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

compressed graphs and the minimum degree algorithm the minimum degree algorithm is well known and widely used to find a permutation matrix p such that a sparse symmetric positive definite matrix a can be factored as pap t ll t to yield a lower triangular matrix l with relatively few nonzero entries in this paper we show that by compressing the graph of a when possible the run time of the minimum degree algorithm can be significantly reduced some empirical results for a collection of test matrices are presented

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

three classes of minimal linear codes over the finite fields of odd characteristic minimal linear codes are a special subclass of linear codes and have significant applications in secret sharing and secure two party computation in this paper we focus on constructing minimal linear codes with frac w min w max leq frac p 1 p for any odd prime p based on a generic construction of linear codes where w min and w max denote the minimum and maximum nonzero weights in a code respectively first we present two new infinite families of minimal linear codes with two or three weights by selecting suitable subcode of linear codes which are not minimal second we also present an infinite family of minimal linear codes by employing partial spreads which can be viewed as a generalization of the construction of ding et al in addition we determine the weight distributions of all these minimal linear codes

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

denumerable constrained markov decision processes and finite approximations the purpose of this paper is two fold first to establish the theory of discounted constrained markov decision processes with a countable state and action spaces with general multi chain structure second to introduce finite approximation methods we define the occupation measures and obtain properties of the set of all achievable occupation measures under the different admissible policies we establish the optimality of stationary policies for the constrained control problem and obtain an lp with a countable number of decision variables through which stationary optimal policies are computed since for such an lp one cannot expect to find an optimal solution in a finite number of operations we present two schemes for finite approximations and establish the convergence of optimal values and policies for both the discounted and the expected average cost with unbounded cost sometimes it turns out to be easier to solve the problem with infinite state space than the problem with finite yet large state space based on the optimal policy for the problem with infinite state space we construct policies which are almost optimal for the problem with truncated state space this method is applied to obtain an e optimal policy for a problem of optimal priority assignment under constraints for a system of k finite queues

the tortoise and the hare restart gmres when solving large nonsymmetric systems of linear equations with the restarted gmres algorithm one is inclined to select a relatively large restart parameter in the hope of mimicking the full gmres process surprisingly cases exist where small values of the restart parameter yield convergence in fewer iterations than larger values here two simple examples are presented where gmres 1 converges exactly in three iterations while gmres 2 stagnates one of these examples reveals that gmres 1 convergence can be extremely sensitive to small changes in the initial residual

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

adaptive techniques for improving the performance of incomplete factorization preconditioning three techniques for improving the robustness and performance of incomplete factorization preconditioners for sparse systems with symmetric positive definite or mildly indefinite coefficient matrices are introduced the primary contribution is two new block algorithms for incomplete factorization that result in an improvement in the performance of both the preconditioner generation and the iterative solution phases one of the algorithms applies to matrices that have a natural block structure in their original form and the other applies to matrices without natural dense blocks additionally two relatively simple but highly effective heuristic strategies are introduced these include selecting the solver based on the definiteness properties of the preconditioner and automatic selection and tuning of incomplete factorization parameters all three techniques have adaptive components i e the preconditioner solver combination chooses parameters or algorithmic components based on the properties of the coefficient matrix and its incomplete factors two of the three techniques are applicable to incomplete lu factorization for unsymmetric systems as well

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

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

enhancing performance and robustness of ilu preconditioners by blocking and selective transposition incomplete factorization is one of the most effective general purpose preconditioning methods for krylov subspace solvers for large sparse systems of linear equations two techniques for enhancing the robustness and performance of incomplete lu factorization for sparse unsymmetric systems are described a block incomplete factorization algorithm based on the crout variation of lu factorization is presented the algorithm is suitable for incorporating threshold based dropping as well as unrestricted partial pivoting and it overcomes several limitations of existing incomplete lu factorization algorithms with and without blocking it is shown that blocking has a three pronged impact it speeds up the computation of incomplete factors and the solution of the associated triangular systems it permits denser and more robust factors to be computed economically and it permits a trade off with the restart parameter of gmres to further improve the overall speed and robustness a highly effective heuristic for imp

finding exact and approximate block structures for ilu preconditioning sparse matrices which arise in many applications often possess a block structure that can be exploited in iterative and direct solution methods these block matrices have as their entries small dense blocks with constant or variable dimensions block versions of incomplete lu factorizations which have been developed to take advantage of such structures give rise to a class of preconditioners that are among the most effective available this paper presents general techniques for automatically determining block structures in sparse matrices a standard graph compression algorithm used in direct sparse matrix methods is considered along with two other algorithms which are also capable of unraveling approximate block structures

five families of three weight ternary cyclic codes and their duals as a subclass of linear codes cyclic codes have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms in this paper five families of three weight ternary cyclic codes whose duals have two zeros are presented the weight distributions of the five families of cyclic codes are settled the duals of two families of the cyclic codes are optimal

ilum a multi elimination ilu preconditioner for general sparse matrices standard preconditioning techniques based on incomplete lu ilu factorizations offer a limited degree of parallelism in general a few of the alternatives advocated so far consist of either using some form of polynomial preconditioning or applying the usual ilu factorization to a matrix obtained from a multicolor ordering in this paper we present an incomplete factorization technique based on independent set orderings and multicoloring we note that in order to improve robustness it is necessary to allow the preconditioner to have an arbitrarily high accuracy as is done with ilus based on threshold techniques the ilum factorization described in this paper is in this category it can be viewed as a multifrontal version of a gaussian elimination procedure with threshold dropping which has a high degree of potential parallelism the emphasis is on methods that deal specifically with general unstructured sparse matrices such as those arising from finite element methods on unstructured meshes

a domain decomposing parallel sparse linear system solver the solution of large sparse linear systems is often the most time consuming part of many science and engineering applications computational fluid dynamics circuit simulation power network analysis and material science are just a few examples of the application areas in which large sparse linear systems need to be solved effectively in this paper we introduce a new parallel hybrid sparse linear system solver for distributed memory architectures that contains both direct and iterative components we show that by using our solver one can alleviate the drawbacks of direct and iterative solvers achieving better scalability than with direct solvers and more robustness than with classical preconditioned iterative solvers comparisons to well known direct and iterative solvers on a parallel architecture are provided

on the covering structures of two classes of linear codes from perfect nonlinear functions in this paper the weight distributions of two classes of linear codes based on all known explicit perfect nonlinear functions from fqm to itself are determined using a unified approach all the minimal codewords of these codes are characterized according to their weights which suggests that their covering structures are determined finally all the minimal access sets of the secret sharing schemes based on their dual codes are obtained

approximations in dynamic zero sum games ii we pursue in this paper our study of approximations of values and epsilon saddle point policies in dynamic zero sum games after extending the general theorem for approximation we study zero sum stochastic games with countable state space and unbounded immediate reward we focus on the expected average payoff criterion we use some tools developed in m m tidball and e altman siam j control optim 34 1996 pp 311 328 to obtain the convergence of the values as well as the convergence of the epsilon saddle point policies in various approximation problems we consider several schemes of truncation of the state space e g finite state approximation and approximations of games with discount factor close to one for the game with expected average cost we use the extension of the general theorem for approximation to study approximations in stochastic games with complete information finally we consider the problem of approximating the sets of policies we obtain some general results that we apply to a pursuit evasion differential game

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

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

finite state approximations for denumerable multidimensional state discounted markov decision processes abstract a finite state iterative scheme introduced by white in recent developments in markov decision processes r hartley l c thomas and d j white eds academic press new york 1980 to approximate the value function of denumerable state markov decision processes is extended to denumerable multidimensional state space under essentially the same assumptions given in op cit a simpler proof of the convergence theorem is obtained together with convergence rates the iterative scheme is used to determine an asymptotically discount optimal policy which in turn can be used to obtain a discount optimal stationary policy

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

a convex analytic approach to markov decision processes this paper develops a new framework for the study of markov decision processes in which the control problem is viewed as an optimization problem on the set of canonically induced measures on the trajectory space of the joint state and control process this set is shown to be compact convex one then associates with each of the usual cost criteria infinite horizon discounted cost finite horizon control up to an exit time a naturally defined occupation measure such that the cost is an integral of some function with respect to this measure these measures are shown to form a compact convex set whose extreme points are characterized classical results about existence of optimal strategies are recovered from this and several applications to multicriteria and constrained optimization problems are briefly indicated

narrow sense bch codes over mathrm gf q with length n frac q m 1 q 1 cyclic codes are widely employed in communication systems storage devices and consumer electronics as they have efficient encoding and decoding algorithms bch codes as a special subclass of cyclic codes are in most cases among the best cyclic codes a subclass of good bch codes are the narrow sense bch codes over mathrm gf q with length n q m 1 q 1 little is known about this class of bch codes when q 2 the objective of this paper is to study some of the codes within this class in particular the dimension the minimum distance and the weight distribution of some ternary bch codes with length n 3 m 1 2 are determined in this paper a class of ternary bch codes meeting the griesmer bound is identified an application of some of the bch codes in secret sharing is also investigated

properties and applications of preimage distributions of perfect nonlinear functions the preimage distributions of perfect nonlinear functions from an abelian group of order n to an abelian group of order 3 or 4 respectively are studied based on the properties of the preimage distributions of perfect nonlinear functions from an abelian group of order 3r to an abelian group of order 3 the weight distributions of the ternary linear codes c pi from the perfect nonlinear functions pi x from f 3 r to itself are determined these results suggest that two open problems proposed by carlet ding and yuan in 2005 and 2006 respectively are answered

an algebraic convergence theory for restricted additive schwarz methods using weighted max norms convergence results for the restrictive additive schwarz ras method of cai and sarkis siam j sci comput 21 1999 pp 792 797 for the solution of linear systems of the form ax b are provided using an algebraic view of additive schwarz methods and the theory of multisplittings the linear systems studied are usually discretizations of partial differential equations in two or three dimensions it is shown that in the case of a symmetric positive definite the projections defined by the methods are not orthogonal with respect to the inner product defined by a and therefore the standard analysis cannot be used here the convergence results presented are for the class of m matrices and more generally for h matrices using weighted max norms comparison between different versions of the ras method are given in terms of these norms a comparison theorem with respect to the classical additive schwarz method makes it possible to indirectly get quantitative results on rates of convergence which otherwise cannot be obtained by the theory several ras variants are considered including new ones and two level schemes

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

stabilized and block approximate inverse preconditioners for problems in solid and structural mechanics the solution of linear systems arising in the finite element analysis of shells and solids by the preconditioned conjugate gradient method is considered stabilized and block versions of the ainv factorized approximate inverse preconditioner are presented and tested on a variety of difficult problems comparisons with other preconditioning methods are also included

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

multilevel preconditioners constructed from inverse based ilus this paper analyzes dropping strategies in a multilevel incomplete lu decomposition context and presents a few strategies for obtaining related ilus with enhanced robustness the analysis shows that the incomplete lu factorization resulting from dropping small entries in gaussian elimination produces a good preconditioner when the inverses of these factors have norms that are not too large as a consequence a few strategies are developed whose goal is to achieve this feature a number of templates for enabling implementations of these factorizations are presented numerical experiments show that the resulting ilus offer a good compromise between robustness and efficiency

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

the superlinear convergence behaviour of gmres abstract gmres is a rather popular iterative method for the solution of nonsingular nonsymmetric linear systems it is well known that gmres often has a so called superlinear convergence behaviour i e the rate of convergence seems to improve as the iteration proceeds for the conjugate gradients method this phenomenon has been related to a modest degree of convergence of the ritz values it has been observed in experiments that for gmres too changes in the convergence behaviour seem to be related to the convergence of ritz values in this paper we prove that as soon as eigenvalues of the original operator are sufficiently well approximated by ritz values gmres from then on converges at least as fast as for a related system in which these eigenvalues and their eigenvector components are missing

characterizations of e duality gap statements for composed optimization problems abstract in this paper we present different regularity conditions that equivalently characterize e duality gap statements for optimization problems consisting of minimizing the sum of a function with the precomposition of a cone increasing function to a vector function these regularity conditions are formulated by using epigraphs and e subdifferentials taking e 0 one can rediscover recent results on stable strong and total duality and zero duality gap from the literature moreover we deliver as byproducts e optimality conditions and e saddle point statements for the aforementioned kind of problems and e farkas statements involving the sum of a function with the precomposition of a cone increasing function to a vector function

on the relationships between perfect nonlinear functions and universal hash families in this paper the relationships between perfect nonlinear in brief pn functions and optimal universal hash families are discussed we point out the equivalence of constructions between them i e from pn functions one can obtain optimal universal hash families and vice versa as an application of our construction a message authentication code is proposed which provides better resistance to substitution attack than a known construction given by carlet et al in 2006 more generally the connections between functions with given differential uniformity and some universal hash families are studied

differential spectrum of kasami power permutations over odd characteristic finite fields functions with low differential uniformity have important applications in cryptography coding theory and sequence design the differential spectrum of a cryptographic function is of great interest for estimating its resistance to some variants of differential cryptanalysis finding power permutations i e monomial bijective mappings over finite fields with low differential uniformity and determining their differential spectra have received a lot of attention over the past two decades the objective of this paper is to study the differential properties of the well known kasami power permutations x p 2k p k 1 over mathrm gf p n where p is an odd prime and k is an integer with gcd n k 1 it turns out that this family of monomials is differentially p 1 uniform our result in the case of p 3 gives an affirmative solution to a recent conjecture by xu cao and xu most notably the differential spectrum of this family of power permutations is completely determined

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

linear codes with two or three weights from weakly regular bent functions linear codes with a few weights have applications in consumer electronics communication data storage system secret sharing authentication codes association schemes and strongly regular graphs this paper first generalizes the method of constructing two weight and three weight linear codes of ding et al and zhou et al to general weakly regular bent functions and determines the weight distributions of these linear codes it solves an open problem proposed by ding et al furthermore this paper constructs new linear codes with two or three weights and presents their weight distributions they contain some optimal codes meeting certain bound on linear codes

multilevel ilu decomposition in this paper the multilevel ilu mlilu decomposition is introduced during an incomplete gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels on these levels some smoothing steps are computed the mlilu decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices convergence rates independent of the number of unknowns are shown numerically for several examples many numerical experiments including unsymmetric and anisotropic problems problems with jumping coefficients as well as realistic problems are presented they indicate a very robust convergence behavior of the mlilu method

algebraic multilevel preconditioning methods i a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance

linear codes from perfect nonlinear mappings and their secret sharing schemes in this paper error correcting codes from perfect nonlinear mappings are constructed and then employed to construct secret sharing schemes the error correcting codes obtained in this paper are very good in general and many of them are optimal or almost optimal the secret sharing schemes obtained in this paper have two types of access structures the first type is democratic in the sense that every participant is involved in the same number of minimal access sets in the second type of access structures there are a few dictators who are in every minimal access set while each of the remaining participants is in the same number of minimal access sets

distributed schur complement techniques for general sparse linear systems this paper presents a few preconditioning techniques for solving general sparse linear systems on distributed memory environments these techniques utilize the schur complement system for deriving the preconditioning matrix in a number of ways two of these preconditioners consist of an approximate solution process for the global system which exploits approximate lu factorizations for diagonal blocks of the schur complement another preconditioner uses a sparse approximate inverse technique to obtain certain local approximations of the schur complement comparisons are reported for systems of varying difficulty

two person zero sum markov games receding horizon approach we consider a receding horizon approach as an approximate solution to two person zero sum markov games with infinite horizon discounted cost and average cost criteria we first present error bounds from the optimal equilibrium value of the game when both players take correlated receding horizon policies that are based on exact or approximate solutions of receding finite horizon subgames motivated by the worst case optimal control of queueing systems by altman we then analyze error bounds when the minimizer plays the approximate receding horizon control and the maximizer plays the worst case policy we finally discuss some state space size independent methods to compute the value of the subgame approximately for the approximate receding horizon control along with heuristic receding horizon policies for the minimizer

cyclotomic constructions of cyclic codes with length being the product of two primes cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms in this paper three types of generalized cyclotomy of order two are described and three classes of cyclic codes of length n1 n2 and dimension n1 n2 1 2 are presented and analyzed where n1 and n2 are two distinct primes bounds on their minimum odd like weight are also proved some of the codes presented in this paper are among the best cyclic codes

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance