efficient block preconditioning for a c1 finite element discretisation of the dirichlet biharmonic problem we present an efficient block preconditioner for the two dimensional biharmonic dirichlet problem discretised by c1 bicubic hermite finite elements in this formulation each node in the mesh has four different degrees of freedom dofs grouping dofs of the same type together leads to a natural blocking of the galerkin coefficient matrix based on this block structure we develop two preconditioners a 2x2 block diagonal preconditioner bd and a block bordered diagonal bbd preconditioner we prove mesh independent bounds for the spectra of the bd preconditioned galerkin matrix under certain conditions the eigenvalue analysis is based on the fact that the proposed preconditioner like the coefficient matrix itself is symmetric positive definite and is assembled from element matrices we demonstrate the effectiveness of an inexact version of the bbd preconditioner which exhibits near optimal scaling in terms of computational cost with respect to the discrete problem size finally we study robustness of this preconditioner with respect to element stretching domain distortion and non convex domains

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

new numerical algorithms for the nonlinear diffusion model of image denoising and segmentation abstract in this paper for the nonlinear diffusion model of image denoising and segmentation we present two high order explicit algorithms and a semi implicit algorithm using algebraic multi grid method the unconditional stability of the semi implicit approximation is given the numerical results of the above three algorithms are demonstrated with comparison to general explicit algorithm in y chen b c vemuri l wang image denoising and segmentation via nonlinear diffusion comput math appl 39 2000 131 149 for real noised images the numerical results show that the implicit algorithm is quicker and sharper than all explicit algorithms

an algebraic multigrid solver for transonic flow problems this article presents the latest developments of an algebraic multigrid amg based on full potential equation fpe solver for transonic flow problems with emphasis on advanced applications the mathematical difficulties of the problem are associated with the fact that the governing equation changes its type from elliptic subsonic flow to hyperbolic supersonic flow the flow solver is capable of dealing with flows from subsonic to transonic and supersonic conditions and is based on structured body fitted grids approach for treating complex geometries the computational method was demonstrated on a variety of problems to be capable of predicting the shock formation and achieving residual reduction of roughly an order of magnitude per cycle both for elliptic and hyperbolic problems through the entire range of flow regimes independent of the problem size resolution

acceleration methods for total variation based image denoising in this paper we apply a fixed point method to solve the total variation based image denoising problem an algebraic multigrid method is used to solve the corresponding linear equations krylov subspace acceleration is adopted to improve convergence in the fixed point iteration a good initial guess for this outer iteration at finest grid is obtained by combining fixed point iteration and geometric multigrid interpolation successively from the coarsest grid to the finest grid numerical experiments demonstrate that this method is efficient and robust even for images with large noise to signal ratios

moduli interpretations for noncongruence modular curves we consider the moduli of elliptic curves with g structures where g is a finite 2 generated group when g is abelian a g structure is the same as a classical congruence level structure there is a natural action of text sl 2 mathbb z on these level structures if gamma is a stabilizer of this action then the quotient of the upper half plane by gamma parametrizes isomorphism classes of elliptic curves equipped with g structures when g is sufficiently nonabelian the stabilizers gamma are noncongruence using this we obtain arithmetic models of noncongruence modular curves as moduli spaces of elliptic curves equipped with nonabelian g structures as applications we describe a link to the inverse galois problem and show how our moduli interpretations explains the bad primes for the unbounded denominators conjecture and allows us to translate the conjecture into the language of geometry and galois theory

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

stability and dispersion analysis for higher order 3 d adi fdtd method in this paper the stability and dispersion analysis for higher order three dimensional 3 d alternating direction implicit adi finite difference time domain fdtd methods is presented starting with the stability and dispersion analysis of the fourth order 3 d adi fdtd method which adopts the fourth order cell centered finite difference scheme for the spatial differential operator we generalize the analysis to other higher order methods based on the sixth and tenth order cell centered finite difference schemes to the best of our knowledge this is the first time that a comprehensive study of the stability and dispersion characteristics for one series of higher order 3 d adi fdtd methods is presented our analysis results show that all the higher order adi fdtd methods that are based on cell centered finite difference schemes are unconditionally stable the generalized form of the dispersion relations for these unconditionally stable adi fdtd methods is also presented using the relations attained the effects of the order of schemes mesh size and time step on the dispersion are illustrated through numerical results this study will be useful for the selection and evaluation of various higher order adi methods

lyapunov and matrix norm stability analysis of adi fdtd schemes for doubly lossy media lyapunov and matrix norm stability analysis is applied on various alternating direction implicit finite difference time domain adi fdtd schemes for doubly lossy media the stability analysis is performed rigorously in both time and fourier domains among the schemes considered are averaging forward backward backward forward forward forward exponential time differencing and backward backward from the analysis it is found that all schemes except backward backward scheme are unconditionally stable for backward backward scheme the condition for stability is determined

efficient algebraic multigrid algorithms and their convergence in this paper efficient algorithms for the algebraic multigrid amg method are advanced and amg methods are proposed for improving the convergence rate a new convergence theorem of amg is given and a complete theoretical study of new algorithms is subsequently presented various numerical results demonstrate the efficiency of the new approaches

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

riesz bases of wavelets and applications to numerical solutions of elliptic equations we investigate riesz bases of wavelets in sobolev spaces and their applications to numerical solutions of the biharmonic equation and general elliptic equations of fourth order first we study bi cubic splines on the unit square with homogeneous boundary conditions the approximation properties of these cubic splines are established and applied to convergence analysis of the finite element method for the biharmonic equation second we develop a fairly general theory for riesz bases of hilbert spaces equipped with induced norms under the guidance of the general theory we are able to construct wavelet bases for sobolev spaces on the unit square the condition numbers of the stiffness matrices associated with the wavelet bases are relatively small and uniformly bounded third we provide several numerical examples to show that the numerical schemes based on our wavelet bases are very efficient finally we extend our study to general elliptic equations of fourth order and demonstrate that our numerical schemes also have superb performance in the general case

stability and dispersion analysis for adi fdtd method in lossy media the stability and dispersion analysis for the alternating direction implicit finite difference time domain adi fdtd method in lossy media is presented although the stability and numerical dispersion have been analyzed for the adi fdtd method most of the analysis is dedicated to the cases of lossless media here the stability and dispersion analysis is performed for the method in lossy media the stability analysis theoretically proves the unconditional stability of the adi fdtd method in lossy media meanwhile the dispersion analysis reveals the numerical loss and dispersion characteristics of this method this will be meaningful for the evaluation and further development of the adi fdtd method in lossy media

acceleration and stabilization properties of minimal residual smoothing technique in multigrid we analyze the standard multigrid method accelerated by a minimal residual smoothing mrs technique we show that mrs can accelerate the convergence of the slow residual components thus accelerates the overall multigrid convergence we prove that under certain hypotheses mrs stabilizes the divergence of certain slow residual components and thus stabilizes the divergent multigrid iteration the analysis is customarily conducted on the two level method

new interpolation formulas of using geometric assumptions in the algebraic multigrid method in this paper new interpolation formulas for using geometric assumptions in the algebraic multigrid amg method are reported the theoretical and convergence analysis will be presented the effectiveness and robustness of these interpolation formulas are demonstrated by numerical experiments not only is a rapid rate of convergence achieved but the amg algorithm used in conjunction with these formulas can also be used to solve various ill conditioned systems of equations the principal contribution of the present method is to extend the range of applications of the amg method developed by ruge and stuben

a fast algebraic multigrid preconditioned conjugate gradient solver abstract this work presents a new approach for selecting the coarse grids allowing a faster algebraic multigrid amg preconditioned conjugate gradient solver this approach is based on an appropriate choice of the parameter considering the matrix density during the coarsening process which implies in a significant reduction in the matrix dimension at all amg levels

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

algebraic multigrid based on element interpolation amge we introduce amge an algebraic multigrid method for solving the discrete equations that arise in ritz type finite element methods for partial differential equations assuming access to the element stiffness matrices we have that amge is based on the use of two local measures which are derived from global measures that appear in existing multigrid theory these new measures are used to determine local representations of algebraically smooth error components that provide the basis for constructing effective interpolation and hence the coarsening process for amg here we focus on the interpolation process choice of the coarse grids based on these measures is the subject of current research we develop a theoretical foundation for amge and present numerical results that demonstrate the efficacy of the method

a split step fdtd method for 3 d maxwell s equations in general anisotropic media a split step finite difference time domain fdtd method is presented for 3 d maxwell s equations in general anisotropic media the general anisotropic media can be characterized by full permittivity and permeability tensors stability analysis of the proposed split step fdtd method using the fourier method is presented the eigenvalues of the fourier amplification matrix are numerically shown to have unity magnitude even for time steps greater than the courant limit time step thereby illustrating the stable and non dissipative nature of the split step fdtd method in general anisotropic media numerical results are presented to further validate the accuracy and stability of the proposed split step fdtd method in general anisotropic media

parallel algebraic multigrid methods on distributed memory computers algebraic multigrid methods are well suited as preconditioners for iterative solvers we consider linear systems of equations which are sparse and symmetric positive definite and which stem from a finite element discretization of a second order self adjoint elliptic partial differential equation or a system of them since preconditioners based on algebraic multigrid are very efficient additional speedup can only be achieved by parallelization in this paper we propose a general parallel algebraic multigrid algorithm for finite element discretizations based on domain decomposition ideas which is well suited for distributed memory computers this paper pays special attention to the coarsening strategy which has to be adapted in the parallel case moreover a general framework of data distribution gives rise to a construction scheme for the prolongation operators results of numerical studies on parallel computers with distributed memory are presented which show the high efficiency of the approach

an algebraic multigrid method with interpolation reproducing rigid body modes for semi definite problems in two dimensional linear elasticity based on the geometric grid information as geometric coordinates an algebraic multigrid amg method with the interpolation reproducing the rigid body modes namely the kernel elements of semi definite operator arising from linear elasticity is constructed and such method is applied to the linear elasticity problems with a traction free boundary condition and crystal problems with free boundary conditions as well the results of various numerical experiments in two dimensions are presented it is shown from the numerical results that the constructed amg method is robust and efficient for such semi definite problems and the convergence is uniformly bounded away from one independent of the problem size furthermore the amg method proposed in this paper has better convergence rate than the commonly used amg methods simultaneously an amg method that can preserve the quotient space which means that if the exact solution of original problem belongs to the quotient space of discrete operator considered then the numerical solution of amg method is convergent in the same quotient space is obtained using the technique of orthogonal decomposition

an arbitrary order lod fdtd method and its stability and numerical dispersion an arbitrary order unconditionally stable three dimensional 3 d locally one dimensional finite difference time method fdtd lod fdtd method is proposed theoretical proof and numerical verification of the unconditional stability are shown and numerical dispersion is derived analytically effects of discretization parameters on the numerical dispersion errors are studied comprehensively it is found that the second order lod fdtd has the same level of numerical dispersion error as that of the unconditionally stable alternating direction implicit finite difference time domain adi fdtd method and other lod fdtd methods but with higher computational efficiency to reduce the dispersion errors either a higher order lod fdtd method or a denser grid can be applied but the choice has to be carefully made in order to achieve best trade off between the accuracy and computational efficiency the work presented in this paper lays the foundations and guidelines for practical uses of the lod method including the potential mixed order lod fdtd methods

adaptive algebraic multigrid efficient numerical simulation of physical processes is constrained by our ability to solve the resulting linear systems prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort overcoming the limitations of geometric multigrid methods to simple geometries and differential equations algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix while this allows for efficient black box solution of the linear systems associated with discretizations of many elliptic differential equations it also results in a lack of robustness due to unsatisfied assumptions made on the near null spaces of these matrices this paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process emphasis is on the principles that guide the adaptivity and their application to algebraic multigrid solution of certain symmetric positive definite linear systems

iterative methods for overflow queuing models ii preconditioned conjugate gradient methods are employed to find the steady state probability distribution of markovian queuing networks that have overflow capacity different singular preconditioners that can be handled by separation of variables are discussed the resulting preconditioned systems are nonsingular numerical results show that the number of iterations required for convergence grows very slowly with the queue sizes

an adaptive algorithm for tv based model of three norms lq q 12 1 2 in image restoration abstract in this paper we present an adaptive method for the tv based model of three norms l q q 1 2 1 2 for the image restoration problem the algorithm with the l2 norm is used in the smooth regions where the value of u is small the algorithm with the l 1 2 norm is applied for the jumps where the value of u is large when the value of u is moderate the algorithm with the l1 norm is employed thus the three algorithms are applied for different regions of a given image such that the advantages of each algorithm are adopted the numerical experiments demonstrate that our adaptive algorithm can not only keep the original edge and original detailed information but also weaken the staircase phenomenon in the restored images specifically in contrast to the l1 norm as in the rudin osher fatemi model the l2 norm yields better results in the smooth and flat regions and the l 1 2 norm is more suitable in regions with strong discontinuities therefore our adaptive algorithm is efficient and robust even for images with large noises

unconditionally stable adi crank nicolson implementation and lossy split error revisited this paper first shows that the existing adi and cn implementations initiated by namiki follows an indirect way of solving a linear system the adi and cn update equations can and should be implemented following the standard fdtd formulation where the update equations translate directly from ampere s and faraday s laws the advantages leading to a more complete lossy split error analysis and possibilities of mitigating the numerical errors are also presented in high loss applications the cn method greatly outperforms the adi method in terms of accuracy the root cause of the adi degradation is discussed

analysis of two alternative adi fdtd formulations for transverse electric waves in lossy materials a numerical dispersion analysis of the alternating direction implicit finite difference time domain method for transverse electric waves in lossy materials is presented two different finite difference approximations for the conduction terms are considered the double average and the synchronized schemes the numerical dispersion relation is derived in a closed form and validated through numerical simulations this study shows that despite its popularity the accuracy of the double average scheme is sensitive to how well the relaxation time constant of the material is resolved by the time step poor resolutions lead to unacceptably large numerical errors on the other hand for good conductors the synchronized scheme allows stability factors as large as 100 to be used without deteriorating the accuracy significantly

an efficient algebraic multigrid preconditioned conjugate gradient solver in this paper we present a robust and efficient algebraic multigrid preconditioned conjugate gradient solver for systems of linear equations arising from the finite element discretization of a scalar elliptic partial differential equation of second order on unstructured meshes the algebraic multigrid amg method is one of most promising methods for solving large systems of linear equations arising from unstructured meshes the conventional amg method usually requires an expensive setup time particularly for three dimensional problems so that generally it is not used for small and medium size systems or low accuracy approximations our solver has a quick setup phase for the amg method and a fast iteration cycle these allow us to apply this solver for not only large systems but also small to medium systems of linear equations and also for systems requiring low accuracy approximations

remark on convergence of algebraic multigrid in the form of matrix decomposition we introduce the convergence of algebraic multigrid in the form of matrix decomposition the convergence is proved in block versions of the multi elimination incomplete lu bilum factorization technique and the approximation of their inverses to preserve sparsity the convergence theorem can be applied to general interpolation operator furthermore we discuss the error caused by the error matrix

extension of the adi fdtd method to debye media this paper describes an extension of the alternating direction implicit finite difference time domain adi fdtd method to analyze problems involving debye media

implicit id fdtd algorithm based on crank nicolson scheme dispersion relation and stability analysis a new implicit isotropic dispersion finite difference time domain id fdtd algorithm is proposed which is formulated based on the crank nicolson cn implicit scheme the update equation of the new scheme is given for a three dimension 3d problem and a general lossy medium including electric and magnetic losses the dispersion relation of the cn id fdtd scheme is obtained based on the eigen analysis technique also the unconditional stability is mathematically proved by using the energy method for a practical application a maximum time limit is proposed for free space to validate the proposed scheme a 2d cavity problem is considered the electric fields inside the cavity which are calculated by the proposed conventional cn fdtd schemes and the exact solution are compared

a massively parallel fractional step solver for incompressible flows this paper presents a parallel implementation of fractional solvers for the incompressible navier stokes equations using an algebraic approach under this framework predictor corrector and incremental projection schemes are seen as sub classes of the same class making apparent its differences and similarities an additional advantage of this approach is to set a common basis for a parallelization strategy which can be extended to other split techniques or to compressible flows the predictor corrector scheme consists in solving the momentum equation and a modified continuity equation namely a simple iteration for the pressure schur complement consecutively in order to converge to the monolithic solution thus avoiding fractional errors on the other hand the incremental projection scheme solves only one iteration of the predictor corrector per time step and adds a correction equation to fulfill the mass conservation as shown in the paper these two schemes are very well suited for massively parallel implementation in fact when compared with monolithic schemes simpler solvers and preconditioners can be used to solve the non symmetric momentum equations gmres bi cgstab and to solve the symmetric continuity equation cg deflated cg this gives good speedup properties of the algorithm the implementation of the mesh partitioning technique is presented as well as the parallel performances and speedups for thousands of processors

a parallel fictitious domain multigrid preconditioner for the solution of poisson s equation in complex geometries abstract a parallel multilevel preconditioner based on domain decomposition and fictitious domain methods has been presented for the solution of the poisson equation in complicated geometries rectangular blocks with matching grids on interfaces on a structured rectangular mesh have been used for the decomposition of the problem domain sloping sides or curved boundary surfaces are approximated using stepwise surfaces formed by the grid cells a seven point stencil based on the central difference scheme has been used for the discretization of the laplacian for both interior and boundary grid points and this results in a symmetric linear algebraic system for any type of boundary condition the preconditioned conjugate gradient method has been used for the solution of this symmetric system the multilevel preconditioner for the cg is based on a v cycle multigrid applied to the poisson equation on a fictitious domain formed by the union of the rectangular blocks used for the domain decomposition numerical results are presented for two typical poisson problems in complicated geometries one related to heat conduction and the other one arising from the les dns of incompressible turbulent flow over a packed array of spheres these results clearly show the efficiency and robustness of the proposed approach

multigrid algorithm from cyclic reduction for markovian queueing networks a multigrid method based on cyclic reduction strategy is proposed to solve huge nonsymmetric singular linear systems arising from markovian queueing networks a simple way to construct the matrix dependent prolongation and restriction operators is presented in this paper numerical results for multiple queues are given to illustrate the efficiency and robustness of our methods

uniformization of deligne mumford curves we compute the fundamental groups of non singular analytic delignemumford curves classify the simply connected ones and classify analytic deligne mumford curves by their uniformization type as a result we find an explicit presentation of an arbitrary deligne mumford curve as a quotient stack along the way we compute the automorphism 2 groups of weighted projective stacks p n1 n2 nr we also discuss connections with the theory of f groups 2 groups and bass serre theory of graphs of groups

a filtered cumulant lattice boltzmann method for violent two phase flows abstract the lattice boltzmann method lbm has been successfully applied to simulate various two phase flow problems such as rayleigh taylor instability rising bubble and droplet dynamics however the method known as two phase lbm faces difficulty in simulating violent two phase flow problems such as dam breaking and liquid jet breakup where the density ratio and reynolds number are high and there are rapid and complex topological changes in this paper we address the problem by using a filtered cumulant lbm the incompressible two phase flow equations are solved by using a velocity based formulation of two phase lbm with a cumulant collision model to stably simulate problems with high density ratio and reynolds number there is no pressure iteration in this formulation and we can keep the simplicity of the cumulant model to further enhance the stability in violent flows a second order filter is applied on the velocity field which can be turned off for non violent flows for interface capturing a conservative phase field lbm which guarantees the mass conservation is employed the proposed method is first validated using five non violent flow problems and the results are in good agreements with experimental and computational references the proposed method is then employed to solve three violent flow problems and the results are in qualitatively good agreements with experimental and computational references the proposed method is more diffusive than the unfiltered model but is stable for all problems simulated in this paper these results can be used as a preliminary study on violent two phase flow simulation using two phase lbm

on the algebraic multigrid method new formulations for the algebraic multigrid amg method are presented a new interpolation operator is developed in which the weighting could be negative numerical experiments demonstrate that the use of negative interpolation weights is necessary in some applications new approaches to construct the restriction operator and the coarse grid equations are discussed two new amg methods are proposed theoretical study and convergence analysis of the amg methods are presented the main contributions of this paper are to improve the convergence rate and to extend the range of applications of an amg method numerical experiments are reported for matrix computations that resulted from partial differential equations signal processing and queueing network problems the success of the proposed new amg algorithms is clearly demonstrated by applications to non diagonally dominant matrix problems for which the standard amg method fails to converge

amg for problems of elasticity algebraic multigrid amg is an efficient iterative method designed as a black box solver that uses the principles of geometric multigrid methods the components necessary for the multigrid cycling process are automatically constructed using information taken from the matrix itself the method does not require uniform grids making it an attractive approach for problems in structures and elasticity here we describe the method and give results for a model plane elasticity problem

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

an algebraic multigrid preconditioner for a class of singular m matrices we apply algebraic multigrid amg as a preconditioner for solving large singular linear systems of the type i t t x 0 with gmres here t is assumed to be the transition matrix of a markov process although amg and gmres were originally designed for the solution of regular systems with adequate adaptation their applicability can be extended to problems as described above

algebraic multigrid theory the symmetric case a rigorous two level theory is developed for general symmetric matrices and nonsymmetric ones using kaczmarz relaxation without assuming any regularity not even any grid structure of the unknowns the theory applies to algebraic multigrid amg processes as well as to the usual geometric multigrid it yields very realistic estimates and precise answers to basic algorithmic questions such as in what algebraic sense does gauss seidel or jacobi kaczmarz line gauss seidel etc relaxation smooth the error when is it appropriate to use block relaxation what algebraic relations must be satisfied by the coarse to fine interpolations what is the algorithmic role of the geometric origin of the problem the theory helps to rigorize local mode analyses and locally analyze cases where the latter is inapplicable

a wavelet based algebraic multigrid preconditioner for sparse linear systems this work considers the use of discrete wavelet transform dwt based in filters in the construction of the hierarchy of matrices in the algebraic multigrid method amg the two dimensional dwt is applied to produce an approximation of the matrix in each level of the wavelets multiresolution decomposition process in this procedure an operator is created formed only by lowpass filters that is applied to the rows and columns of the matrix capturing this approximation this same operator is used as an intergrid transfer in the amg wavelet based algebraic multigrid method wamg was implemented with daubechies wavelets of orders 6 4 and 2 and used as a preconditioner for the generalized minimal residual method gmres numerical results are presented comparing this new approach with the standard algebraic multigrid as preconditioner for sparse linear systems

parallel multigrid computation of unsteady incompressible viscous flows using a matrix free implicit method and high resolution characteristics based scheme abstract a three dimensional parallel unstructured non nested multigrid solver for solutions of unsteady incompressible viscous flow is developed and validated the finite volume navier stokes solver is based on the artificial compressibility approach with a high resolution method of characteristics based scheme for handling convection terms the unsteady flow is calculated with a matrix free implicit dual time stepping scheme the parallelization of the multigrid solver is achieved by multigrid domain decomposition approach mg dd using single program multiple data spmd and multiple instruction multiple data mimd programming paradigm there are two parallelization strategies proposed in this work first strategy is a one level parallelization strategy using geometric domain decomposition technique alone second strategy is a two level parallelization strategy that consists of a hybrid of both geometric domain decomposition and data decomposition techniques message passing interface mpi and openmp standard are used to communicate data between processors and decompose loop iterations arrays respectively the parallel multigrid code is used to simulate both steady and unsteady incompressible viscous flows over a circular cylinder and a lid driven cavity flow a maximum speedup of 22 5 could be achieved on 32 processors for instance the lid driven cavity flow of re 1000 the results obtained agree well with numerical solutions obtained by other researchers as well as experimental measurements a detailed study of the time step size and number of pseudo sub iterations per time step required for simulating unsteady flow are presented in this paper

convergence of algebraic multigrid methods for symmetric positive definite matrices with weak diagonal dominance for symmetric positive definite matrices with weak diagonal dominance the algebraic sense of smooth errors is discussed and convergence of algebraic multigrid amg methods is proved it is found that the results of amg for symmetric positive definite m matrices with weak diagonal dominance also hold without the m matrix assumption the key result is the new formula x email protected ia i email protected j email protected email protected j a i j a j i x 2 email protected email protected j email protected email protected j a i j x email protected email protected ia i j a i j x j 2