the dual ghr zeros and residues abstract the dual generalized hessenberg representation dual ghr is a canonic form for single input single output systems which identifies the input output properties of the system here it is shown how the dual ghr can be used to identify almost pole zero cancellations and small residues of the corresponding poles this connection between the state space representation and transfer function is established using geometrical arguments near unobservability and near uncontrollability measures relying on angles between subspaces play a key role

a semiparametric approach for the two sample comparison of survival times with long term survivors summary in the two sample comparison of survival times with long term survivors the overall difference between the two distributions reflects differences occurring in early follow up for susceptible subjects and in long term follow up for nonsusceptible subjects in this setting we propose statistics for testing i no overall ii no short term and iii no long term difference between the two distributions to be compared the statistics are derived as follows a semiparametric model is defined that characterizes a short term effect and a long term effect by approximating this model about no difference in early survival a time dependent proportional hazards model is obtained the statistics are obtained from this working model the asymptotic distributions of the statistics for testing no overall or no short term effects are ascertained while that of the statistic for testing no long term effect is valid only when the short term effect is small simulation studies investigate the power properties of the proposed tests for different configurations the results show the interesting behavior of the proposed tests for situations where a short term effect is expected an example investigating the impact of progesterone receptors status on local tumor relapse for patients with early breast cancer illustrates the use of the proposed tests

robustness of adaptive control for stochastic systems robustness of the stochastic adaptive systems with unmodelled dynamics and external disturbances is discussed in this paper an adaptive controller is designed using the els algorithm except for the finite switches it is shown that the closed loop system is globally stable both the estimation error and the tracking error approach their minima when the unmodelled dynamics diminish that is the asymptotically adaptive system operating with the els algorithm has a good robustness with respect to unmodelled dynamics and external disturbances

a multigrid algorithm for the lowest order raviart thomas mixed triangular finite element method an optimal order multigrid method for the lowest order raviart thomas mixed triangular finite element is developed the algorithm and the convergence analysis are based on the equivalence between raviart thomas mixed methods and certain nonconforming methods both the dirichlet and singular neumann boundary value problems for second order elliptic equations are discussed

robustness analysis of identification and adaptive control for stochastic systems abstract this paper gives a preliminary robustness analysis for stochastic systems with unmodelled dynamics the extended least squares algorithm and stochastic adaptive control for tracking are defined on the basis of the modelled part of the system and it is shown that the parameter estimate is closed to the true one and the tracking error is small if the unmodelled dynamics is small

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

multilevel preconditioning of rotated bilinear non conforming fem problems preconditioners based on various multilevel extensions of two level finite element methods fem lead to iterative methods which often have an optimal order computational complexity with respect to the number of degrees of freedom of the system such methods were first presented in o axelsson p s vassilevski algebraic multilevel preconditioning methods i numer math 56 1989 157 177 o axelsson p s vassilevski algebraic multilevel preconditioning methods ii siam j numer anal 27 1990 1569 1590 on recursive two level splittings of the finite element space the key role in the derivation of optimal convergence rate estimates is played by the constant c in the so called strengthened cauchy bunyakowski schwarz cbs inequality associated with the angle between the two subspaces of the splitting more precisely the value of the upper bound for c 0 1 is a part of the construction of various multilevel extensions of the related two level methods in this paper algebraic two level and multilevel preconditioning algorithms for second order elliptic boundary value problems are constructed where the discretization is done using rannacher turek non conforming rotated bilinear finite elements on quadrilaterals an important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinement are not nested in general to handle this a proper two level basis is required to enable us to fit the general framework for the construction of two level preconditioners for conforming finite elements and to generalize the method to the multilevel case the proposed variants of the hierarchical two level basis are first introduced in a rather general setting then the parameters involved are studied and optimized the major contribution of the paper is the derived estimates of the constant c in the strengthened cbs inequality which is shown to allow the efficient multilevel extension of the related two level preconditioners representative numerical tests well illustrate the optimal complexity of the resulting iterative solver

robust algebraic multilevel preconditioning in h curl and h div abstract an algebraic multilevel iteration method for solving a system of linear algebraic equations arising in h curl and h div spaces is presented the algorithm is developed for the discrete problem obtained by using the space of lowest order nedelec and raviart thomas nedelec elements the theoretical analysis of the method is based only on some algebraic sequences and generalized eigenvalues of local element wise problems in the hierarchical basis framework explicit recursion formulas are derived to compute the element matrices and the constant which measures the quality of the space splitting at any given level it is proved that the proposed method is robust with respect to the problem parameters and is of optimal order complexity supporting numerical results including the case when the parameters have jumps are also presented

multilevel preconditioning of two dimensional elliptic problems discretized by a class of discontinuous galerkin methods we present optimal order preconditioners for certain discontinuous galerkin dg finite element discretizations of elliptic boundary value problems we consider two variants of hierarchical splittings where the underlying finite element meshes are geometrically nested a specific assembling process is proposed which facilitates the local analysis of the angle between the related subspaces by applying the corresponding two level basis transformation recursively a sequence of algebraic problems is generated that can be associated with a hierarchy of coarse versions of dg approximations of the original problem new bounds for the constant gamma in the strengthened cauchy bunyakowski schwarz inequality are derived the presented numerical results support the theoretical analysis and demonstrate the potential of this approach

adaptive stabilization of unstable and nonminimum phase stochastic systems abstract the system under consideration is the discrete time stochastic system a z yn zb z un wn driven by a martingale difference sequence wn where a z are polynomilas in backward shift operator z with unknown coefficients and both a z and b z may be unstable with the purpose of demonstrating theoretical possibility rather than designing a practically applicable control law this paper constructs an adaptive control that stabilizes the system and simultaneously guarantees strong consistency of the least squares estimates for unknown coefficients

quadrilateral h div finite elements we consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the piola transform extending results previously obtained for scalar approximation the finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element which is then transformed to a space of vector fields on each convex quadrilateral element via the piola transform associated to a bilinear isomorphism of the square onto the element for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms the situation is more complicated and we give a precise characterization of what is needed for optimal order l2 approximation of the function and of its divergence as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of h div we also derive new estimates for approximation by quadrilateral raviart thomas elements requiring less regularity and propose a new quadrilateral finite element space which provides optimal order approximation in h div finally we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of poisson s equation

algebraic multilevel iteration method for lowest order raviart thomas space and applications an optimal order algebraic multilevel iterative method for solving system of linear algebraic equations arising from the finite element discretization of certain boundary value problems that have their weak formulation in the space h div is presented the algorithm is developed for the discrete problem obtained by using the lowest order raviart thomas space the method is theoretically analyzed and supporting numerical examples are presented furthermore as a particular application the algorithm is used for the solution of the discrete minimization problem which arises in the functional type a posteriori error estimates for the discontinuous galerkin approximation of elliptic problems copyright 2011 john wiley sons ltd

stochastic adaptive prediction and model reference control guo and chen 1991 have recently shown how to establish the self optimality and mean square stability of a self tuning regulator the idea allows us to proceed with the development of a more comprehensive theory of stochastic adaptive filtering control and identification in adaptive filtering we examine both indirect and noninterlaced direct schemes for prediction using both least squares and gradient parameter estimation algorithms in addition to analyzing similar direct adaptive control algorithms we propose new generalized certainty equivalence adaptive model reference control laws with simultaneous disturbance rejection we also establish that the parameters converge to the null space of a certain matrix from this one may deduce the convergence of several adaptive controllers

analysis of coefficient quantization errors in state space digital filters in this paper we examine the sensitivity of state space digital filters to coefficient quantization errors a cost function using an l p norm criteria is used to measure the deviation in frequency response of the filter the cost function is used to indicate the connection between roundoff noise and filter sensitivity to coefficient quantization errors then based on this cost function it is shown that low roundoff noise state space digital filters filters in balanced coordinates and filters based on polynomials orthogonal on the unit circle have low sensitivity to coefficient quantization errors furthermore the sensitivity of the filters is shown to depend on the second order modes thereby exhibiting certain robustness and immunity to filter bandwidth

on the additive version of the algebraic multilevel iteration method for anisotropic elliptic problems in this paper a recently proposed additive version of the algebraic multilevel iteration method for iterative solution of elliptic boundary value problems is studied the method constructs a nearly optimal order parameter free preconditioner which is robust with respect to anisotropy and discontinuity of the problem coefficients it uses a new strategy for approximating the blocks corresponding to new basis functions on each discretization level to cope with the difficulties arising from the anisotropy the problem on the coarsest mesh is solved using a bordering technique with a special choice of bordering vectors the aim is to find a parameter free black box robust solver the results are derived in the framework of a hierarchical basis linear finite element discretization of an elliptic problem on arbitrary triangular meshes and a hierarchical basis bilinear finite element discretization on cartesian meshes a comparison of the method with some other iterative solution techniques is presented robustness and high efficiency of the proposed algorithm are demonstrated on several model type problems

nodal auxiliary space preconditioning in h curl and h div spaces in this paper we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of h curl and h div elliptic variational problems the preconditioners exclusively rely on solvers for discrete poisson problems we prove mesh independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning the main tools are discrete analogues of so called regular decomposition results in the function spaces h curl and h div our preconditioner for h curl is similar to an algorithm proposed in r beck algebraic multigrid by component splitting for edge elements on simplicial triangulations tech rep sc 99 40 zib berlin germany 1999

nonparametric estimation and testing in a cure model nonparametric generalized maximum likelihood product limit point estimators and confidence intervals are given for a cure model with random censorship one two and k sample likelihood ratio tests for inference on the cure rates are developed in the two sample case its power is compared to the power of several alternatives including the log rank and gray and tsiatis 1989 biometrics 45 899 904 tests implications for the use of the likelihood ratio test in a clinical trial designed to compare cure rates are discussed

the effect of identifier structure on parameter convergence the effects of a chosen system structure on the identification of its parameters are considered in particular convergence rates are compared for the parameters of three state space structures direct form ii parallel and dual generalized hessenberg representation structures it is shown that the chosen structure does influence identification algorithms and that this influence is measured by examining the information contained in the structural parameters a conjecture about the relative convergence of the parameters is offered and evidence in its support is provided an important result is that identification of the usually identified direct from ii parameters the standard arma parameters does not necessarily yield the fastest parameter convergence for the system being identified

a mixture model combining logistic regression with proportional hazards regression summary a model is proposed for the analysis of censored data which combines a logistic formulation for the probability of occurrence of an event with a proportional hazards specification for the time of occurrence of the event the proposed model is a semiparametric generalization of a parametric model due to farewell 1982 estimates of the regression parameters are obtained by maximizing a monte carlo approximation of a marginal likelihood and the em algorithm is used to estimate the baseline survivor function we present some simulation results to verify the validity of the suggested estimation procedure it appears that the semiparametric estimates are reasonably efficient with acceptable bias whereas the parametric estimates can be highly dependent on the parametric assumptions

the aastrom wittenmark self tuning regulator revisited and els based adaptive trackers although considerable progress has been made in stochastic adaptive control the problem concerning the convergence of the original self tuning regulator proposed by k j aastrom and b wittenmark 1973 is still open since it is attractive in theory and important in applications this problem is examined and a rigorous proof of its stability and optimality is given related problems such as convergence of the extended least squares els based adaptive tracker are also considered

parametric versus non parametric methods for estimating cure rates based on censored survival data if a patient s failure time is incorrectly recorded as being too early the correction will lower the plateau of the kaplan meier curve and hence the associated estimated cure rate implications of this counter intuitive observation are discussed in addition a parametric approach based on the gompertz distribution to the problem of cure rate estimation is presented

optimal adaptive control and consistent parameter estimate for deterministic systems with quadratic cost in this paper an adaptive control is designed for deterministic systems on the basis of the projection algorithm so that the quadratic loss function approaches its minimum at a rate of n e e 0 and the parameter estimate converges to its true value with a near exponential rate

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

the strong consistency of the stochastic gradient algorithm of adaptive control by use of a technique due to chen sufficient conditions are established for the strong consistency of the stochastic gradient sg algorithm for mimo armax stochastic systems without monitoring this result is then used in conjunction with the method of disturbed adaptive controls introduced in 1 2 hence it is shown that the sg algorithm generates strongly consistent parameter estimates while it is operating as a part of the sg algorithm of adaptive control of goodwin ramadge and caines

indirect adaptive pole placement control of mimo stochastic systems self tuning results in this paper we consider indirect adaptive pole placement control appc of linear multivariable stochastic systems instead of the canonical representation often used in the literature we propose using a non minimal but otherwise uniquely identifiable pseudo canonical parameterization that is more suitable for multivariable armax model identification to identify the plant we use the weighted extended least squares wels algorithm a least squares method with slowly decreasing weights which was introduced in bercu 1995 the pole placement controller parameters are then calculated by using a certain perturbation of the parameter estimates such that the linear models corresponding to the perturbed estimates are uniformly controllable and observable we prove that with a reasonable amount of prior information the resulting appc scheme is globally stabilizing and asymptotically self tuning regardless of the degree of persistency of external excitation these results represent the most complete study of stochastic multivariable appc systems to this date

auxiliary space preconditioning in h 0 curl ω we adapt the principle of auxiliary space preconditioning as presented in j xu the auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids computing 56 1996 pp 215 235 to h curl elliptic variational problems discretized by means of edge elements the focus is on theoretical analysis within the abstract framework of subspace correction employing a helmholtz type splitting of edge element vector fields we can establish asymptotic h uniform optimality of the preconditioner defined by our auxiliary space method

a linear rank test for use when the main interest is in differences in cure rates for diseases with a positive probability of being cured a family of alternatives to the null hypothesis of equality of survival distributions is introduced which is designed to focus power against alternatives with differences in cure rates the optimal linear rank test for this alternative is derived and found to be substantially more efficient than the log rank test for this alternative when cure rates are less than 50 while there is little difference between the tests if the cure rates are 50 or greater the simple test based on the difference of kaplan meier estimates of the proportion cured is also examined and found to be fully efficient for this alternative with no censoring while its efficiency rapidly drops as censoring is increased the new test is not a pure test of equality of cure rates when the data are censored but rather is a test of equality of survival distributions that focuses power against late differences in the survival curves

steady state and parameter tracking properties of self tuning minimum variance regulators abstract the self tuning regulator with a fixed gain and a fixed forgetting factor applied to control of an arx plant is considered the ordinary differential equation analysis of the closed loop system is performed in the neighbourhood of the stable equilibrium point and the approximate expressions for the steady state mean square parameter tracking and regulation errors are derived for the adaptive regulators based on the exponentially weighted least squares and stochastic approximation identification respectively it is shown that the concept of the effective memory length of the estimation algorithm has to be revised if the closed loop system identification with the gain fixing technique is used

cure rate models power of the logrank and generalized wilcoxon tests this paper reports the results of simulations that compare the power of the logrank and generalized wilcoxon tests in situations where the cure rate model is appropriate

nonstationary time series identification abstract in this paper strongly consistent estimates are given for both unknown parameters and orders of the nonstationary time series where the nonstationarity arises because 1 at any time n a feedback control is added to the usual arma process which is successfully applied to modelling economic systems and 2 det a z may have zeros on the unit circle in addition to those located outside the unit disk where a z is the matrix polynomial corresponding to the ar part of the arma process

convergence rate of least squares identification and adaptive control for stochastic systems the strong consistency and the convergence rate of least squares identification for the multidimensional armax model are established under some decaying excitation conditions which are satisfied if both input and output do not grow too fast and the attenuating excitation technique is applied the parameter identification results are applied to adaptive control systems with a quadratic loss function the rate of convergence of the loss function to its minimum is also obtained

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

a note on continuous time els abstract the parameter estimation of continuous time finite dimensional linear stochastic systems is a problem of long standing interest the method usually used is the extended least squares els algorithm described by a nonlinear stochastic differential equation sld with the existence of the global strong solution assumed this paper shows that the els estimate does exist in 0 and at the same time presents a number of convergence results paralleling those for the discrete time case

flexible conjugate gradients we analyze the conjugate gradient cg method with preconditioning slightly variable from one iteration to the next to maintain the optimal convergence properties we consider a variant proposed by axelsson that performs an explicit orthogonalization of the search directions vectors for this method which we refer to as flexible cg we develop a theoretical analysis that shows that the convergence rate is essentially independent of the variations in the preconditioner as long as the latter are kept sufficiently small we further discuss the real convergence rate on the basis of some heuristic arguments supported by numerical experiments depending on the eigenvalue distribution corresponding to the fixed reference preconditioner several situations have to be distinguished in some cases the convergence is as fast with truncated versions of the algorithm or even with the standard cg method whereas quite large variations are allowed without too much penalty in other cases the flexible variant effectively outperforms the standard method while the need for truncation limits the size of the variations that can be reasonably allowed

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

adaptive lqg control of input output systems a cost biased approach in this paper we consider linear systems in input output form and introduce a new adaptive linear quadratic gaussian lqg control scheme which is shown to be self optimizing the identification algorithm incorporates a cost biasing term which favors the parameters with smaller lqg optimal cost and a second term that aims at moderating the time variability of the estimate the corresponding closed loop scheme is proven to be stable and to achieve an asymptotic lqg cost equal to the one obtained under complete knowledge of the true system self optimization the results of this paper extend in a nontrivial way previous results established along the cost biased approach in other settings

a comparison of tests of the difference in the proportion of patients who are cured we compared by simulation the likelihood ratio wald and score tests based on a mixture model similar to that proposed by farewell 1982 biometrics 38 1041 1046 and a simple nonparametric test based on the plateau value of the product limit estimate for testing the difference in cured proportions between two groups the parametric tests obtained their asymptotic properties even in small samples provided that one could assume equal failure rates in the two groups otherwise good agreement with predictions required that essentially all potential failures had been observed the comparative properties of the parametric tests depended on whether the population survival functions crossed with the power of the wald test as good as or better than the others in the common situation when the survival functions do not cross however its size was sometimes less than nominal the score test was often not defined and is therefore of limited value the product limit test often performed as well as the parametric tests and despite being biased in some circumstances may be a useful alternative to these especially in small samples when some potential failures have not been observed

multilevel iterative methods for mixed finite element discretizations of elliptic problems for solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed the methods are within the general framework of the product multiplicative scheme for operators in a hilbert space proposed recently by bramble pasciak wang and xu 5 6 26 27 and make use of certain multilevel decomposition of the corresponding spaces for the flux variable

preconditioning and two level multigrid methods of arbitrary degree of approximation let h be a mesh parameter corresponding to a finite element mesh for an elliptic problem we describe preconditioning methods for two level meshes which for most prob lems solved in practice behave as methods of optimal order in both storage and computa tional complexity namely per mesh point these numbers are bounded above by relatively small constants for all h ho where ho is small enough to cover all but excessively fine meshes we note that in practice multigrid methods are actually solved on a finite often even a fixed number of grid levels in which case also these methods are not asymptotically optimal as h 0 numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems variable coefficients general domains and boundary conditions for all but excessively fine meshes furthermore most of the latter methods have been implemented only for difference schemes of second order of accuracy whereas our methods are applicable to higher order approximations we claim that our scheme could be added fairly easily to many existing finite element codes 1 introduction consider the numerical solution of elliptic boundary value prob lems discretized by finite element methods we assume that the boundary is polygonal or consists of planes we note that in practical problems one often has a fine enough grid already after the definition of the boundary and the minimal number of vertices needed for a first coarse triangulation anyhow if not so in most cases one makes only a few steps of mesh refinement hence the power of multigrid methods their optimal order of computational complexity is most often not achieved fully because optimality requires a large number of recursively defined meshes for details see e g 4 and for further references see 7 hence one might as well consider other methods perhaps simpler and more effective on a fixed mesh but which are not asymptotically optimal here we shall describe a method which uses only a fixed mesh but for which one nevertheless achieves a low order of computational complexity and of seemingly optimal order except for from a practical viewpoint excessively small meshes to be more precise the computational cost per mesh point is bounded by clog n for n no where n is the number of mesh points no is large enough to cover most applications and c is small enough that the method is competitive with multigrid methods as is well known the latter need recursion and the usual smoothing followed by corrections of the solutions on the different mesh levels we claim that the new method is more suitable for implementation in existing finite element packages in fact most packages for the multigrid methods are only for second order difference methods

the role of the strengthened cauchy buniakowskii schwarz inequality in multilevel methods the basic theory of the strengthened cauchy buniakowskii schwarz inequality and its applications in multilevel methods for the solution of linear systems arising from finite element or finite difference discretisation of elliptic partial differential equations is surveyed proofs are given both in a finite element context and in purely algebraic form

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