preconditioner in this paper we follow the second approach and propose a class of preconditioners both for unsymmetric and for symmetric linear systems that can also be adapted for symmetric positive definite problems we effectively solve the preconditioned system exactly in the low dimensional space associated with the smallest eigenvalues and use this to update the preconditioned residual this update results in shifting eigenvalues from close to the origin to near to one for the new preconditioner this is ideal when there are only a few eigenvalues near the origin while all the others are close to one because the updated preconditioned system becomes close to the identity we illustrate the performance of our method through extensive numerical experiments on a set of general linear systems finally we show the advantages of the preconditioners for solving dense linear systems arising in electromagnetism applications which were the main motivation for this work

modeling electrocortical activity through improved local approximations of integral neural field equations neural field models of firing rate activity typically take the form of integral equations with space dependent axonal delays under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation pde model that properly treats the axonal delay terms of the integral formulation our analysis avoids the so called long wavelength approximation that has previously been used to formulate pde models for neural activity in two spatial dimensions direct numerical simulations of this pde model show instabilities of the homogeneous steady state that are in full agreement with a turing instability analysis of the original integral model we discuss the benefits of such a local model and its usefulness in modeling electrocortical activity in particular we are able to treat patchy connections whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion in this case the emergence of a lattice directed traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled pdes

a direct projection method for markov chains a direct method based on oblique projections is adapted to compute the stationary distribution vector of a finite markov chain the algorithm can also be used to compute the group inverse of the corresponding generator matrix it is shown how to update the stationary vector and other quantities of interest when one row of the transition probability matrix is modified a gth like variant that appears to compute the stationary probabilities to high relative accuracy is developed

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

riemannian geometry of bicovariant group lattices group lattices cayley digraphs of a discrete group are in natural correspondence with differential calculi on the group on such a differential calculus geometric structures can be introduced following general recipes of noncommutative differential geometry despite the noncommutativity between functions and generalized differential forms for the subclass of bicovariant group lattices considered in this work it is possible to understand central geometric objects like metric torsion and curvature as tensors with left covariance properties this ensures that tensor components with respect to a basis of the space of 1 forms transform in the familiar homogeneous way under a change of basis there is a natural compatibility condition for a metric and a linear connection the resulting pseudo riemannian geometry is explored in this work it is demonstrated that the components of the metric are indeed able to properly describe properties of discrete geometries like lengths and angles a simple g

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

some theoretical and numerical results for delayed neural field equations in this paper we study neural field models with delays which define a useful framework for modeling macroscopic parts of the cortex involving several populations of neurons nonlinear delayed integro differential equations describe the spatio temporal behavior of these fields using methods from the theory of delay differential equations we show the existence and uniqueness of a solution of these equations a lyapunov analysis gives us sufficient conditions for the solutions to be asymptotically stable we also present a fairly detailed study of the numerical computation of these solutions this is to our knowledge the first time that a serious analysis of the problem of the existence and uniqueness of a solution of these equations has been performed another original contribution of ours is the definition of a lyapunov functional and the result of stability it implies we illustrate our numerical schemes on a variety of examples that are relevant to modeling in neuroscience

on the properties of nonlinear nonlocal operators arising in neural field models abstract we study the existence and continuous dependence of stationary solutions of the one population wilson cowan model on the steepness of the firing rate functions we investigate the properties of the nonlinear nonlocal operators which arise when formulating the stationary one population wilson cowan model as a fixed point problem the theory is used to study the existence and continuous dependence of localized stationary solutions of this model on the steepness of the firing rate functions the present work generalizes and complements previously obtained results as we relax on the assumptions that the firing rate functions are given by smoothed heaviside functions

differential geometry of group lattices in a series of publications we developed differential geometry on discrete sets based on concepts of noncommutative geometry in particular it turned out that first order differential calculi over the algebra of functions on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set a particular class of digraphs are cayley graphs also known as group lattices they are determined by a discrete group g and a finite subset s there is a distinguished subclass of bicovariant cayley graphs with the property ad s s s we explore the properties of differential calculi which arise from cayley graphs via the above correspondence the first order calculi extend to higher orders and then allow us to introduce further differential geometric structures furthermore we explore the properties of discrete vector fields which describe deterministic flows on group lattices a lie derivative with respect to a discrete vector field and an inner

differential calculi on commutative algebras a differential calculus on an associative algebra a is an algebraic analogue of the calculus of differential forms on a smooth manifold it supplies a with a structure on which dynamics and field theory can to some extent be formulated in very much the same way we are used to in the geometrical arena underlying classical physical theories and models in previous work certain differential calculi on a commutative algebra exhibited relations with lattice structures stochastics and parametrized quantum theories this motivated the present systematic investigation of differential calculi on commutative and associative algebras various results about their structure are obtained in particular it is shown that there is a correspondence between first order differential calculi on such an algebra and commutative and associative products in the space of i forms an example of such a product is provided by the ito calculus of stochastic differentials for the case where the algebra a is freely generated by coordinates xi i 1 n we study calculi for which the differentials dxi constitute a basis of the space of 1 forms as a left a module these may be regarded as deformations of the ordinary differential calculus on rn for n or 3 a classification of all orbits under the general linear group of such calculi with constant structure functions is presented we analyse whether these calculi are reducible i e a skew tensor product of lower dimensional calculi or whether they are the extension as defined in this paper of a one dimension lower calculus furthermore generalizations to arbitrary n are obtained for all these calculi

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

dynamical mechanism for sharp orientation tuning in an integrate and fire model of a cortical hypercolumn orientation tuning in a ring of pulse coupled integrate and fire if neurons is analyzed in terms of spontaneous pattern formation it is shown how the ring bifurcates from a synchronous state to a non phase locked state whose spike trains are characterized by clustered but irregular fluctuations of the interspike intervals isis the separation of these clusters in phase space results in a localized peak of activity as measured by the time averaged firing rate of the neurons this generates a sharp orientation tuning curve that can lock to a slowly rotating weakly tuned external stimulus under certain conditions the peak can slowly rotate even to a fixed external stimulus the ring also exhibits hysteresis due to the subcritical nature of the bifurcation to sharp orientation tuning such behavior is shown to be consistent with a corresponding analog version of the if model in the limit of slow synaptic interactions for fast synapses the deterministic fluctuations of the isis associated with the tuning curve can support a coefficient of variation of order unity

a robust ilu with pivoting based on monitoring the growth of the inverse factors abstract an incomplete lu decomposition with pivoting is presented that progressively monitors the growth of the inverse factors of l u the information on the growth of the inverse factors is used as feedback for dropping entries in l and u this method often yields a robust and effective preconditioner especially when the system is highly indefinite numerical examples demonstrate the effectiveness of this approach

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

multiscale compressed block decomposition for fast direct solution of method of moments linear system the multiscale compressed block decomposition algorithm ms cbd is presented for highly accelerated direct non iterative solution of electromagnetic scattering and radiation problems with the method of moments mom the algorithm is demonstrated to exhibit n2 computational complexity and storage requirements scaling with n1 5 for electrically large objects several numerical examples illustrate the efficiency of the method in particular for problems with multiple excitation vectors the largest problem presented in this paper is the monostatic rcs of the nasa almond at 50 ghz for one thousand incidence angles discretized using 442 089 rwg basis functions being entirely algebraic ms cbd is independent of the greens function of the problem

effects of distributed transmission speeds on propagating activity in neural populations motivated by experimental evidence a distribution of axonal transmission speeds is introduced into a standard field model of neural populations the resulting field dynamics is analytically studied by a systematic investigation of the stability and bifurcations of equilibrium solutions using a perturbation approach the effect of distributed speeds on bifurcations of equilibria are determined for general connectivity and speed distributions in addition a nonlinear analysis of traveling fronts is given it is shown that the variance of the speed distribution affects the frequency of bifurcating periodic solutions and the phase speed of traveling waves moreover a new effect is discovered where the introduction of axonal speed distributions leads to the maximization of the traveling front speed doi 10 1103 physreve 73 021906 pacs numbers 87 18 hf 02 60 nm 87 19 la the measured activity of the neural system indicates that the spatiotemporal dynamics of the brain is intimately related to the various brain functions and cognition as examples we mention the space time instabilities during several types of hallucinations 1 and epileptic seizures 2 hence the study of the dynamics of neural models is important for understanding neural and cognitive processes in recent years several models of coupled neurons have attracted much attention including networks involving spatial structures 3 8 describing a continuous synaptically coupled neural field extended in space the dynamics of these fields is governed by

stability and bifurcations in neural fields with finite propagation speed and general connectivity a stability analysis is presented for neural field equations in the presence of finite propagation speed along axons and for a general class of connectivity kernels and synaptic proper ties sufficient conditions are given for the stability of equilibrium solutions it is shown that the propagation delays play a significant role in nonstationary bifurcations of equilibria whereas the stationary bifurcations depend only on the connectivity kernel in the case of nonstationary bifurca tions bounds are determined on the frequencies of the resulting oscillatory solutions a perturbative scheme is used to calculate the types of bifurcations leading to spatial patterns oscillations and traveling waves for high propagation speeds a simple method is derived that allows the determi nation of the bifurcation type by visual inspection of the fourier transforms of the kernel and its first moment results are numerically illustrated on a class of neurologically plausible systems with combinations of gaussian excitatory and inhibitory connections

a robust and efficient ilu that incorporates the growth of the inverse triangular factors in this paper we present a new ilu decomposition which is based on an existing sparse direct solver in contrast to many incomplete lu decompositions this ilu incorporates information about the inverse factors l 1 and u 1 which have direct influence on the dropping strategy we demonstrate in several large scale examples that this implementation constructs a robust preconditioner

multilevel sparse approximate inverse preconditioners for adaptive mesh refinement abstract we present an efficient and effective preconditioning method for time dependent simulations with dynamic adaptive mesh refinement and implicit time integration adaptive mesh refinement greatly improves the efficiency of simulations where the solution develops steep gradients in small regions of the computational domain that change over time unfortunately adaptive mesh refinement also introduces a number of problems for preconditioning in parallel iterative linear solvers as the changes in the mesh lead to structural changes in the linear systems we must solve hence we may need to compute a new preconditioner at every time step since this would be expensive we propose preconditioners that are cheap to adapt for dynamic changes to the mesh more specifically we propose preconditioners that require only localized changes to the preconditioner for localized changes in the mesh our preconditioners combine sparse approximate inverses and multilevel techniques we demonstrate significant improvements in convergence rates of krylov subspace methods and significant reductions of the overall runtime furthermore we demonstrate experimentally level independent convergence rates for various problems

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

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

application of a gpu accelerated hybrid preconditioned conjugate gradient approach for large 3d problems in computational geomechanics this paper presents a hybrid preconditioning technique for conjugate gradient method and discusses its parallel implementation on graphic processing unit gpu for solving large sparse linear systems arising from application of interior point methods to conic optimization problems in the context of nonlinear finite element limit analysis fela for computational geomechanics for large 3d problems the use of direct solvers in general becomes prohibitively expensive due to exponentially growing memory requirements and computational time besides the so called saddle point systems resulting from use of optimization framework is not an exemption on the other hand although preconditioned iterative methods have moderate storage requirements and therefore can be applied to much larger problems than direct methods they usually exhibit high number of iterations to reach convergence in present paper we show that this problem can be effectively tackled using the proposed hybrid preconditioner along with an elaborate implementation on gpu furthermore numerical results verify the robustness and efficiency of the proposed technique

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

adaptively preconditioned gmres algorithms the restarted gmres algorithm proposed by saad and schultz siam j sci statist comput 7 1986 pp 856 869 is one of the most popular iterative methods for the solution of large linear systems of equations ax b with a nonsymmetric and sparse matrix this algorithm is particularly attractive when a good preconditioner is available the present paper describes two new methods for determining preconditioners from spectral information gathered by the arnoldi process during iterations by the restarted gmres algorithm these methods seek to determine an invariant subspace of the matrix a associated with eigenvalues close to the origin and to move these eigenvalues so that a higher rate of convergence of the iterative methods is achieved

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

a direct projection method for sparse linear systems an oblique projection method is adapted to solve large sparse unstructured systems of linear equations this row projection technique is a direct method which can be interpreted as an oblique kaczmarz type algorithm and is also related to other standard solution methods when a sparsity preserving pivoting strategy is incorporated it is demonstrated that the technique can be superior in terms of both fill in and arithmetic complexity to more standard sparse algorithms based on gaussian elimination this is especially true for systems arising from stiff ordinary differential equations problems in chemical kinetics studies

mixed constraint preconditioning in computational contact mechanics contact mechanics can be addressed numerically by finite elements using either a penalty formulation or the lagrange multipliers the penalty approach leads to a linearized symmetric positive definite system which can prove severely ill conditioned with the iterative solution to large 3d problems requiring expensive preconditioners to accelerate or even to allow for convergence in the present paper a mixed constraint preconditioner mcp is developed and implemented for the iterative solution of contact problems based on the penalty approach a theoretical bound for the eigenspectrum of the preconditioned matrix is provided depending on the user specified parameters that control the preconditioner fill in the performance of mcp combined with the preconditioned conjugate gradient is studied in two real 3d test cases describing the geomechanics of faulted porous media and compared to a well established preconditioner such as the incomplete cholesky decomposition illt with optimal fill in numerical results show that typically mcp outperforms illt especially in the most challenging problems where very large penalty parameters must be used finally a modification aimed at simplifying the mcp implementation and increasing its robustness is developed and experimented with showing that the modified mcp can be viewed as an efficient and viable alternative to both mcp and illt especially in problems where the size of the contact surfaces is large with respect to the global domain

the fast multipole method i error analysis and asymptotic complexity this paper is concerned with the application of the fast multipole method fmm to the maxwell equations this application differs in many aspects from other applications such as the n body problem laplace equation and quantum chemistry etc the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products this leads to faster resolution of scattering of harmonic plane waves from perfectly conducting obstacles emphasis here is on a rigorous mathematical approach to the problem we focus on the estimation of the error introduced by the fmm and a rigorous analysis of the complexity o n log n of the algorithm we show that error estimates reported previously are not entirely satisfactory and provide sharper and more precise estimates

new breakdown free variant of ainv method for nonsymmetric positive definite matrices this paper proposes a new breakdown free preconditioning technique called sainv ns of the ainv method of benzi and tuma for nonsymmetric positive definite matrices the resulting preconditioner which is an incomplete factorization of the inverse of a nonsymmetric matrix will be used as an explicit right preconditioner for qmr bicgstab and gmres m methods the preconditoner is reliable pivot breakdown can not occur and effective at reducing the number of iterations some numerical experiments on test matrices are presented to show the efficiency of the new method and comparing to the ainv a algorithm

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

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

a hierarchical low rank schur complement preconditioner for indefinite linear systems nonsymmetric and highly indefinite linear systems can be quite difficult to solve by iterative methods this paper combines ideas from the multilevel schur low rank preconditioner developed by y x

a two phase preconditioning strategy of sparse approximate inverse for indefinite matrices a two phase preconditioning strategy based on a factored sparse approximate inverse is proposed for solving sparse indefinite matrices in each phase the strategy first makes the original matrix diagonally dominant to enhance the stability by a shifting method and constructs an inverse approximation of the shifted matrix by utilizing a factored sparse approximate inverse preconditioner the two inverse approximation matrices produced from each phase are then combined to be used as a preconditioner experimental results show that the presented strategy improves the accuracy and the stability of the preconditioner on solving indefinite sparse matrices furthermore the strategy ensures that convergence rate of the preconditioned iterations of the two phase preconditioning strategy is much better than that of the standard sparse approximate inverse ones for solving indefinite matrices

an approximate inverse preconditioner for toeplitz systems with multiple right hand sides abstract we are interested in preconditioning techniques for symmetric positive definite but ill conditioned toeplitz systems with multiple right hand sides tx b here the right hand sides are available simultaneously our main idea is to use an approximate inverse of t as a preconditioner for t the approximate inverse of t is generated by the inverse formula given by lv and huang x g lv t z huang a note on inversion of toeplitz matrices appl math lett 20 2007 1189 1193 the construction of the preconditioner is of low computational cost requires only the entries of t and does not require explicit knowledge of the generating function f of t we show that if given a proper parameter then the approximate inverse preconditioner approximates t 1 accurately and thus the spectra of these preconditioned matrices are clustered around 1 it follows that the conjugate gradient method converges very quickly when applied to solve the preconditioned systems numerical results are given to demonstrate the effectiveness of our preconditioner

stochastic preconditioning for diagonally dominant matrices this paper presents a new stochastic preconditioning approach for large sparse matrices for the class of matrices that are rowwise and columnwise irreducibly diagonally dominant we prove that an incomplete rm ldl t factorization in a symmetric case or an incomplete ldu factorization in an asymmetric case can be obtained from random walks and used as a preconditioner it is argued that our factor matrices have better quality i e better accuracy size trade offs than preconditioners produced by existing incomplete factorization methods therefore a resulting preconditioned krylov subspace iterative solver requires less computation than traditional methods to solve a set of linear equations with the same error tolerance the advantage increases for larger and denser matrices these claims are verified by numerical tests and we provide techniques that can potentially extend the theory to non diagonally dominant matrices

restarted gmres preconditioned by deflation this paper presents a new preconditioning technique for the restarted gmres algorithm it is based on an invariant subspace approximation which is updated at each cycle numerical examples show that this deflation technique gives a more robust scheme than the restarted algorithm at a low cost of operations and memory

an extension of the conjugate residual method to nonsymmetric linear systems the conjugate gradient cg method and the conjugate residual cr method are krylov subspace methods for solving symmetric positive definite linear systems to solve nonsymmetric linear systems the bi conjugate gradient bi cg method has been proposed as an extension of cg bi cg has attractive short term recurrences and it is the basis for the successful variants such as bi cgstab in this paper we extend cr to nonsymmetric linear systems with the aim of finding an alternative basic solver numerical experiments show that the resulting algorithm with short term recurrences often gives smoother convergence behavior than bi cg hence it may take the place of bi cg for the successful variants

pitchfork hopf bifurcations in 1d neural field models with transmission delays recently local bifurcation theory for delayed neural fields was developed in this paper we show how symmetry arguments and residue calculus can be used to simplify the computation of the spectrum in special cases and the evaluation of the normal form coefficients respectively this is done hand in hand with an extensive study of two pitchfork hopf bifurcations for a 1d neural field model with wizard hat type connectivity

standing and travelling waves in a spherical brain model the nunez model revisited the nunez model for the generation of electroencephalogram eeg signals is naturally described as a neural field model on a sphere with space dependent delays for simplicity dynamical realisations of this model either as a damped wave equation or an integro differential equation have typically been studied in idealised one dimensional or planar settings here we revisit the original nunez model to specifically address the role of spherical topology on spatio temporal pattern generation we do this using a mixture of turing instability analysis symmetric bifurcation theory centre manifold reduction and direct simulations with a bespoke numerical scheme in particular we examine standing and travelling wave solutions using normal form computation of primary and secondary bifurcations from a steady state interestingly we observe spatio temporal patterns which have counterparts seen in the eeg patterns of both epileptic and schizophrenic brain conditions

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

a transformation approach that makes spai psai and rsai procedures efficient for large double irregular nonsymmetric sparse linear systems abstract a sparse matrix is called double irregular sparse if it has at least one relatively dense column and row and it is double regular sparse if all the columns and rows of it are sparse the sparse approximate inverse preconditioning procedures spai psai t o l and rsai t o l are costly and even impractical to construct preconditioners for a large sparse nonsymmetric linear system with the coefficient matrix being double irregular sparse but they are efficient for double regular sparse problems double irregular sparse linear systems have a wide range of applications and 24 4 of the nonsymmetric matrices in the florida university collection are double irregular sparse for this class of problems we propose a transformation approach which consists of four steps i transform a given double irregular sparse problem into a small number of double regular sparse ones with the same coefficient matrix a ii use spai psai t o l and rsai t o l to construct sparse approximate inverses m of a iii solve the preconditioned double regular sparse linear systems by krylov solvers and iv recover an approximate solution of the original problem with a prescribed accuracy from those of the double regular sparse ones a number of theoretical and practical issues are considered on the transformation approach numerical experiments on a number of real world problems confirm the very sharp superiority of the transformation approach to the standard approach that preconditions the original double irregular sparse problem by spai psai t o l or rsai t o l and solves the resulting preconditioned system by krylov solvers

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

numerical solution of the neural field equation in the two dimensional case we are concerned with the numerical solution of a class of integro differential equations known as neural field equations which describe the large scale dynamics of spatially structured networks of neurons these equations have many applications in neuroscience and robotics we describe a numerical method for the approximation of solutions in the two dimensional case including a space dependent delay in the integrand function compared with known algorithms for this type of equation we propose a scheme with higher accuracy in the time discretization since computational efficiency is a key issue in this type of calculation we use a new method for reducing the complexity of the algorithm the convergence issues are discussed in detail and a number of numerical examples are presented which illustrate the performance of the method

differential calculi on the quantum group glp q 2 the author presents a systematical study of left covariant differential calculus on the quantum group glp q 2 a two parameter deformation of the general linear group in particular the author explicitly constructs the most general bicovariant calculus it depends on a new parameter s the corresponding lie algebra of left coinvariant vector fields is a two parameter deformation of the classical lie algebra p and q appear only in the combination r pq in the limit p q to 1 one obtains non standard bicovariant differential calculi on the classical lie group

discrete differential calculus graphs topologies and gauge theory differential calculus on discrete sets is developed in the spirit of noncommutative geometry any differential algebra on a discrete set can be regarded as a reduction of the universal differential algebra and this allows a systematic exploration of differential algebras on a given set associated with a differential algebra is a di graph where two vertices are connected by at most two antiparallel arrows the interpretation of such a graph as a hasse diagram determining a locally finite topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure it is shown that field theories and in particular gauge theories can be formulated on a discrete set in close analogy with the continuum case the framework presented generalizes ordinary lattice theory which is recovered from an oriented hypercubic lattice graph it also includes e g the two point space used by connes and lott and others in models of elementary particle physics the formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an internal discrete space a la connes et al to models of dimensionally reduced gauge fields furthermore a symmetric lattice is also studied which in a certain continuum limit turns out to be related to a noncommutative differential calculus on manifolds

on the quantum differential calculus and the quantum holomorphicity under some natural assumptions less restrictive than in the paper by wess and zumino preprint cern th 5697 90 lapp th 284 90 differential calculi on the quantum plane are found and investigated complex structure complex derivatives and holomorphic functions on the quantum plane are defined generalized cauchy riemann equations are given and solved

a complete pivoting strategy for the right looking robust incomplete factorization preconditioner in this paper we use a complete pivoting strategy for the right looking version of robust incomplete factorization preconditioner benzi and tuma 2003 7 the new preconditioner has been used as the right preconditioner for several linear systems and its effectiveness has been studied

computing and deflating eigenvalues while solving multiple right hand side linear systems with an application to quantum chromodynamics we present a new algorithm that computes eigenvalues and eigenvectors of a hermitian positive definite matrix while solving a linear system of equations with conjugate gradient cg traditionally all the cg iteration vectors could be saved and recombined through the eigenvectors of the tridiagonal projection matrix which is equivalent theoretically to unrestarted lanczos our algorithm capitalizes on the iteration vectors produced by cg to update only a small window of vectors that approximate the eigenvectors while this window is restarted in a locally optimal way the cg algorithm for the linear system is unaffected yet in all our experiments if the window has more than a properly chosen but small number of vectors it converges to the required eigenvectors at a rate identical to unrestarted lanczos after the solution of the linear system eigenvectors that have not accurately converged can be improved in an incremental fashion by solving additional linear systems in this case eigenvectors identified in earlier systems can be used to deflate and thus accelerate the convergence of subsequent systems we have used this algorithm with excellent results in lattice quantum chromodynamics applications where hundreds of right hand sides may be needed specifically about 70 eigenvectors are obtained to full accuracy after solving 24 right hand sides deflating these from the large number of subsequent right hand sides removes the dreaded critical slowdown where the conditioning of the matrix increases as the quark mass reaches a critical value our experiments show almost a constant number of iterations for our method regardless of quark mass and speedups of 8 over original cg for light quark masses

combining kronecker product approximation with discrete wavelet transforms to solve dense function related linear systems a new solution technique is proposed for linear systems with large dense matrices of a certain class including those that come from typical integral equations of potential theory this technique combines kronecker product approximation and wavelet sparsification for the kronecker product factors the user is only required to supply a procedure for computation of each entry of the given matrix the main sources of efficiency are the incomplete cross approximation procedure adapted from the mosaic skeleton method of the second author and data sparse preconditioners the incomplete lu decomposition with dynamic choice of the fill in structure with a prescribed threshold and the inverse kronecker product preconditioner constructed for the sum of kronecker products of sparsified finger like matrices computed by the discrete wavelet transform in some model but quite representative examples the new technique allowed us to solve dense systems with more than 1 million unknowns in a few minutes on a personal computer with 1 gbyte operative memory

a block qmr algorithm for non hermitian linear systems with multiple right hand sides abstract many applications require the solution of multiple linear systems that have the same coefficient matrix but differ in their right hand sides instead of applying an iterative method to each of these systems individually it is often more efficient to employ a block version of the method that generates iterates for all the systems simultaneously in this paper we propose a block version of freund and nachtigal s quasi minimal residual qmr method for the iterative solution of non hermitian linear systems the block qmr method uses a novel lanczos type process for multiple starting vectors which was recently developed by aliaga boley freund and hernandez to compute suitable basis vectors for the underlying block krylov subspaces we describe the basic block qmr method and also give important implementation details in particular we show how to incorporate deflation to drop converged linear systems and to delete linearly and almost linearly dependent vectors in the underlying block krylov sequences numerical results are reported that illustrate typical features of the block qmr method

left looking version of ainv preconditioner with complete pivoting strategy abstract in this paper we apply a complete pivoting strategy to compute the left looking version of ainv preconditioner for linear systems as the preprocessing the multilevel nested dissection reordering has also been applied we have used this preconditioner as the right preconditioner for several linear systems where the coefficient matrices have been downloaded from the university of florida sparse matrix collection numerical experiments presented in this paper indicate the effectiveness of such a complete pivoting on the quality of left looking version of ainv preconditioner

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

a rank one reduction formula and its applications to matrix factorizations let a in r m times n denote an arbitrary matrix if x in r n and y in r m are vectors such that omega y t ax ne 0 then the matrix b a omega 1 axy t a a has rank exactly one less than the rank of a this wedderburn rank one reduction formula is easy to prove yet the idea is so powerful that perhaps all matrix factorizations can be derived from it the formula also appears in places such as the positive definite secant updates bfgs and dfp as well as the abs methods by repeatedly applying the formula to reduce ranks a biconjugation process analogous to the gram schmidt process with oblique projections can be developed this process provides a mechanism for constructing factorizations such as text ldm t qr and svd under a common framework of a general biconjugate decomposition v t au omega that is diagonal and nonsingular two characterizations of biconjugation provide new insight into the lanczos method and its breakdown one characterization shows that

differential calculi and linear connections a method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra as an example the generalized quantum plane is studied it is found that there is a strong correlation but not a one to one correspondence between the module structure of the 1 forms and the metric torsion free connections on it in the commutative limit the connection remains as a shadow of the algebraic structure of the 1 forms

an element based spectrally optimized approximate inverse preconditioner for the euler equations we introduce a method for constructing an element by element sparse approximate inverse sai preconditioner designed to be effective in a massively parallel spectral element modeling environment involving nonsymmetric systems this new preconditioning approach is based on a spectral optimization of a low resolution preconditioned system matrix we show that the local preconditioning matrices obtained via this element based spectrum optimized ebso approach may be applied to arbitrarily high resolution versions of the same system matrix without appreciable loss of preconditioner performance we demonstrate the performance of the ebso preconditioning approach using two dimensional spectral element method formulations for a simple linear conservation law and for the fully compressible two dimensional euler equations with various boundary conditions for the latter model running at sufficiently large courant number the ebso preconditioner significantly reduces both iteration count and wall clock time regar

sparse solution of the lyapunov equation for large scale interconnected systems we consider the problem of computing an approximate banded solution of the continuous time lyapunov equation a x x a t p where the coefficient matrices a and p are large symmetric banded matrices the sparsity pattern of a describes the interconnection structure of a large scale interconnected system recently it has been shown that the entries of the solution x are spatially localized or decaying away from a banded pattern we show that the decay of the entries of x is faster if the condition number of a is smaller by exploiting the decay of entries of x we develop two computationally efficient methods for approximating x by a banded matrix for a well conditioned and sparse banded a the computational and memory complexities of the methods scale linearly with the state dimension we perform extensive numerical experiments that confirm this and that demonstrate the effectiveness of the developed methods the methods proposed in this paper can be generalized to sparsity patterns of a and p that are more general than banded matrices the results of this paper open the possibility for developing computationally efficient methods for approximating the solution of the large scale riccati equation by a sparse matrix

multilevel block ilu preconditioner for sparse nonsymmetric m matrices we develop in this paper a multilevel block ilu preconditioner for solving sparse nonsymmetric m matrices arising from convection diffusion equations discretized by finite difference scheme this preconditioner is based on recursive block ilu factorization where the schur complement matrices are successively approached by some sparse matrices satisfying some hypothesis we study the preconditioned gmres iterative method and multigrid method for solving linear systems arising from this scheme

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

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

analysis of projection methods for solving linear systems with multiple right hand sides we analyze a class of krylov projection methods but mainly concentrate on a specific conjugate gradient cg implementation by smith peterson and mittra ieee transactions on antennas and propogation 37 1989 pp 1490 1493 to solve the linear system ax b where a is symmetric positive definite and b is a multiple of right hand sides this method generates a krylov subspace from a set of direction vectors obtained by solving one of the systems called the seed system by the cg method and then projects the residuals of other systems orthogonally onto the generated krylov subspace to get the approximate solutions the whole process is repeated with another unsolved system as a seed until all the systems are solved we observe in practice a superconvergence behavior of the cg process of the seed system when compared with the usual cg process we also observe that only a small number of restarts is required to solve all the systems if the right hand sides are close to each other these two features together make the method particularly effective in this paper we give theoretical proof to justify these observations furthermore we combine the advantages of this method and the block cg method and propose a block extension of this single seed method

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

classification of bicovariant differential calculi on quantum groups of type a b c and d under the assumptions thatq is not a root of unity and that the differentialsduji of the matrix entries span the left module of first order forms we classify bicovariant differential calculi on quantum groupsan 1 bn cn anddn we prove that apart one dimensional differential calculi and from finitely many values ofq there are precisely2n such calculi on the quantum groupan 1 slq n forn 3 all these calculi have the dimensionn2 for the quantum groupsbn cn anddn we show that except for finitely manyq there exist precisely twon2 dimensional bicovariant calculi forn 3 wheren 2n 1 forbn andn 2n forcn dn the structure of these calculi is explicitly described and the corresponding ad invariant right ideals of ker e are determined in the limitq 1 two of the 2n calculi foran 1 and one of the two calculi forbn cn anddn contain the ordinary classical differential calculus on the corresponding lie group as a quotient

noncommutative differential calculus and lattice gauge theory the authors study consistent deformations of the classical differential calculus on algebras of functions and more generally commutative algebras such that differentials and functions satisfy nontrivial commutation relations for a class of such calculi it is shown that the deformation parameters correspond to the spacings of a lattice these differential calculi generate a lattice on a space continuum the whole setting of a lattice theory can then be deduced from the continuum theory via deformation of the standard differential calculus in this framework one just has to express the lagrangian for the continuum theory in terms of differential forms this expression then also makes sense for the deformed differential calculus there is a natural integral associated with the latter integration of the lagrangian over a space continuum then produces the correct lattice action for a large class of theories this is explicitly shown for the scalar field action and the action for su m gauge theory

using spectral low rank preconditioners for large electromagnetic calculations for solving large dense complex linear systems that arise in electromagnetic calculations we perform experiments using a general purpose spectral low rank update preconditioner in the context of the gmres method preconditioned by an approximate inverse preconditioner the goal of the spectral preconditioner is to improve the convergence properties by shifting by one the smallest eigenvalues of the original preconditioned system numerical experiments on parallel distributed memory computers are presented to illustrate the efficiency of this technique on large and challenging real life industrial problems copyright 2004 john wiley sons ltd

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

an ilu preconditioner for nonsymmetric positive definite matrices by using the conjugate gram schmidt process the present paper investigates the approach mentioned but not developed in benzi tuma numer linear algebra appl 10 2003 in this approach using a conjugate gram schmidt algorithm an ilu preconditioner whose existence is guaranteed for any nonsingular symmetric and nonsymmetric real positive definite matrix is obtained numerical results on some nonsymmetric positive definite matrices demonstrate the efficiency of the method

absolute stability and synchronization in neural field models with transmission delays abstract neural fields model macroscopic parts of the cortex which involve several populations of neurons we consider a class of neural field models which are represented by integro differential equations with transmission time delays which are space dependent the considered domains underlying the systems can be bounded or unbounded a new approach called sequential contracting instead of the conventional lyapunov functional technique is employed to investigate the global dynamics of such systems sufficient conditions for the absolute stability and synchronization of the systems are established several numerical examples are presented to demonstrate the theoretical results

approximate inverse techniques for block partitioned matrices this paper proposes some preconditioning options when the system matrix is in block partitioned form this form may arise naturally for example from the incompressible navier stokes equations or may be imposed after a domain decomposition reordering approximate inverse techniques are used to generate sparse approximate solutions whenever these are needed in forming the preconditioner the storage requirements for these preconditioners may be much less than for incomplete lu factorization ilu preconditioners for tough large scale computational fluid dynamics cfd problems the numerical experiments show that these preconditioners can help solve difficult linear systems whose coefficient matrices are highly indefinite

bicovariant differential calculus on quantum groups su q n and so q n following woronowicz s proposal the bicovariant differential calculus on the quantum groupssu q n andso q n is constructed a systematic construction of bicovariant bimodules by using the hat r q matrix is presented the relation between the hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules and theq deformed universal enveloping algebras is given imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined as an application the maurer cartan equations and theq analogue of the structure constants are formulated

automorphisms of associative algebras and noncommutative geometry a class of differential calculi is explored which is determined by a set of automorphisms of the underlying associative algebra several examples are presented in particular differential calculi on the quantum plane the h deformed plane and the quantum group glp q 2 are recovered in this way geometric structures such as metrics and compatible linear connections are introduced

spectral two step preconditioning of multilevel fast multipole algorithm for the fast monostatic rcs calculation a new spectral two step preconditioning of multilevel fast multipole algorithm mlfma is proposed to solve large dense linear systems with multiple right hand sides arising in monostatic radar cross section rcs calculations the first system is solved with a deflated generalized minimal residual gmres method and the eigenvector information is generated at the same time based on this eigenvector information a spectral preconditioner is defined and combined with a previously constructed sparse approximate inverse sai preconditioner in a two step manner resulting in the proposed spectral two step preconditioner restarted gmres with the newly constructed spectral two step preconditioner is considered as the iterative method for solving subsequent systems and the mlfma is used to speed up the matrix vector product operations numerical experiments indicate that the new preconditioner is very effective with the mlfma and can reduce both the iteration number and the computational time significantly