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

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

a branch and cut approach to the cardinality constrained circuit problem the cardinality constrained circuit problem cccp is the problem of finding a minimum cost circuit in a graph where the circuit is constrained to have at most k edges the cccp is np hard we present classes of facet inducing inequalities for the convex hull of feasible circuits and a branch and cut solution approach using these inequalities

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

on signed incomplete cholesky factorization preconditioners for saddle point systems limited memory incomplete cholesky factorizations can provide robust preconditioners for sparse symmetric positive definite linear systems in this paper the focus is on extending the approach to sparse symmetric indefinite systems in saddle point form a limited memory signed incomplete cholesky factorization of the form ldl t is proposed where the diagonal matrix d has entries pm 1 the main advantage of this approach is its simplicity as it avoids the use of numerical pivoting instead a global shift strategy involving two shifts one for the 1 1 block and one for the 2 2 block of the saddle point matrix is used to prevent breakdown and to improve performance the matrix is optionally prescaled and preordered using a standard sparse matrix ordering scheme that is then postprocessed to give a constrained ordering that reduces the likelihood of breakdown and need for shifts the use of intermediate memory memory used in the construction of the incomplete factorization but subsequently d

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

symmetric submodular system contractions and gomory hu tree abstract let s v f be a symmetric submodular system for two distinct elements s and l of v let s l denote the set of all subsets of v which separate s from l by using any gomory hu tree of s we can obtain an element of s l which has minimum value among all the elements of s l this tree can be constructed iteratively by solving v 1 minimum s l separator problem pendant pairs of a symmetric submodular system play a key role in finding a minimizer of this system in this paper we obtain a gomory hu tree of a contraction of s with respect to some subsets of v only by using contraction in a gomory hu tree of s furthermore we obtain some pendant pairs of s and of its contractions by using a gomory hu tree of s

an in depth study of graph partitioning measures for perceptual organization in recent years one of the effective engines for perceptual organization of low level image features is based on the partitioning of a graph representation that captures gestalt inspired local structures such as similarity proximity continuity parallelism and perpendicularity over the low level image features mainly motivated by computational efficiency considerations this graph partitioning process is usually implemented as a recursive bipartitioning process where at each step the graph is broken into two parts based on a partitioning measure we focus on three such measures namely the minimum average and normalized cuts the minimum cut partition seeks to minimize the total link weights cut the average cut measure is proportional to the total link weight cut normalized by the sizes of the partitions the normalized cut measure is normalized by the product of the total connectivity valencies of the nodes in each partition we provide theoretical and empirical insight into the nature of the three partitioning measures in terms of the underlying image statistics in particular we consider for what kinds of image statistics would optimizing a measure irrespective of the particular algorithm used result in correct partitioning are the quality of the groups significantly different for each cut measure are there classes of images for which grouping by partitioning does not work well also can the recursive bipartitioning strategy separate out groups corresponding to k objects from each other in the analysis we draw from probability theory and the rich body of work on stochastic ordering of random variables our major conclusion is that optimization of none of the three measures is guaranteed to result in the correct partitioning of k objects in the strict stochastic order sense for all image statistics

on the intersection of the f maximal subgroups and the generalized f hypercentre of a finite group abstract let f be a class of groups a chief factor h k of a group g is called f central in g provided h k g c g h k f we write z f g to denote the product of all normal subgroups of g whose g chief factors of order divisible by at least one prime in are f central we call z f g the f hypercentre of g a subgroup u of a group g is called f maximal in g provided that a u f and b if u v g and v f then u v in this paper we study the properties of the intersection of all f maximal subgroups of a finite group in particular we analyze the condition under which z f g coincides with the intersection of all f maximal subgroups of g

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

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

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

improved scaling for quantum monte carlo on insulators quantum monte carlo qmc methods are often used to calculate properties of many body quantum systems the main cost of many qmc methods for example the variational monte carlo vmc method is in constructing a sequence of slater matrices and computing the ratios of determinants for successive slater matrices recent work has improved the scaling of constructing slater matrices for insulators so that the cost of constructing slater matrices in these systems is now linear in the number of particles whereas computing determinant ratios remains cubic in the number of particles with the long term aim of simulating much larger systems we improve the scaling of computing the determinant ratios in the vmc method for simulating insulators by using preconditioned iterative solvers the main contribution of this paper is the development of a method to efficiently compute for the slater matrices a sequence of preconditioners that make the iterative solver converge rapidly this involves cheap preconditioner updates an effective reordering strategy and a cheap method to monitor instability of incomplete lu decomposition with threshold and pivoting ilutp preconditioners using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive slater matrices reduces the scaling of qmc algorithms from o n 3 per sweep to roughly o n 2 where n is the number of particles and a sweep is a sequence of n steps each attempting to move a distinct particle we demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors our results show that preconditioned iterative solvers can dramatically reduce the cost of vmc for large r systems

the evolution of the minimum degree ordering algorithm over the past fifteen years the implementation of the minimum degree algorithm has received much study and many important enhancements have been made to it this paper describes these various enhancements their historical development and some experiments showing how very effective they are in improving the execution time of the algorithm a shortcoming is also presented that exists in all of the widely used implementations of the algorithm namely that the quality of the ordering provided by the implementations is surprisingly sensitive to the initial ordering for example changing the input ordering can lead to an increase or decrease of as much as a factor of three in the cost of the subsequent numerical factorization this sensitivity is caused by the lack of an effective tie breaking strategy and the authors experiments illustrate the importance of developing such a strategy

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

approximation and hardness results for the max k uncut problem abstract in the study of the homophily law of large scale complex networks we get a combinatorial optimization problem which we call the max k uncut problem given an n vertex undirected graph g v e with nonnegative weights w e e e defined on edges and a positive integer k the max k uncut problem asks to find a partition v 1 v 2 v k of v such that the total weight of edges that are not cut is maximized intuitively an edge that is not cut connects two vertices with the same or similar attributes since they are in the same part of the partition interestingly the max k uncut problem is just the complement of the classic min k cut problem for max k uncut we present a randomized 1 k n 2 approximation algorithm a greedy 1 2 k 1 n approximation algorithm and an 1 2 approximation algorithm by reducing it to densest k subgraph where is the approximation ratio of the densest k subgraph problem more importantly we show that max k uncut and densest k subgraph are in fact equivalent in approximability up to a factor of 2 we also prove an approximation hardness result for max k uncut under the assumption p np

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

enhanced multi level block ilu preconditioning strategies for general sparse linear systems abstract this paper introduces several strategies to deal with pivot blocks in multi level block incomplete lu factorization bilum preconditioning techniques these techniques are aimed at increasing the robustness and controlling the amount of fill ins of bilum for solving large sparse linear systems when large size blocks are used to form block independent set techniques proposed in this paper include double dropping strategies approximate singular value decomposition variable size blocks and use of an arrowhead block submatrix we point out the advantages and disadvantages of these strategies and discuss their efficient implementations numerical experiments are conducted to show the usefulness of the new techniques in dealing with hard to solve problems arising from computational fluid dynamics in addition we discuss the relation between multi level ilu preconditioning methods and algebraic multi level methods

a modified ssor preconditioner for sparse symmetric indefinite linear systems of equations the standard ssor preconditioner is ineffective for the iterative solution of the symmetric indefinite linear systems arising from finite element discretization of the biot s consolidation equations in this paper a modified block ssor preconditioner combined with the eisenstat trick implementation is proposed for actual implementation a pointwise variant of this modified block ssor preconditioner is highly recommended to obtain a compromise between simplicity and effectiveness numerical experiments show that the proposed modified ssor preconditioned symmetric qmr solver can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime moreover the proposed modified ssor preconditioners can be generalized to non symmetric biot s systems copyright 2005 john wiley sons ltd

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

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

weighted matchings for preconditioning symmetric indefinite linear systems maximum weight matchings have become an important tool for solving highly indefinite unsymmetric linear systems especially in direct solvers in this study we investigate the benefit of reorderings and scalings based on symmetrized maximum weight matchings as a preprocessing step for incomplete mathrm ldl t factorizations the reorderings are constructed such that the matched entries form 1 times 1 or 2 times 2 diagonal blocks in order to increase the diagonal dominance of the system during the incomplete factorization only tridiagonal pivoting is used we report results for this approach and comparisons with other solution methods for a diverse set of symmetric indefinite matrices ranging from nonlinear elasticity to interior point optimization

a hybridized iterative algorithm of the bicorstab and gpbicor methods for solving non hermitian linear systems in this study we derive a new iterative algorithm including its preconditioned version which is a hybridized variant of the biconjugate a orthogonal residual stabilized bicorstab method and the generalized product type solvers based on bicor gpbicor method the proposed method which is named gpbicor m similarly to the gpbicg m method proposed by fujino 2002 can be regarded as an extension of the bicorstab2 method introduced by zhao and huang 2013 inspired by fujino s idea for improving the bicgstab2 method in the established gpbicor m method the parameters computed by the bicorstab method are chosen at successive m iteration steps and afterwards the parameters of the gpbicor method are utilized in the subsequent iteration steps therefore the proposed method can inherit the low computational cost of bicorstab and the attractive convergence of gpbicor extensive numerical convergence results on selected real and complex matrices are shown to assess the performances of the proposed gpbicor m method also against other popular non hermitian krylov subspace methods

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

greedy splitting algorithms for approximating multiway partition problems given a system v t f k where v is a finite set open image in new window is a submodular function and k 2 is an integer the general multiway partition problem mpp asks to find a k partition open image in new window v1 v2 vk of v that satisfiesopen image in new windowfor all i and minimizes f v1 f v2 f vk where open image in new window is a k partition of open image in new window hold mpp formulation captures a generalization in submodular systems of many np hard problems such as k way cut multiterminal cut target split and their generalizations in hypergraphs this paper presents a simple and unified framework for developing and analyzing approximation algorithms for various mpps

a spectral multigrid method combined with mlfmm for solving electromagnetic wave scattering problems a new spectral multigrid method smg combined with the multilevel fast multipole method mlfmm is proposed for solving electromagnetic wave scattering problems the mlfmm is used to speed up the matrix vector product operations and the smg is employed to accelerate the convergence rate of the krylov iteration unlike traditional algebraic multigrid methods amg the spectral multigrid method is an algebraic two grid cycle built on a preconditioned krylov iterative method that is used as the smoother and the grid transfer operators are defined using the spectral information of the preconditioned matrix numerical experiments indicate that this class of multigrid method is very effective with the mlfmm and can reduce both the iteration number and the overall simulation time significantly

a note on intersections of maximal f subgroups abstract let g be a finite solvable group and f a saturated formation we characterize the intersection of all maximal f subgroups of g

on generalized u hypercentral subgroups of a finite group abstract we give some conditions under which a normal subgroup of a finite group g is contained in the intersection of all maximal supersoluble subgroups of g

bicgcr2 a new extension of conjugate residual method for solving non hermitian linear systems abstract in the present paper we introduce a new extension of the conjugate residual cr method for solving non hermitian linear systems with the aim of developing an alternative basic solver to the established biconjugate gradient bicg biconjugate residual bicr and biconjugate a orthogonal residual bicor methods the proposed krylov subspace method referred to as the bicgcr2 method is based on short term vector recurrences and is mathematically equivalent to both bicr and bicor we demonstrate by extensive numerical experiments that the proposed iterative solver has often better convergence performance than bicg bicr and bicor hence it may be exploited for the development of new variants of non optimal krylov subspace methods

on the intersection of the normalizers of derived subgroups of all subgroups of a finite group given a finite group g we define the subgroup d g to be the intersection of the normalizers of derived subgroups of all subgroups of g set d0 1 define di 1 g di g d g di g for i 1 by d g denote the terminal term of the ascending series it is proved that the derived subgroup g is nilpotent if and only if g d g furthermore if all elements of prime order of g are in d g then g is soluble with fitting length at most 3 in section 3 it is proved that if the group g satisfies g d g then g is nilpotent and g has nilpotency class at most 2

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

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

finding k cuts within twice the optimal two simple approximation algorithms for the minimum k cut problem are presented each algorithm finds a k cut having weight within a factor of 2 2 k of the optimal one of our algorithms is particularly efficient it requires a total of only n 1 maximum flow computations for finding a set of near optimal k cuts one for each value of k between 2 and n

approximating k cuts using network strength as a lagrangean relaxation abstract given an undirected edge weighted connected graph the k cut problem is to partition the vertex set into k non empty connected components so as to minimize the total weight of edges whose end points are in different components we present a combinatorial polynomial time 2 approximation algorithm for the k cut problem we use a lagrangean relaxation also suggested by barahona f barahona on the k cut problem operations research letters 26 2000 99 105 to reduce the problem to the attack problem for which a polynomial time algorithm was provided by cunningham w cunningham optimal attack and reinforcement of a network journal of the acm 32 3 1985 549 561 we prove several structural results of the relaxation and use these results to develop an approximation algorithm we provide analytical comparisons of our algorithm and lower bound with two others saran and vazirani h saran v vazirani finding k cuts within twice the optimal siam journal of computing 24 1 1995 101 108 and naor and rabani j naor y rabani tree packing and approximating k cuts in proceedings of the 12th annual acm siam symposium on discrete algorithms 2001 pp 26 27 we also provide computational results comparing the performance of our algorithm on random graphs with respect to the lower bound provided by the attack problem as well as an alternate 2 approximation algorithm provided by saran and vazirani cited above

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

multipoint and multi objective aerodynamic shape optimization a newton krylov algorithm is presented for the aerodynamic optimization of singleand multi element airfoil configurations the flow is governed by the compressible navier stokes equations in conjunction with a one equation turbulence model the preconditioned generalized minimum residual method is applied to solve the discreteadjoint equation leading to a fast computation of accurate objective function gradients optimization constraints are enforced through a penalty formulation and the resulting unconstrained problem is solved via a quasi newton method design examples include lift enhancement and multi point lift constrained drag minimization problems furthermore the new algorithm is used to compute a pareto front for a multi objective problem and the results are validated using a genetic algorithm overall the new algorithm provides an ecient and robust approach for addressing the issues of complex aerodynamic

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

beating the 2 approximation factor for global bicut in the fixed terminal bicut problem the input is a directed graph with two specified nodes s and t and the goal is to find a smallest subset of edges whose removal ensures that s cannot reach tandt cannot reach s in the global bicut problem the input is a directed graph and the goal is to find a smallest subset of edges whose removal ensures that there exist two nodes s and t such that s cannot reach tandt cannot reach s fixed terminal bicut and global bicut are natural extensions of s t min cut and global min cut respectively from undirected graphs to directed graphs fixed terminal bicut is np hard admits a simple 2 approximation and does not admit a 2 epsilon approximation for any constant epsilon 0 assuming the unique games conjecture in this work we show that global bicut admits a 2 1 448 approximation thus improving on the approximability of the global variant in comparison to the fixed terminal variant

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

vbarms a variable block algebraic recursive multilevel solver for sparse linear systems sparse matrices arising from the solution of systems of partial differential equations often exhibit a perfect block structure meaning that the nonzero blocks in the sparsity pattern are fully dense and typically small e g when several unknown quantities are associated with the same grid point similar block orderings can be sometimes unravelled also on general unstructured matrices by ordering consecutively rows and columns with a similar sparsity pattern and treating some zero entries of the reordered matrix as nonzero elements with a little sacrifice of memory we show how we can take advantage of these frequently occurring structures in the design of the multilevel incomplete lu factorization preconditioner arms saad and suchomel 2002 14 and maximize computational efficiency

on the πf norm and the h f norm of a finite group abstract let h be a fitting class and f a formation we call a subgroup n h f g of a finite group g the h f norm of g if n h f g is the intersection of the normalizers of the products of the f residuals of all subgroups of g and the h radical of g let denote a set of primes and let g denote the class of all finite groups we call the subgroup n g f g of g the f norm of g a normal subgroup n of g is called f hypercentral in g if either n 1 or n 1 and every g chief factor below n of order divisible by at least one prime in is f central in g let z f g denote the f hypercentre of g that is the product of all f hypercentral normal subgroups of g in this paper we study the properties of the h f norm especially of the f norm of a finite group g in particular we investigate the relationship between the f norm and the f hypercentre of g

a markov chain based multi elimination preconditioner for elliptic pde problems abstract preconditioning is a simple but very powerful technique for accelerating the convergence of an iterative method for solving large sparse linear systems traditionally to design a good preconditioner implemented on a personal computer or a cluster of computer systems with cpus the preconditioner should meet the following three conditions a the preconditioner is a good approximation of the inverse of the original linear coefficient matrix in some sense b setting up the components needed for the preconditioning operator is easy and c the cost for the application of the preconditioner with a given vector is low however as new computer systems are rapidly invented such designing strategies need to be reinvestigated motivated by the random walk technique which is a computationally intensive task we proposed a different preconditioner based on a multi elimination algorithm where the inversion operator is approximated explicitly by a partial product of the neumann series of the iterative matrix for the original system the computation of such an approximation takes advantages of the strengths of the gpu system we report some numerical results implemented by cuda petsc in our implementation we adopt the hybrid gpu cpu approach that preconditioning setup and application are performed on gpu while krylov subspace iteration which requires a certain degree of communications is done on cpu this method minimizes data communication between gpu cpu which is a bottleneck for many existing algorithms

a novel hierarchical two level spectral preconditioning technique for electromagnetic wave scattering a new set of higher order hierarchical basis functions based on curvilinear triangular patch is proposed for expansion of the current in electrical field integral equations solved by method of moments the hierarchical two level spectral preconditioning technique is developed for the generalized minimal residual iterative method in which the multilevel fast multipole method is used to accelerate matrix vector product the sparse approximate inverse sai preconditioner based on the higher order hierarchical basis functions is used to damp the high frequencies of the error and the low frequencies is eliminated by a spectral preconditioner in a two level manner defined on the lower order basis functions the spectral preconditioner is combined with sai preconditioner to obtain a hierarchical two level spectral preconditioner numerical experiments indicate that the new preconditioner can significantly reduce both the iteration number and computational time

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

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

on the intersection of all maximal f subgroups of a finite group abstract let f be a class of groups a subgroup h of a group g is said to be a maximal f subgroup of g if h f and g has no a subgroup e f such that h e the symbol f g denotes the intersection of all maximal f subgroups of g we study the influence of the subgroup f g on the structure of a finite group g

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

a grid based multilevel incomplete lu factorization preconditioning technique for general sparse matrices we design a grid based multilevel incomplete lu preconditioner gilum for solving general sparse matrices this preconditioner combines a high accuracy ilu factorization with an algebraic multilevel recursive reduction the gilum preconditioner is a compliment to the domain based multilevel block ilut preconditioner a major difference between these two preconditioners is the way that the coarse level nodes are chosen the approach of gilum is analogous to that of algebraic multigrid method the gilum approach avoids some controversial issues in algebraic multigrid method such as how to construct the interlevel transfer operators and how to compute the coarse level operator numerical experiments are conducted to compare gilum with other ilu preconditioners

algorithmic aspects of homophyly of networks abstract we investigate the algorithmic problems of the homophyly phenomenon in networks given an undirected graph g v e and a vertex coloring c v 1 2 k of g we say that a vertex v v is happy if v shares the same color with all its neighbors and unhappy otherwise and that an edge e e is happy if its two endpoints have the same color and unhappy otherwise supposing c is a partial vertex coloring of g we define the maximum happy vertices problem mhv for short as to color all the remaining vertices such that the number of happy vertices is maximized and the maximum happy edges problem mhe for short as to color all the remaining vertices such that the number of happy edges is maximized let k be the number of colors allowed in the problems we show that both mhv and mhe can be solved in polynomial time if k 2 and that both mhv and mhe are np hard if k 3 we devise a max 1 k 3 approximation algorithm for the mhv problem where is the maximum degree of vertices in the input graph and a 1 2 approximation algorithm for the mhe problem this is the first theoretical progress of these two natural and fundamental new problems

fast unsteady flow computations with a jacobian free newton krylov algorithm despite the advances in computer power and numerical algorithms over the last decades solutions to unsteady flow problems remain computing time intensive especially for large reynolds number flows nonlinear multigrid which is commonly used to solve the nonlinear systems of equations converges slowly the stiffness induced by the large aspect ratio cells and turbulence is not tackled well by this solution method in previous work we showed that a jacobian free newton krylov jfnk algorithm preconditioned with an approximate factorization of the jacobian that approximately matches the target residual operator enables a speed up of a factor of 10 compared to standard nonlinear multigrid for two dimensional large reynolds number unsteady flow computations the goal of this paper is to demonstrate that the jfnk algorithm is also suited to tackle the stiffness induced by the maximum aspect ratio the grid density the physical time step and the reynolds number compared to standard nonlinear multigrid speed ups up to a factor of 25 are achieved

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 new stabilization strategy for incomplete lu preconditioning of indefinite matrices a two step combined preconditioning strategy is proposed to construct stable and accurate incomplete lu factorization of indefinite matrices arising from cfd applications this preconditioning procedure is divided into two steps each step is a factorization of a shifted matrix numeral experiments show that a preconditioner can be computed with high accuracy and low fill in and that the new strategy is robust on some difficult indefinite matrix test problems

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

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

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

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

rational approximation preconditioners for sparse linear systems this paper presents a class of preconditioning techniques which exploit rational function approximations to the inverse of the original matrix the matrix is first shifted and then an incomplete lu factorization of the resulting matrix is computed the resulting factors are then used to compute a better preconditioner for the original matrix since the incomplete factorization is made on a shifted matrix a good lu factorization is obtained without allowing much fill in the result needs to be extrapolated to the nonshifted matrix thus the main motivation for this process is to save memory the method is useful for matrices whose incomplete lu factorizations are poor e g unstable

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

on the sum max graph partitioning problem abstract this paper tackles the following problem given a connected graph g v e with a weight function on its edges and an integer k v find a partition of v into k clusters such that the sum over all pairs of clusters of the heaviest edges between the clusters is minimized we first prove that this problem and even the unweighted variant cannot be approximated within a factor of o n 1 unless p np and cannot admit an fpt algorithm unless fpt w 1 next we develop several algorithms we first present a greedy algorithm and prove that it achieves a tight approximation ratio smaller than k 2 concerning the unweighted version of the problem we study how the diameter of the input graph can influence its complexity by obtaining negative results for graphs of low diameter positive results for graphs of high diameter and a polynomial time approximation algorithm with an approximation ratio smaller than k 2 and depending on the diameter we also highlight a link between the k sparsest subgraph problem and the unweighted version this allows us to develop several constant ratio approximation algorithms running in exponential time for general graphs and in polynomial time in some restricted graph classes finally we discuss the exact resolution for fixed k and the connections to a graph homomorphism problem

on the norm of the nilpotent residuals of all subgroups of a finite group abstract r baer and wielandt in 1934 and 1958 respectively considered the intersection of the normalizers of all subgroups of g and the intersection of the normalizers of all subnormal subgroups of g in this paper for a finite group g we define the subgroup s g to be the intersection of the normalizers of the nilpotent residuals of all subgroups of g set s 0 1 define s i 1 g s i g s g s i g for i 1 by s g denote the terminal term of the ascending series it is proved that g s g if and only if the nilpotent residual g n is nilpotent furthermore if all elements of prime order of g are in s g then g is solvable and l p g 1 where l p g is p length of g for p g where g means the set of prime divisors of g

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