equivalent relation between normalized spatial entropy and fractal dimension abstract fractal dimension is defined on the base of entropy including macro state entropy and information entropy the generalized correlation dimension of multifractals is based on renyi entropy however the mathematical transform from entropy to fractal dimension is not clear in both theory and practice this paper is devoted to revealing the new equivalence relation between spatial entropy and fractal dimension using the box counting method by means of a set of regular fractals the numerical relationship between spatial entropy and fractal dimension is examined the results show that the ratio of actual entropy m q to the maximum entropy m max equals the ratio of actual dimension d q to the maximum dimension d max the spatial entropy and fractal dimension of complex spatial systems can be converted into one another by using functional box counting method the theoretical inference is verified by observational data of urban form a conclusion is that the normalized spatial entropy is equal to the normalized fractal dimension fractal dimensions proved to be the characteristic values of entropies in empirical studies if the linear size of spatial measurement is small enough a normalized entropy value is infinitely approximate to the corresponding normalized fractal dimension value based on the theoretical result new spatial indexes of urban space filling can be defined and multifractal parameters can be generalized to describe both simple systems and complex systems

multi scaling allometric analysis for urban and regional development abstract the concept of allometric growth is based on scaling relations and it has been applied to urban and regional analysis for a long time however most allometric analyses were devoted to the single proportional relation between two elements of a geographical system few researches focus on the allometric scaling of multielements in this paper a process of multiscaling allometric analysis is developed for the studies on spatio temporal evolution of complex systems by means of linear algebra general system theory and by analogy with the analytical hierarchy process the concepts of allometric growth can be integrated with the ideas from fractal dimension thus a new methodology of geo spatial analysis and the related theoretical models emerge based on the least squares regression and matrix operations a simple algorithm is proposed to solve the multiscaling allometric equation applying the analytical method of multielement allometry to chinese cities and regions yields satisfying results a conclusion is reached that the multiscaling allometric analysis can be employed to make a comprehensive evaluation for the relative levels of urban and regional development and explain spatial heterogeneity the notion of multiscaling allometry may enrich the current theory and methodology of spatial analyses of urban and regional evolution

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

iterative solution of a 3 d scattering problem from arbitrary shaped multidielectric and multiconducting bodies we present an iterative algorithm used for the computation of the scattering from arbitrary shaped bodies made of dielectric and conducting material the harmonic problem is discretized by a hybrid boundary integral method finite element method to solve the discretized linear system a modified version of a minimal residual algorithm has been developed for complex matrices it takes into account the lagrange multiplier constraint in a special way this modified version also allows the linear system to be solved for many right hand sides in an efficient manner

the equivalence of multi axis spine systems recommended stiffness limits using a standardized testing protocol abstract the complexity of multi axis spine testing often makes it challenging to compare results from different studies the aim of this work was to develop and implement a standardized testing protocol across three six axis spine systems compare them and provide stiffness and phase angle limits against which other test systems can be compared standardized synthetic lumbar specimens n 5 comprising three springs embedded in polymer at each end were tested on each system using pure moments in flexion extension lateral bending and axial rotation tests were performed using sine and triangle waves with an amplitude of 8 nm a frequency of 0 1 hz and with axial preloads of 0 and 500 n the stiffness phase angle and r2 value of the moment against rotation in the principal axis were calculated at the center of each specimen the tracking error was adopted as a measure of each test system to minimize non principal loads defined as the root mean squared difference between actual and target loads all three test systems demonstrated similar stiffnesses with small

efficient numerical differentiation of implicitly defined curves for sparse systems a numerical technique is developed for the efficient numerical differentiation of regular implicitly defined curves existing in high dimensional real space such as those representing homotopies where the system of equations which defines the curve implicitly is assumed to be sparse the calculation is verified numerically through its application to the curve defined implicitly by a homotopy constructed based on a discretization of the equations governing compressible aerodynamic fluid flow consideration is given to computational cost data storage and accuracy this method is applicable to any implicitly defined curves or trajectories which can occur for example in dynamical systems analysis or control applications also exist in the area of homotopy continuation where implicitly defined curves are approximately traced numerically such applications include the analysis of curve traceability and the construction of higher order predictors the latter is investigated numerically and it is found that increasing the order of accuracy of the predictor can significantly improve the curve tracing accuracy within a limited radius

accelerated gcro dr method for solving sequences of systems of linear equations by recycling some error approximations generated during a previous cycle of iterations we present a technique for improving the convergence of gcro dr the scheme is able to mitigate the occurrence of small skip angles occurring in gcro dr and thus considerably accelerates the convergence the alternative phenomenon of the residual vectors is examined numerically and the effectiveness of the new method is illustrated by several examples from practical applications

preconditioned conjugate gradients for solving singular systems abstract in this paper the preconditioned conjugate gradient method is used to solve the system of linear equations ax b where a is a singular symmetric positive semi definite matrix the method diverges if b is not exactly in the range r a of a if the null space n a of a is explicitly known then this divergence can be avoided by subtracting from b its orthogonal projection onto n a as well as analysing this subtraction conditions necessary for the existence of a nonsingular incomplete cholesky decomposition are given finally the theory is applied to the discretized semi definite neumann problem

recycling subspace information for diffuse optical tomography we discuss the efficient solution of a long sequence of slowly varying linear systems arising in computations for diffuse optical tomographic imaging the reconstruction of three dimensional absorption and scattering information by matching computed solutions from a parameterized model to measured data leads to a nonlinear least squares problem that we solve using the gauss newton method with a line search this algorithm requires the solution of a long sequence of linear systems each choice of parameters in the nonlinear least squares algorithm results in a different matrix describing the optical properties of the medium these matrices change slowly from one step to the next but may change significantly over many steps for each matrix we must solve a set of linear systems with multiple shifts and multiple right hand sides for this problem we derive strategies for recycling krylov subspace information that exploit properties of the application and the nonlinear optimization algorithm to significantly reduce the total number of iterations over all linear systems furthermore we introduce variants of gcro that exploit symmetry and that allow simultaneous solution of multiple shifted systems using a single krylov subspace in combination with recycling although we focus on a particular application and optimization algorithm our approach is applicable generally to problems where sequences of linear systems must be solved this may guide other researchers to exploit the opportunities of tunable solvers we provide results for two sets of numerical experiments to demonstrate the effectiveness of the resulting method

a simplified and flexible variant of gcrot for solving nonsymmetric linear systems there is a need for flexible iterative solvers that can solve large scale 10 6 unknowns nonsymmetric sparse linear systems to a small tolerance among flexible solvers flexible gmres fgmres is attractive because it minimizes the residual norm over a particular subspace in practice fgmres is often restarted periodically to keep memory and work requirements reasonable however like restarted gmres restarted fgmres can suffer from stagnation this has led us to develop a flexible variant of the krylov linear solver gcrot generalized conjugate residual with inner orthogonalization and outer truncation unlike the original gcrot algorithm the proposed gcrot variant uses a simplified truncation strategy similar to loose gmres lgmres this modification is motivated by numerical experiments that suggest the specific subspace retained in the outer iteration of gcrot is less important than its size the flexible gcrot variant appears to be well suited for advection dominated problems in particular when applied to an adjoint problem from computational aerodynamics the proposed gcrot variant is robust and efficient compared with several popular truncated krylov subspace methods finally a flexible version of lgmres is easily constructed by recognizing algorithmic similarities to gcrot

a note on pareto tails and lognormal body of us cities size distribution abstract luckstead and devadoss 2017 fitted a distribution made up of lower tail pareto lognormal body and upper tail pareto to city sizes in the us here we propose a class of distributions providing better fits the fits are assessed in terms of akaike information criterion bayesian information criterion and probability plots

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

accelerated simultaneous iterations for large finite element eigenproblems abstract an accelerated simultaneous iteration method is presented for the solution of the generalized eigenproblem ax bx where a and b are real sparse symmetric positive definite matrices the approach is well suited for the determination of the leftmost eigenpairs of problems with large size n the procedure relies on the optimization of the rayleigh quotient over a subspace of orthogonal vectors by a conjugate gradient technique effectively preconditioned with the pointwise incomplete cholesky factorization the method is applied to the evaluation of the smallest 15 eigenpairs of finite element models with size ranging between 150 and 2300 the numerical experiments show that while the simultaneous conjugate gradient scheme fails to converge the accelerated iterations yield accurate results in a number of steps which is much smaller than n the new approach does not require the a priori estimate of any empirical parameter and appears to be a robust reliable and efficient tool for the partial eigensolution of large finite element problems

mstab stabilized induced dimension reduction for krylov subspace recycling we introduce mathcal m stab a krylov subspace recycling method for the iterative solution of sequences of linear systems where the system matrix is fixed and is large sparse and nonsymmetric and the right hand side vectors are available in sequence mathcal m stab utilizes the short recurrence principle of induced dimension reduction type methods adapted to solve sequences of linear systems using idrstab for solving the linear system with the first right hand side the proposed method then recycles the petrov space constructed throughout the solution of that system generating a larger initial space for subsequent linear systems the richer space potentially produces a rapidly convergent scheme numerical experiments demonstrate that mathcal m stab often enters the superlinear convergence regime faster than other krylov type recycling methods

deflated gmres for systems with multiple shifts and multiple right hand sides abstract we consider solution of multiply shifted systems of nonsymmetric linear equations possibly also with multiple right hand sides first for a single right hand side the matrix is shifted by several multiples of the identity such problems arise in a number of applications including lattice quantum chromodynamics where the matrices are complex and non hermitian some krylov iterative methods such as gmres and bicgstab have been used to solve multiply shifted systems for about the cost of solving just one system restarted gmres can be improved by deflating eigenvalues for matrices that have a few small eigenvalues we show that a particular deflated method gmres dr can be applied to multiply shifted systems in quantum chromodynamics it is common to have multiple right hand sides with multiple shifts for each right hand side we develop a method that efficiently solves the multiple right hand sides by using a deflated version of gmres and yet keeps costs for all of the multiply shifted systems close to those for one shift an example is given showing this can be extremely effective with a quantum chromodynamics matrix

the evolution of zipf s law indicative of city development abstract zipf s law of city size distributions can be expressed by three types of mathematical models one parameter form two parameter form and three parameter form the one parameter and one of the two parameter models are familiar to urban scientists however the three parameter model and another type of two parameter model have not attracted attention this paper is devoted to exploring the conditions and scopes of application of these zipf models by mathematical reasoning and empirical analysis new discoveries are made as follows first if the size distribution of cities in a geographical region cannot be described with the one or two parameter model maybe it can be characterized by the three parameter model with a scaling factor and a scale translational factor second all these zipf models can be unified by hierarchical scaling laws based on cascade structure third the patterns of city size distributions seem to evolve from three parameter mode to two parameter mode and then to one parameter mode four year census data of chinese cities are employed to verify the three parameter zipf s law and the corresponding hierarchical structure of rank size distributions this study is revealing for people to understand the scientific laws of social systems and the property of urban development

integral equation based domain decomposition method for solving electromagnetic wave scattering from non penetrable objects the integral equation ie method is commonly utilized to model time harmonic electromagnetic em problems one of the greatest challenges in its applications arises in the solution of the resulting ill conditioned matrix equation we introduce a new domain decomposition method ddm for the ie solution of em wave scattering from non penetrable objects the proposed method is a non overlapping non conformal ddm and it provides a computationally efficient and effective preconditioner for the ie matrix equations moreover the proposed approach is very suitable for dealing with multi scale electromagnetic problems since each sub domain has its own characteristics length and will be meshed independently furthermore for each sub domain we are free to choose the most effective ie sub domain solver based on its local geometrical features and electromagnetic characteristics additionally the multilevel fast multi pole algorithm mlfma is utilized to accelerate the computations of couplings between sub domains numerical results demonstrate that the proposed method yields rapid convergence in the outer krylov iterative solution process finally simulations of several large scale examples testify to the effectiveness and robustness of the proposed ie based ddm

large scale topology optimization using preconditioned krylov subspace methods with recycling summary the computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis we propose fast iterative solvers for large threedimensional topology optimization problems to address this problem since the linear systems in the sequence of optimization steps change slowly from one step to the next we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems in addition we introduce a minres minimum residual method version with recycling and a short term recurrence to make recycling more efficient for symmetric problems furthermore we discuss preconditioning to ensure fast convergence we show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density we demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast pc

measurement of a spinal motion segment stiffness matrix the six degrees of freedom elastic behavior of spinal motion segments can be approximated by a stiffness matrix a method is described to measure this stiffness matrix directly with the motion segment held under physiological conditions of axial preload and in an isotonic fluid bath by measuring the forces and moments associated with each of the six orthogonal translations and rotations the stiffness matrix was obtained from the load displacement measurements by linear least squares assuming a symmetric matrix results from a pig lumbar spinal motion segment in an isotonic bath with and without a 500 n axial preload showed a large stiffening effect with axial preload

aerodynamic optimization algorithm with integrated geometry parameterization and mesh movement an efficient gradient based algorithm for aerodynamic shape optimization is presented the algorithm consists of several components including a novel integrated geometry parameterization and mesh movement a parallel newton krylov flow solver and an adjoint based gradient evaluation to integrate geometry parameterization and mesh movement generalized b spline volumes are used to parameterize both the surface and volume mesh the volume mesh of b spline control points mimics a coarse mesh a linear elasticity mesh movement algorithm is applied directly to this coarse mesh and the fine mesh is regenerated algebraically using this approach mesh movement time is reduced by two to three orders of magnitude relative to a node based movement the mesh adjoint system also becomes smaller and is thus amenable to complex step derivative approximations when solving the flow adjoint equations using restarted krylov subspace methods a nested subspace strategy is shown to be more robust than truncating the entire subspace optimization is accomplished using a sequential quadratic programming algorithm the effectiveness of the complete algorithm is demonstrated using a lift constrained induced drag minimization that involves large changes in geometry

recent developments in the algebraic multiblock method for euler equations abstract in this paper the algebraic multiblock method is presented and applied to the solution of the euler equations used to model inviscid compressible flow problems the method employs two extreme grids aimed at combining multigrid and domain decomposition ideas the fine grid is used by the discretization method the coarse grid results from the fine grid and defines the domain partitioning of the solution procedure the multigrid concept is practically simulated by applying the idea of the substructuring technique in which the inner finite volumes of a block are expressed by their boundary volumes through a matrix inversion process the conjugate gradient squared method is used for the approximate solution of numerous large block pentadiagonal systems emerging at each time step additionally efficient preconditioning and smoothing are employed to accelerate the iterative solution the proposed method is applied for the solution of the time marching euler equations using an implicit upwind finite volume method euflex numerical results and comparisons show the efficiency of the proposed method even for sequential processing

a preconditioned semi implicit method for magnetohydrodynamics equations the paper describes the development of a semi implicit solver designed for the need to follow slow evolution flows encountered in nonlinear resistive computational plasma dynamics cpd spatial discretization uses a second order finite difference approximation while temporal advance is achieved by a second order semi implicit predictor corrector scheme that reduces the severe time step constraints imposed by the fast magnetosonic and alfven waves on standard explicit schemes an efficient preconditioner adapted to magnetohydrodynamics mhd problems and associated with conjugate gradient like linear solvers considerably increases the cpu saving when compared with an explicit advance this scheme enables the use of a large time step as well as the necessary high spatial resolution an application of this scheme to a class of astrophysical nonlinear mhd problems allows one to perform numerical experiments relevant to the slow mhd evolution of the magnetic field dominating the outer atmosphere of the sun that leads to small scales formation

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

a multi level adaptive mesh refinement for an integrated finite element level set formulation to simulate multiphase flows with surface tension abstract a multi level adaptive mesh refinement technique for an integrated finite element level set formulation was implemented to simulate the incompressible interfacial flows with surface tension by using unstructured p2p1 triangular and tetrahedral meshes linear quadratic elements near the interface were dynamically refined as an interface evolves in order to produce an accurate solution with a lower computational cost an adaptive mesh refinement algorithm for a linear element has been extended to a quadratic element by using a node reordering the proposed algorithm was successfully validated by solving several benchmark problems simulation of a static bubble problem was found to give an accurate solution with negligible spurious currents compared with existing results simulation results of rising bubble problems in 2d and 3d were also compared against existing numerical and experimental results in terms of terminal rising speed and terminal bubble shape we have found that the present algorithm provided stable and accurate solutions even for a high density ratio and that the solution accuracy strongly depended on the mesh resolution near the interface further the present algorithm was shown to be very efficient because the computational overhead by mesh refinement algorithm was very small compared to the elapsed times consumed for solving the incompressible navier stokes equations and the advection re initialization equations of level set method lastly the coalescence scenario of two oblique bubbles was well reproduced by the present 3d simulation

truncation strategies for optimal krylov subspace methods optimal krylov subspace methods like gmres and gcr have to compute an orthogonal basis for the entire krylov subspace to compute the minimal residual approximation to the solution therefore when the number of iterations becomes large the amount of work and the storage requirements become excessive in practice one has to limit the resources the most obvious ways to do this are to restart gmres after some number of iterations and to keep only some number of the most recent vectors in gcr this may lead to very poor convergence and even stagnation therefore we will describe a method that reveals which subspaces of the krylov space were important for convergence thus far and exactly how important they are this information is then used to select which subspace to keep for orthogonalizing future search directions numerical results indicate this to be a very effective strategy

deflated and restarted symmetric lanczos methods for eigenvalues and linear equations with multiple right hand sides a deflated restarted lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors the restarting limits the storage so that finding eigenvectors is practical meanwhile the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting some reorthogonalization is necessary to control roundoff error and several approaches are discussed the eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right hand sides experiments are given with large matrices from quantum chromodynamics that have many right hand sides

an improved iterative optimization technique for the leftmost eigenpairs of large symmetric matrices abstract an accelerated optimization technique combined with a stepwise deflation procedure is presented for the efficient evaluation of the p p 20 leftmost eigenvalues and eigenvectors of finite element symmetric positive definite p d matrices of very large size the optimization is performed on the rayleigh quotient of the deflated matrices by the aid of a conjugate gradient cg scheme effectively preconditioned with the incomplete cholesky factorization no a priori estimate of acceleration parameters is required numerical experiments on large arbitrarily sparse problems taken from the engineering finite elements f e practice show a very fast convergence rate for any value of p within the explored interval and particularly so for the minimal eigenpair in this case the number of iterations needed to achieve an accurate solution may be as much as 2 orders of magnitude smaller than the problem size several results concerning the spectral behavior of the cg preconditioning matrices are also given and discussed

dual consistency and functional accuracy consider the discretization of a partial differential equation pde and an integral functional that depends on the pde solution the discretization is dual consistent if it leads to a discrete dual problem that is a consistent approximation of the corresponding continuous dual problem consequently a dual consistent discretization is a synthesis of the so called discrete adjoint and continuous adjoint approaches we highlight the impact of dual consistency on summation by parts sbp finite difference discretizations of steady state pdes specifically superconvergent functionals and accurate functional error estimates in the case of functional superconvergence the discrete adjoint variables do not need to be computed since dual consistency on its own is sufficient numerical examples demonstrate that dual consistent schemes significantly outperform dual inconsistent schemes in terms of functional accuracy and error estimate effectiveness the dual consistent and dual inconsistent discretizations have similar computational costs so dual consistency leads to improved efficiency to illustrate the dual consistency analysis of sbp schemes we thoroughly examine a discretization of the euler equations of gas dynamics including the treatment of the boundary conditions numerical dissipation interface penalties and quadrature by sbp norms

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

a block gcrot m k method for linear systems with multiple right hand sides this study is mainly focused on iterative solution to multiple linear systems with several right hand sides for solving such systems efficiently we explore a new block gcrot m k bgcrot m k method which is derived by extending gcrot m k method jason e hicken david w zingg a simplified and flexible variant of gcrot for solving nonsymmetric linear systems siam j sci comput 32 2010 1672 1694 we analyze its main properties it is shown that under the condition of full rank of block residual the frobenius norm of the block residual generated by the proposed method is always nonincreasing moreover we also present its block flexible version bfgcrot m k finally numerical examples demonstrate that the bgcrot m k method and its flexible variant can achieve a smoothed residual and can be more competitive than some other block solvers

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

city size distributions for india and china this paper studies the size distributions of urban agglomerations for india and china we have estimated the scaling exponent for zipf s law with the indian census data for the years of 1981 2001 and the chinese census data for 1990 and 2000 along with the biased linear fit estimate the maximum likelihood estimate for the pareto and tsallis q exponential distribution has been computed for india the scaling exponent is in the range of 1 88 2 06 and for china it is in the interval 1 82 2 29 the goodness of fit tests of the estimated distributions are performed using the kolmogorov smirnov statistic

comparison of krylov subspace methods with preconditioning techniques for solving boundary value problems abstract in this paper we made an attempt to establish the usefulness of lanczos solver with preconditioning technique over the preconditioned conjugate gradient cg solvers we have presented here a detail comparative study with respect to convergence speed as well as cpu time by considering appropriate boundary value problems

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

drag minimization based on the navier stokes equations using a newton krylov approach a methodology is presented for performing numerical aerodynamic shape optimization based on the three dimensional reynolds averaged navier stokes rans equations an initial multiblock structured mesh is first fit with b spline volumes that form the basis for a hybrid mesh movement scheme that is tightly integrated with the geometry parameterization based on b spline surfaces the rans equations and the one equation spalart allmaras turbulence model are solved in a fully coupled manner using an efficient parallel newton krylov algorithm with approximate schur preconditioning gradient evaluations are performed using the discrete adjoint approach with analytical differentiation of the discrete flow and mesh movement equations the overall methodology remains robust even in the presence of large shape changes several examples of lift constrained drag minimization are provided including a study of the common research model wing geometry a wing body tail geometry with a prescribed spanwise load distributi

conjugate gradient type methods and preconditioning in this paper we consider various iterative methods for the numerical solution of very large sparse linear systems of equations which arise in the discretization of partial differential equations as the performance of the methods generally depends on the characteristics of the problems to be solved a judicious choice between the methods will require knowledge about the system the aim of this paper is to review the properties of the various iteration methods in order to assist the user in making a deliberate selection

relaxation schemes for spectral multigrid methods the effectiveness of relaxation schemes for solving the systems of algebraic equations which arise from spectral discretizations of elliptic equations is examined iterative methods are an attractive alternative to direct methods because fourier transform techniques enable the discrete matrix vector products to be computed almost as efficiently as for corresponding but sparse finite difference discretizations preconditioning is found to be essential for acceptable rates of convergence preconditioners based on second order finite difference methods are used a comparison is made of the performance of different relaxation methods on model problems with a variety of conditions specified around the boundary the investigations show that iterations based on incomplete lu decompositions provide the most efficient methods for solving these algebraic systems

a comparative study of the behaviour of a direct solver and a preconditioned iterative solver for the equations arising from the discretization by chebyshev collocation of a second order partial differential equation on a square the behaviour is compared of two solvers for the discrete equations arising from the discretization using chebyshev collocation of a second order linear partial differential equation on a square the alternative solvers considered are a direct solver and an iterative solver based on preconditioning with the matrix arising from finite difference discretization of the governing equation the total error of the collocation derivatives and the separate contributions from round off and discretization error are examined the efficiency of the two solvers is compared the iterative solver is more efficient than the direct solver on fine grids for equations similar to the poisson equation provided that there are dirichlet boundary conditions on at least three of the sides of the square

incomplete block matrix factorization preconditioning methods the ultimate answer abstract a considerable interest has been devoted to block matrix incomplete factorization preconditioning methods in recent time see 12 2 3 4 6 7 8 10 and 21 the purpose of this paper is to survey some of these results and to present new results showing how much more efficient and robust such methods can be as compared to other admittingly also efficient methods such as pointwise incomplete factorization preconditioning methods and multilevel methods with hierarchical basis functions both on scalar as well as on parallel or vectorizable computers

flexible inner outer krylov subspace methods flexible krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next given a krylov subspace method such as cg gmres qmr etc for the solution of a linear system ax b instead of having a fixed preconditioner m and the right preconditioned equation am 1 y b mx y one may have a different matrix say mk at each step in this paper the case where the preconditioner itself is a krylov subspace method is studied there are several papers in the literature where such a situation is presented and numerical examples given a general theory is provided encompassing many of these cases including truncated methods the overall space where the solution is approximated is no longer a krylov subspace but a subspace of a larger krylov space we show how this subspace keeps growing as the outer iteration progresses thus providing a convergence theory for these inner outer methods numerical tests illustrate some important implementation aspects that make the discussed inner outer methods very appealing in practical circumstances

on improving linear solver performance a block variant of gmres the increasing gap between processor performance and memory access time warrants the re examination of data movement in iterative linear solver algorithms for this reason we explore and establish the feasibility of modifying a standard iterative linear solver algorithm in a manner that reduces the movement of data through memory in particular we present an alternative to the restarted gmres algorithm for solving a single right hand side linear system ax b based on solving the block linear system ax b algorithm performance i e time to solution is improved by using the matrix a in operations on groups of vectors experimental results demonstrate the importance of implementation choices on data movement as well as the effectiveness of the new method on a variety of problems from different application areas

boundary orthogonal biharmonic grids via preconditioned gradient methods a finite difference approximation of the biharmonic equation is solved using three preconditioned gradient methods for the generation of curvilinear boundary orthogonal grids in two dimensions the developed algorithms based on the conjugated gradient the steepest descent and the minimal residual method are applied in a number of domains of engineering interest for the purpose of comparison in terms of cpu time and number of iterations to convergence

accelerating the induced dimension reduction method using spectral information the induced dimension reduction method idr s sonneveld and van gijzen 2008 is a short recurrences krylov method to solve systems of linear equations in this work we accelerate this method using spectral information we construct a hessenberg relation from the idr s residual recurrences formulas from which we approximate the eigenvalues and eigenvectors using the ritz values we propose a self contained variant of the ritz idr s method simoncini and szyld 2010 for solving a system of linear equations in addition the ritz vectors are used to speed up idr s for the solution of sequence of systems of linear equations 2018 elsevier b v

restarting the nonsymmetric lanczos algorithm for eigenvalues and linear equations including multiple right hand sides a restarted nonsymmetric lanczos algorithm is given for computing eigenvalues and both right and left eigenvectors the restarting limits the storage so that finding eigenvectors is practical restarting also makes it possible to deal with roundoff error in new ways we give a scheme for avoiding near breakdown and discuss maintaining biorthogonality a system of linear equations can be solved simultaneously with the eigenvalue computations deflation from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting the right and left eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right hand sides

nested krylov methods based on gcr abstract recently the gmresr method for the solution of linear systems of equations has been introduced by vuik and van der vorst 1991 similar methods have been proposed by axelsson and vassilevski 1991 and saad 1993 fgmres 1 gmresr involves an outer and an inner method the outer method is gcr which is used to compute the optimal approximation over a given set of search vectors in the sense that the residual is minimized the inner method is gmres which computes a new search vector by approximately solving the residual equation this search vector is then used by the outer algorithm to compute a new approximation however the optimality of the approximation over the space of search vectors is ignored in the inner gmres algorithm this leads to suboptimal corrections to the solution in the outer algorithm therefore we propose to preserve the orthogonality relations of gcr in the inner gmres algorithm this gives optimal corrections to the solution and also leads to solving the residual equation in a smaller subspace and with an improved operator which should also lead to faster convergence however this involves using krylov methods with a singular nonsymmetric operator we will discuss some important properties of this we will show by experiments that in terms of matrix vector products this modification almost always leads to better convergence because we do more orthogonalizations it does not always give an improved performance in time this depends on the costs of the matrix vector products relative to the costs of the orthogonalizations of course we can also use methods other than gmres as the inner method methods with short recurrences like bicgstab seem especially interesting the experimental results indicate that especially for such methods it is advantageous to preserve the orthogonality in the inner method

recycling krylov subspaces for cfd applications and a new hybrid recycling solver we focus on robust and efficient iterative solvers for the pressure poisson equation in incompressible navier stokes problems preconditioned krylov subspace methods are popular for these problems with bicgstab and gmres m most frequently used for nonsymmetric systems bicgstab is popular because it has cheap iterations but it may fail for stiff problems especially early on as the initial guess is far from the solution restarted gmres is better more robust in this phase but restarting may lead to very slow convergence therefore we evaluate the rgcrot method for these systems this method recycles a selected subspace of the search space called recycle space after a restart this generally improves the convergence drastically compared with gmres m recycling subspaces is also advantageous for subsequent linear systems if the matrix changes slowly or is constant however rgcrot iterations are still expensive in memory and computation time compared with those of bicgstab hence we propose a new hybrid approach that combines the cheap iterations of bicgstab with the robustness of rgcrot for the first few time steps the algorithm uses rgcrot and builds an effective recycle space and then it recycles that space in the rbicgstab solver we evaluate rgcrot on a turbulent channel flow problem and we evaluate both rgcrot and the new hybrid combination of rgcrot and rbicgstab on a porous medium flow problem we see substantial performance gains for both the problems

comparison of b spline surface and free form deformation geometry control for aerodynamic optimization two aerodynamic shape optimization geometry control methods b spline surface control and free form deformation are applied to three optimization problems and compared on the bases of optimal shap

the mathematical relationship between zipf s law and the hierarchical scaling law abstract the empirical studies of city size distribution show that zipf s law and the hierarchical scaling law are linked in many ways the rank size scaling and hierarchical scaling seem to be two different sides of the same coin but their relationship has never been revealed by strict mathematical proof in this paper the zipf s distribution of cities is abstracted as a q sequence based on this sequence a self similar hierarchy consisting of many levels is defined and the numbers of cities in different levels form a geometric sequence an exponential distribution of the average size of cities is derived from the hierarchy thus we have two exponential functions from which follows a hierarchical scaling equation the results can be statistically verified by simple mathematical experiments and observational data of cities a theoretical foundation is then laid for the conversion from zipf s law to the hierarchical scaling law and the latter can show more information about city development than the former moreover the self similar hierarchy provides a new perspective for studying networks of cities as complex systems a series of mathematical rules applied to cities such as the allometric growth law the 2 n principle and pareto s law can be associated with one another by the hierarchical organization

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

a dynamic model for city size distribution beyond zipf s law abstract we present a growth model for a system of cities this model recovers not only zipf s law but also other kinds of city size distributions csds a new positive exponent which yields zipf s law only when equal to 1 was introduced we define three classes of csd depending on the value of larger than smaller than or equal to 1 the model is based on a random growth of the city population together with the variation of the number of cities in the system the striking result is the peculiar behavior of the model it is only statistical deterministic moreover we found that the exponent may be larger smaller or equal to 1 just like in real systems of cities depending on the rate of creation of new cities and the time elapsed during the growth it is to our knowledge the first time that the influence of the time on the type of the distribution is investigated the results of the model are in very good agreement with real csd the classification and model can be also applied to other entities like countries incomes firms etc

a multigrid conjugate residual method for the numerical solution of the hartree fock equation for diatomic molecules abstract discretization of the hartree fock equations in operational form leads to unsymmetric positive definite and indefinite linear equations to solve these equations a combination of the multigrid method and the orthomin method with gauss seidel relaxation as preconditioner is used the differential equations are approximated to the sixth order and the solution is extrapolated to the eighth order the method is fully parallelized the largest molecule treated is cuh

pcg methods in transient fe analysis part i first order problems the performances of some pcg preconditioned conjugate gradient algorithms are evaluated in the solution of first order time dependent parabolic partial differential equations such as the heat conduction equation which have been spatially discretized using finite elements consistent mass discretizations are preferred by the authors to lumped mass ones and various preconditioners are then compared diagonal incomplete choleski and ebe element by element recommendations are made and implications for parallel computation outlined

a numerical study of various algorithms related to the preconditioned conjugate gradient method we present the results of a numerical study of the preconditioned conjugate gradient algorithm the minimal residual algorithm the biconjugate gradient algorithm and the bi minimal residual algorithm using both simple test matrices and more realistic test matrices derived from physical problems the application of the methods to unsymmetric matrices is considered we emphasize the importance of a good preconditioning look at various methods including icgg n and incomplete block factorization and make some practical recommendations some of the folk lore surrounding the semi iterative methods is dispelled

gmres with deflated restarting a modification is given of the gmres iterative method for nonsymmetric systems of linear equations the new method deflates eigenvalues using wu and simon s thick restarting approach siam j matrix anal appl 22 2000 pp 602 616 it has the efficiency of implicit restarting but is simpler and does not have the same numerical concerns the deflation of small eigenvalues can greatly improve the convergence of restarted gmres also it is demonstrated that using harmonic ritz vectors is important because then the whole subspace is a krylov subspace that contains certain important smaller subspaces

characterization of the l4 l5 s1 motion segment using the stepwise reduction method the two aims of this study were to generate data for a more accurate calibration of finite element models including the l5 s1 segment and to find mechanical differences between the l4 l5 and l5 s1 segments then the range of motion rom and facet forces for the l4 s1 segment were measured using the stepwise reduction method this consists of sequentially testing and reducing each segment in nine stages by cutting the ligaments facet capsules and removing the nucleus five l4 s1 human segments median 65 years range 53 84 years sd 11 0 years were loaded under a maximum pure moment of 8nm the rom was measured using stereo photogrammetry via tracking of three markers and the facet contact forces cf were measured using a tekscan system the rom for the l4 l5 segment and all stages showed good agreement with published data the major differences in rom between the l4 l5 and l5 s1 segments were found for lateral bending and all stages for which the l4 l5 rom was about 1 5 3 times higher than that of the l5 s1 segment consistent with l5 s1 facet cf about 1 3 to 4 times higher than those measured for the l4 l5 segment for the other movements and few stages the l4 l5 rom was significantly lower that of the l5 s1 segment rom and cf provide important baseline data for more accurate calibration of fe models and to understand the role that their structures play in lower lumbar spine mechanics

recycling krylov subspaces for sequences of linear systems many problems in science and engineering require the solution of a long sequence of slowly changing linear systems we propose and analyze two methods that significantly reduce the total number of matrix vector products required to solve all systems we consider the general case where both the matrix and right hand side change and we make no assumptions regarding the change in the right hand sides furthermore we consider general nonsingular matrices and we do not assume that all matrices are pairwise close or that the sequence of matrices converges to a particular matrix our methods work well under these general assumptions and hence form a significant advancement with respect to related work in this area we can reduce the cost of solving subsequent systems in the sequence by recycling selected subspaces generated for previous systems we consider two approaches that allow for the continuous improvement of the recycled subspace at low cost we consider both hermitian and non hermitian problems and we analyze our algorithms both theoretically and numerically to illustrate the effects of subspace recycling we also demonstrate the effectiveness of our algorithms for a range of applications from computational mechanics materials science and computational physics

complete stagnation of gmres we study problems for which the iterative method gmres for solving linear systems of equations makes no progress in its initial iterations our tool for analysis is a nonlinear system of equations the stagnation system that characterizes this behavior we focus on complete stagnation for which there is no progress until the last iteration we give necessary and sufficient conditions for complete stagnation of systems involving unitary matrices and show that if a normal matrix completely stagnates then so does an entire family of nonnormal matrices with the same eigenvalues finally we show that there are real matrices for which complete stagnation occurs for certain complex right hand sides but not for real ones 2003 elsevier science inc all rights reserved

multiply broken power law densities as survival functions an alternative to pareto and lognormal fits abstract survival functions defined by multiply broken power law densities are introduced to model firm size data sets of four historical censuses reported in montebruno et al physica a 523 2019 858 the survival functions complementary cumulative distributions are obtained by least squares regression the probability distributions are inferred from the analytic survival functions the mean firm growth the growth volatility and entropy evolution are calculated over a 30 year period covered by the censuses a representation of the survival functions as single power law density with varying exponent depending on firm size is derived index functions i e rescaled logarithmic derivatives of survival functions are used to demonstrate that neither pareto power laws nor lognormal densities can accurately reproduce the tails of the firm size distributions like the survival functions the empirical rank size relations of the four censuses also admit representation by broken power law densities a generating mechanism for empirically obtained broken power law densities based on the fokker planck equation is proposed as well

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

non conformal domain decomposition method with mixed true second order transmission condition for solving large finite antenna arrays a non overlapping and non conformal domain decomposition method ddm is presented for modeling large finite antenna arrays two major benefits of the proposed ddm a a mixed true second order transmission condition sotc with corner edge penalty terms is developed to facilitate fast convergence in the ddm iterations numerical experiments indicate that the convergence of ddm with the aforementioned sotc is insensitive to the sizes of arrays and b the non conformal property of the proposed ddm permits the use of completely independent discretization for each of the sub domains we demonstrate the performance of the proposed approach through several large scale problems

output error estimation for summation by parts finite difference schemes the paper develops a posteriori error estimates of integral output functionals for summation by parts finite difference methods the error estimates are based on the adjoint weighted residual method and take advantage of a variational interpretation of summation by parts discretizations the estimates are computed on a fixed grid and do not require an embedded grid or explicit interpolation operators for smooth boundary value problems containing first and second derivatives the error estimates converge to the exact error as the mesh is refined the theory is verified using linear boundary value problems and the euler equations

a new preconditioning technique for solving large sparse linear systems abstract a new sparse approximate triangular factorization technique for solving large sparse linear system by iterative methods is proposed the method is based on the description of the triangular factorization of a matrix as a product of elementary matrices and provides a general scheme for constructing incomplete preconditioners for the given matrix in particular the familiar incomplete choleski decomposition can be incorporated into this scheme the algorithm based on choice by value compares favorably with the incomplete choleski preconditioner and equipped with a user controlled parameter is able to tackle extremely ill conditioned problems arising in structural analysis semiconductor simulation oil reservoir modelling and other applications when applied to a positive definite symmetric matrix the algorithm produces a preconditioning matrix preserving that property

analysis of acceleration strategies for restarted minimal residual methods we provide an overview of existing strategies which compensate for the deterioration of convergence of minimum residual mr krylov subspace methods due to restarting we evaluate the popular practice of using nearly invariant subspaces to either augment krylov subspaces or to construct preconditioners which invert on these subspaces in the case where these spaces are exactly invariant the augmentation approach is shown to be superior we further show how a strategy recently introduced by de sturler for truncating the approximation space of an mr method can be interpreted as a controlled loosening of the condition for global mr approximation based on the canonical angles between subspaces for the special case of krylov subspace methods we give a concise derivation of the role of ritz and harmonic ritz values and vectors in the polynomial description of krylov spaces as well as of the use of the implicitly updated arnoldi method for manipulating krylov spaces

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

an ergodic theorem for classes of preconditioned matrices abstract we consider abstract classes of matrices a satisfying some structural conditions and in particular satisfying a crucial assumption about the asymptotic distribution of eigenvalues we prove a similar distribution property for classes of preconditioned matrices constructed by using representants of a as a particular case this result applies to preconditioned matrices coming from several important contexts finite differences and faedo ritz galerkin linear systems associated with elliptic and semielliptic boundary value problems very general hermitian toeplitz structures generated by multivariate ll functions this result answers in the positive some structural questions raised by tyrtyshnikov e tyrtyshnikov linear algebra appl 207 1994 225 249 and by the author s serra linear algebra appl 267 1997 139 161 s serra siam j numer anal in press in the toeplitz context

a new deflated block gcrot m k method for the solution of linear systems with multiple right hand sides linear systems with multiple right hand sides arise in many applications to solve such systems efficiently a new deflated block gcrot m k method is explored in this paper by exploiting a modified block arnoldi deflation incorporating this modified block arnoldi deflation the new algorithm can address the possible linear dependence at each iteration during the block arnoldi procedure and reduces expensive computational operations moreover as a block version of gcrot m k the new method inherits the property of easy operability finally some numerical examples also illustrate the effectiveness of the proposed method

preconditioning techniques for non symmetric matrices with application to temperature calculations of cooled concrete preconditioning techniques based on incomplete lu decomposition are described for large sparse non symmetric matrix systems in which the largest part of the coefficient matrix is a symmetric m matrix with a very regular sparsity pattern some methods are described in which a small part of the matrix with an irregular sparsity pattern is treated very efficiently the methods are combined with cgstab and tested on a non stationary diffusion problem in three dimensions for this type of problems the modified incomplete decomposition as suggested by gustafsson performs significantly better than the standard incomplete factorization also some methods are described to construct preconditioners for large sparse non symmetric matrices with arbitrary sparsity patterns methods in which the sparsity pattern of the preconditioning matrix is based on a drop tolerance are very effective and relatively cheap

a domain decomposition method with nonconformal meshes for finite periodic and semi periodic structures we present a domain decomposition method as a preconditioner for krylov type solvers to model complex electromagnetic problems containing periodicities the method reduces memory requirements by decomposing the original problem into several nonoverlapping sub domains the 1st order transmission condition is employed on interfaces between adjacent sub domains to enforce continuity of electromagnetic fields and to ensure the sub domain problems are well posed by following the spirit of duality paring a symmetric system is obtained to reduce the computational burdens of the present method the finite element tearing and interconnecting like algorithm is adopted this algorithm results in the computation of the so called numerical green s function which can be compressed efficiently via a rank revealing matrix factorization algorithm the final system matrix is solved by krylov solvers instead of classical stationary solvers to improve the convergence of iterative solvers several robust implementation details are discussed and the choice of some popular krylov subspace solvers is studied through numerical examples

fast computation of angular responses of large scale three dimensional electromagnetic wave scattering many applications in electromagnetic engineering require predicting the currents induced by a wide range of incident plane waves on a given object this paper proposes a way to efficiently and accurately compute the angular responses for electrically large targets since the targets of concern are electrically large we have employed the newly developed non conformal domain decomposition method ddm as the direct maxwell equation solver the proposed fast angular sweep algorithm is then applied in conjunction with the ddm solver to produce wide range of angular responses there are basically three main ingredients in the proposed fast angular sweep algorithm the use of fourier harmonics to interpolate the right hand sides rhss over a given angular range the number of independent coefficient vectors in the fourier harmonics can be further reduced through the application of singular value decomposition svd and finally the matrix solution of multiple rhss can be solved efficiently using a recently proposed krylov recycling method gcro dr three numerical results are presented and compared to measurements and direct computations to validate the numerical solutions and to illustrate the effectiveness of the proposed approach

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