rational singularities and almost split sequences the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the finite dimen sional representations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules of the quotient singularity associated with g over the past decade almost split sequences have been playing an increasingly important role in the representation theories of finite dimensional algebras and classical orders see 5 and 3 8 for basic existence theorems in these contexts while they have been known for some time to exist in higher dimensional situations 3 it has not been at all clear how they related to singularity theory if at all the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the fimte dimensional complex represen tations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules over the quotient singularity r associated with g as in the case of finite dimensional algebras the ar quiver of reflexive r modules is defined in terms of the almost split sequences of reflexive r modules in the case g c sl 2 c mckay observed that the underlying graph of the mckay quiver with the trivial module removed is isomorphic to the desingulariza tion graph of the associated singularity various explanations of this phenomenon have been given by knorrer gonzalez sprinberg verdier 6 and artin verdier 1 which along the way have established most explicitly in 1 a natural one to one correspondence between the indecomposable reflexive r modules and the nodes of the desingularization graph but why the almost split sequences describe the edge of the desingularization graph still remains to be explained an effort has been made to make this paper as self contained as possible in particular no prior knowledge of almost split sequences is required before describing the contents of the six sections of this paper we fix some notation throughout this paper g is a finite group k an algebraically closed field of characteristic not dividing the order of g and v a two dimensional k representation of g setting s k x y the k algebra of formal power series the two dimen sional representation v gives a linear action of g on s as a group of k algebra automorphisms we denote by s g the skew group ring given by this action

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

on the stability of relaxed incomplete lu factorizations when solving large linear systems of equations arising from the discretization of elliptic boundary value problems a combination of iterative methods and preconditioners based on incomplete lu factorizations is frequently used given a model problem with variable coefficients we investigate a class of incomplete lu factorizations depending on a relaxation parameter we show that the associated preconditioner and the factorization itself both are numerically stable the theoretical results are complemented by numerical experiments

indecomposable cohen macaulay modules and their multiplicities the main aim of this paper is to find a large class of rings for which there are indecomposable maximal cohen macaulay modules of arbitrary high multiplicity or rank in the case of domains

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

maximal subgroups of groups of intermediate growth abstract finding the number of maximal subgroups of infinite index of a finitely generated group is a natural problem that has been solved for several classes of geometric groups linear groups hyperbolic groups mapping class groups etc here we provide a solution for a family of groups with a different geometric origin groups of intermediate growth that act on rooted binary trees in particular we show that the non torsion iterated monodromy groups of the tent map a special case of some groups first introduced by sunic in 32 as siblings of the grigorchuk group have exactly countably many maximal subgroups of infinite index and describe them up to conjugacy this is in contrast to the torsion case e g grigorchuk group where there are no maximal subgroups of infinite index it is also in contrast to the above mentioned geometric groups where there are either none or uncountably many such subgroups along the way we show that all the groups defined by sunic have the congruence subgroup property and are just infinite

weighted matrix ordering and parallel banded preconditioners for iterative linear system solvers the emergence of multicore architectures and highly scalable platforms motivates the development of novel algorithms and techniques that emphasize concurrency and are tolerant of deep memory hierarchies as opposed to minimizing raw flop counts while direct solvers are reliable they are often slow and memory intensive for large problems iterative solvers on the other hand are more efficient but in the absence of robust preconditioners lack reliability while preconditioners based on incomplete factorizations whenever they exist are effective for many problems their parallel scalability is generally limited in this paper we advocate the use of banded preconditioners instead and introduce a reordering strategy that enables their extraction in contrast to traditional bandwidth reduction techniques our reordering strategy takes into account the magnitude of the matrix entries bringing the heaviest elements closer to the diagonal thus enabling the use of banded preconditioners when used with effective banded solvers in our case the spike solver we show that banded preconditioners i are more robust compared to the broad class of incomplete factorization based preconditioners ii deliver higher processor performance resulting in faster time to solution and iii scale to larger parallel configurations we demonstrate these results experimentally on a large class of problems selected from diverse application domains

the incomplete factorization multigraph algorithm we present a new family of multigraph algorithms ilu mg based upon an incomplete sparse matrix factorization using a particular ordering and allowing a limited amount of fill in while much of the motivation for multigraph comes from multigrid ideas ilu mg is distinctly different from algebraic multilevel methods the graph of the sparse matrix a is recursively coarsened by eliminating vertices using a graph model similar to gaussian elimination incomplete factorizations are obtained by allowing only the fill in generated by the vertex parents associated with each vertex multigraph is numerically compared with algebraic multigrid on some examples arising from discretizations of partial differential equations on unstructured grids

a scalable parallel algorithm for incomplete factor preconditioning we describe a parallel algorithm for computing incomplete factor ilu preconditioners the algorithm attains a high degree of parallelism through graph partitioning and a two level ordering strategy both the subdomains and the nodes within each subdomain are ordered to preserve concurrency we show through an algorithmic analysis and through computational results that this algorithm is scalable experimental results include timings on three parallel platforms for problems with up to 20 million unknowns running on up to 216 processors the resulting preconditioned krylov solvers have the desirable property that the number of iterations required for convergence is insensitive to the number of processors

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

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

indecomposable generalized cohen macaulay modules the aim of this paper is to study the indecomposable modules which are cohen macaulay on the punctured spectrum of a local ring and to describe some of their invariants such as their local cohomology groups and ranks one of our main concerns is to find indecomposable quasi buchsbaum modules of high rank with prescribed cohomology over a regular local ring

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

imf an incomplete multifrontal lu factorization for element structured sparse linear systems we propose an incomplete multifrontal lu factorization imf preconditioner that extends supernodal multifrontal methods to incomplete factorizations it can be used as a preconditioner in a krylov subspace method to solve large scale sparse linear systems with an element structure e g those arising from a finite element discretization of a partial differential equation the fact that the element matrices are dense is exploited to increase the computational performance and the robustness of the factorization through efficient partial pivoting imf is compared with the multilevel arms2 the level of fill in ilu and the threshold based ilutp preconditioners our experiments indicate that imf is competitive with arms2 on saddle point problems arising in the solution of the steady state navier stokes equation experiments with element structured matrices arising from structural engineering applications found in the university of florida sparse matrix collection illustrate the robustness of imf final

iterative solution methods for certain sparse linear systems with a non symmetric matrix arising from pde problems abstract in this paper methods are described for the solution of certain sparse linear systems with a non symmetric matrix the power of these methods is demonstrated by extensive numerical experiments application of the methods is limited to problems where the matrix has only eigenvalues with positive real part an important class of this type of matrix arises from discretisation of second order partial differential equations with first order derivative terms

multiscale finite element method for linear elastic geomechanics the demand for accurate and efficient simulation of geomechanical effects is widely increasing in the geoscience community high resolution characterizations of the mechanical properties of subsurface formations are essential for improving modeling predictions such detailed descriptions impose severe computational challenges and motivate the development of multiscale solution strategies we propose a multiscale solution framework for the geomechanical equilibrium problem of heterogeneous porous media based on the finite element method after imposing a coarse scale grid on the given fine scale problem the coarse scale basis functions are obtained by solving local equilibrium problems within coarse elements these basis functions form the restriction and prolongation operators used to obtain the coarse scale system for the displacement vector then a two stage preconditioner that couples the multiscale system with a smoother is derived for the iterative solution of the fine scale linear system various numerical experiments are presented to demonstrate accuracy and robustness of the method

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

performance analysis of preconditioners based on broyden method solving nonlinear equations is a problem often needed to be dealt with in the practical engineering application this paper discusses preconditioning methods based on block broyden method it first introduces the block broyden method and preconditioning technique then it presents four different preconditioners for the block broyden method and discusses the implementation process it also analyzes the time complexity of the four algorithms finally it solves the nonlinear systems arising from the bratu problem by the four preconditioning algorithms experimental results show that the preconditioning technique is effective for the block broyden method and that the preconditioner ssor has the fastest solving speed and the best performance among the four preconditioners therefore it can be used in the large scale problems arising from scientific and engineering computing

weakly maximal subgroups in regular branch groups abstract let g be a finitely generated regular branch group acting by automorphisms on a regular rooted tree t it is well known that stabilizers of infinite rays in t aka parabolic subgroups are weakly maximal subgroups in g that is maximal among subgroups of infinite index we show that given a finite subgroup q g g possesses uncountably many automorphism equivalence classes of weakly maximal subgroups containing q in particular for grigorchuk gupta sidki type groups this implies that they have uncountably many automorphism equivalence classes of weakly maximal subgroups that are not parabolic

preconditioned iterative methods for the nine point approximation to the convection diffusion equation iterative methods preconditioned by incomplete factorizations and sparse approximate inverses are considered for solving linear systems arising from fourth order finite difference schemes for convection diffusion problems simple recurrences for implementing the ilu 0 factorization of the nine point scheme are derived different sparsity patterns are considered for computing approximate inverses for the coefficient matrix and the quality of the preconditioner is studied in terms of plots of the field of values of the preconditioned matrices in terms of algebraic properties of the preconditioned matrices our experimental results show that incomplete factorizations give a preconditioner of better quality than approximate inverses comparison of the convergence rates of gmres applied to the preconditioned linear systems is done with respect to the field of values ritz and harmonic ritz values of the preconditioned matrices numerical results show that the gmres residual norm decreases rapidly when the difference between the ritz and harmonic ritz values becomes small we also describe the results of experiments when some well known krylov subspace methods are used to solve the linear system arising from the compact fourth order discretizations

the effect of ordering on preconditioned gmres algorithm for solving the compressible navier stokes equations we investigate the effect of the ordering of the blocks of unknowns on the rate of convergence of a preconditioned non linear gmres algorithm for solving the navier stokes equations for compressible flows using finite element methods on unstructured grids the gmres algorithm is preconditioned by an incomplete ldu block factorization of the jacobian matrix associated with the non linear problem to solve we examine a wide range of ordering methods including minimum degree reverse cuthill mckee and snake and consider preconditionings without fill in we show empirically that there can be a significant difference in the number of iterations required by the preconditioned non linear gmres method and suggest a criterion for choosing a good ordering algorithm according to the problem to solve we also consider the effect of orderings when an incomplete factorization which allows some fill in is performed we consider the effect of automatically controlling the sparsity of the incomplete factorization through the level of fill in finally following the principal ideas of non linear gmres algorithm we suggest other inexact newton methods

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

nested grids ilu decomposition ngilu a preconditioning technique is described which shows in many cases grid independent convergence this technique only requires an ordering of the unknowns based on the different levels of multigrid and an incomplete lu decomposition based on a drop tolerance the method is demonstrated on a variety of well known elliptic test problems including strongly varying coefficients advective terms and grid refinement

preconditioners for nonsymmetric indefinite linear systems abstract in this paper we develop algorithms for solving nonsymmetric indefinite linear systems by considering the augmented linear systems resulting from a weighted linear least squares problem even though the augmented system is more ill conditioned than the original linear system one can construct preconditioned gmres methods for solving these augmented systems capable of obtaining reasonable approximation of the solution in fewer iterations than the classical ilu preconditioned gmres method for solving the original linear system more specifically we present two different preconditioners for these augmented systems examine the spectral properties of these preconditioned augmented systems and report numerical results to illustrate the effectiveness of these preconditioners

dimension and singularity theory for local rings of finite embedding dimension abstract in this paper we develop an algebraic theory for local rings of finite embedding dimension several extensions of krull dimension are proposed which are then used to generalize singularity notions from commutative algebra finally variants of the homological theorems are shown to hold in equal characteristic this theory is then applied to noetherian local rings in order to get i over a cohen macaulay local ring uniform bounds on the betti numbers of a cohen macaulay module in terms of dimension and multiplicity and similar bounds for the bass numbers of a finitely generated module ii a characterization for being respectively analytically unramified analytically irreducible unmixed quasi unmixed normal cohen macaulay pseudo rational or weakly f regular in terms of certain uniform arithmetic behavior iii in mixed characteristic the improved new intersection theorem when the residual characteristic or ramification index is large with respect to dimension and some other numerical invariants

incomplete lu preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems we consider the preconditioned iterative solution of large dense linear systems where the coefficient matrix is a complex valued matrix arising from discretizing the integral equation of electromagnetic scattering for some scattering structures this matrix can be poorly conditioned the main purpose of this study is to evaluate the efficiency of a class of incomplete lu ilu factorization preconditioners for solving this type of matrices we solve the electromagnetic wave equations using the bicg method with an ilu preconditioner in the context of a multilevel fast multipole algorithm mlfma the novelty of this work is that the ilu preconditioner as constructed using the near part block diagonal submatrices generated from the mlfma experimental results show that the ilu preconditioner reduces the number of bicg iterations substantially compared to the block diagonal preconditioner the preconditioned iteration scheme also maintains the computational complexity of the mlfma and consequently reduces the total cpu time

automated adaptive multilevel solver abstract this paper presents an automated adaptive multilevel solver for linear or linearized system of equations the multilevel aspect of the solver is aimed at securing an optimal rate of convergence while keeping the size of the coarsest problem sufficiently small to ensure that the direct portion of the solution does not dominate the total computational cost adaptivity in terms of a priori selection of the number of levels one or more and construction of the optimal multilevel preconditioner is the key to the robustness of the method the number of levels is selected on the basis of estimated conditioning sparsity of the factor and available memory the auxiliary coarse models if required are automatically constructed on the basis of spectral characteristics of individual aggregates groups of neighboring elements an obstacle test consisting of twenty industry and model problems was designed to i determine the optimal values of computational parameters and to ii compare the adaptive multilevel scheme with existing state of the art equation solvers including the multifrontal solver 17 with the mmd reordering scheme and the pcg solver with the nearly optimal modified incomplete cholesky factorization preconditioner

global basis two level method for indefinite systems part 1 convergence studies a robust two level solver for high indefinite system of equations arising from the finite element discretization is developed it is shown that the optimal coarse model is spanned by the spectrum of the highest eigenmodes of the smoothing iteration matrix convergence studies conducted on a model prolongation operator reveal pathological sensitivity to any deviation from the optimal coarse model copyright 2000 john wiley sons ltd

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

deformations of maximal cohen macaulay modules let r m be a complete cohen macaulay isolated singularity over a field k which is either perfect or k k p 0 according to dieterich di and yo ch 6 see also pol pr or more generally chp there exists a system of parameters x xl xr of r such that the base change functor r x rdefines an injection v preserving the indecomposability from the set of isomorphism classes of maximal cohen macaulay r modules to the set of isomorphism classes of r x modules in this case the ideal generated by x is called after di a reduction ideal trying to describe the image of v we noticed in po2 that a finitely generated r x module is in imv if and only if it has the form xl x p for a finitely generated r2 r x 2 x module satisfying

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

on the burnside problem for periodic groups narain gupta 1 and said sidki 2 1 university of manitoba department of mathematics winnipeg manitoba r3t 2n2 canada 2 university of brasilia department of mathematics brasilia d f brazil introduction the generalized burnside problem refers to the question are finitely generated periodic groups finite this was answered in the negative by golod 1 who proved that for each prime p there exists a finitely generated infinite p group golod s construction in not however direct and is based on his celebrated work with safarevi recently grigor uk 2 has given a direct and elegant construction of an infinite 2 group which is generated by three elements of order 2 in this paper we give for each odd prime p a direct construction of an infinite p group on two generators each of order p our group is a subgroup of the automorphism group of a regular tree of degree p and as might be expected it is residually finite and has infinite exponent preliminaries let p be an odd prime and let t 0 be the infinite regular tree of degree p with vertex 0 so that through each vertex u of t 0 there are p regular subtrees

on accuracy conditions for the numerical computation of waves abstract the helmholtz equation k 2 n 2 u f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems such problems can be solved numerically by first truncating the given unbounded domain imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization for example using a finite element method in practical applications the mesh size h and the wave number k are not independent but are constrained by the accuracy of the desired computation it will be shown that the number of points per wavelength measured by kh 1 is not sufficient to determine the accuracy of a given discretization for example the quantity k 3 h 2 is shown to determine the accuracy in the l 2 norm for a second order discretization method applied to several propagation models

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 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

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

multifrontal incomplete factorization for indefinite and complex symmetric systems a new class of preconditioners based on the adaptive threshold incomplete multifrontal factorization for indefinite and complex symmetric systems is developed numerical experiments consisting of the 3d helmholtz equations fluid structure interaction and localization problems demonstrate the excellent performance of the preconditioner copyright 2001 john wiley sons ltd

boundary behavior for groups of subexponential growth in this paper we introduce a method for partial description of the poisson boundary for a certain class of groups acting on a segment as an application we find among the groups of subexponential growth those that admit nonconstant bounded harmonic functions with respect to some symmetric infinitely supported measure of finite entropy h this implies that the entropy h of the corresponding random walk is finite and positive as another application we exhibit certain discontinuity for the recurrence property of random walks finally as a corollary of our results we get new estimates from below for the growth function of a certain class of grigorchuk groups in particular we exhibit the first example of a group generated by a finite state automaton such that the growth function is subexponential but grows faster than exp n for any 1 we show that in some of our examples the growth function satisfies exp n ln

on maximal subgroups of infinite index in branch and weakly branch groups abstract we generalise a technical tool originally developed by pervova for the study of maximal subgroups in grigorchuk and ggs groups to all weakly branch groups satisfying a natural condition and in particular to all branch groups we then use this tool to prove that every maximal subgroup of infinite index of a branch group is also a branch group as a further application of this result we show that every maximal subgroup of the basilica group is of finite index

analysis of incomplete factorizations for a nine point approximation to a convection diffusion model problem we study the stability of zero fill incomplete lu factorizations of a nine point coefficient matrix arising from a high order compact discretisation of a two dimensional constant coefficient convection diffusion problem nonlinear recurrences for computing entries of the lower and upper triangular matrices are derived and we show that the sequence of diagonal entries of the lower triangular factor is unconditionally convergent a theoretical estimate of the limiting value is derived and we show that this estimate is a good predictor of the computed value the unconditional convergence of the diagonal sequence of the lower triangular factor to a positive limit implies that the incomplete factorization process never encounters a zero pivot and that the other diagonal sequences are also convergent the characteristic polynomials associated with the lower and upper triangular solves that occur during the preconditioning step are studied and conditions for the stability of the triangular solves are derived in terms of the entries of the tridiagonal matrices appearing in the lower and upper subdiagonals of the block triangular system matrix and a triplet of parameters which completely determines the solution of the nonlinear recursions results of ilu preconditioned gmres iterations and the effects of orderings on their convergence are also described

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

iterative solutions to the steady state density matrix for optomechanical systems we present a sparse matrix permutation from graph theory that gives stable incomplete lower upper preconditioners necessary for iterative solutions to the steady state density matrix for quantum optomechanical systems this reordering is efficient adding little overhead to the computation and results in a marked reduction in both memory and runtime requirements compared to other solution methods with performance gains increasing with system size either of these benchmarks can be tuned via the preconditioner accuracy and solution tolerance this reordering optimizes the condition number of the approximate inverse and is the only method found to be stable at large hilbert space dimensions this allows for steady state solutions to otherwise intractable quantum optomechanical systems

asymmetric hermitian and skew hermitian splitting methods for positive definite linear systems in this paper efficient iterative methods for the large sparse non hermitian positive definite systems of linear equations based on the hermitian and skew hermitian splitting of the coefficient matrix are studied these methods include an asymmetric hermitian skew hermitian ahss iteration and its inexact version the inexact asymmetric hermitian skew hermitian iahss iteration which employs some krylov subspace methods as its inner process we theoretically study the convergence properties of the ahss method and the iahss method moreover the contraction factor of the ahss iteration is derived numerical examples illustrating the effectiveness of both ahss and iahss iterations are presented

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