optimal order preconditioning of finite difference matrices a new multilevel preconditioner is proposed for the iterative solution of linear systems whose coefficient matrix is a symmetric m matrix arising from the discretization of a second order elliptic pde it is based on a recursive block incomplete factorization of the matrix partitioned in a two by two block form in which the submatrix related to the fine grid nodes is approximated by a modified incomplete milu factorization and the schur complement computed from a diagonal approximation of the same submatrix a general algebraic analysis proves optimal order convergence under mild assumptions related to the quality of the approximations of the schur complement on the one hand and of the fine grid submatrix on the other hand this analysis does not require a red black ordering of the fine grid nodes nor a transformation of the matrices in hierarchical form considering more specifically 5 point finite difference approximations of two dimensional problems we prove that the spectrum of the preconditioned system is contained in the interval 0 46 3 independently of possible large jumps or anisotropy in the pde coefficients as long as the latter are piecewise constant on the coarsest mesh numerical results illustrate the efficiency and the robustness of the proposed method

preconditioning for pde constrained optimization with total variation regularization abstract the efficient solution of pde constrained optimization problems is of great significance in many scientific and engineering applications the key point of achieving this goal is to solve the linear systems arising from such problems more efficiently by preconditioning in this paper we consider preconditioning the linear systems arising from pde constrained optimization problems with total variation tv regularization based on the block structure of the coefficient matrix or its row and column transformed forms we propose three kinds of new preconditioners by approximating the schur complements in different ways we analyze the eigenvalue properties of the preconditioned coefficient matrices and provide the eigenvalue bounds independent of mesh size we also show the relations of eigenvalue bounds corresponding to different preconditioners numerical results illustrate the efficiency of the proposed preconditioners by comparison with other existing methods

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

the hierarchical basis multigrid method we derive and analyze the hierarchical basis multigrid method for solving discretizations of self adjoint elliptic boundary value problems using piecewise linear triangular finite elements the method is analyzed as a block symmetric gau seidel iteration with inner iterations but it is strongly related to 2 level methods to the standard multigridv cycle and to earlier jacobi like hierarchical basis methods the method is very robust and has a nearly optimal convergence rate and work estimate it is especially well suited to difficult problems with rough solutions discretized using highly nonuniform adaptively refined meshes

a mesh adapted closed form regular kernel for 3d singular integral equations the green s functions employed in the method of moments mom diverge when observation and source points coincide this is at the origin of the difficulties in computing the mom matrix entries and in handling the near field interactions in fast fourier transform fft based fast methods and other sampling based methods in this paper we show that this singularity can be avoided and a modified regular green s function can be used instead this latter is obtained from the spectral representation of the usual green s function via windowing of its spectrum the width of the spectral window depends on the size of the mesh employed for discretizing the problem so that the proposed regular green s function is a mesh adapted regular kernel we address a general 3d problem we relate the mom reaction integrals to the 2d fourier spectrum of the green s function that allows to discuss the necessary spectral bandwidth for the windowed green s function we employ a tapered window and present a closed form expression for the spatial green s function numerical results are presented for 3d antenna and scattering problems discretized with rao wilton glisson rwg functions and for uniform and nonuniform meshing they show that the proposed method yields accurate solutions also for the antenna input impedance the meaning of the regularized green s function is also discussed and put in perspective

the contraction number of a multigrid method for solving the poisson equation the treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrid algorithm but only for fast convergence for the linear equations which arise from the discretization of the poisson equation a convergence factor 0 5 is established independent of the shape of the domain and of the regularity of the solution

on deriving a locally corrected nystrom scheme from a quadrature sampled moment method a novel high order method of moment procedure with quadrature point based discretization is presented the scheme is equivalent to a moment method employing smooth basis and testing functions applying a fixed point numerical quadrature approximation for the outer integral mapping the current to the quadrature points then leads to a formulation that is equivalent to that derived via the locally corrected nystrom method the convergence properties of the current density and rcs for smooth and singular geometries in two and three dimensions are also studied

a greedy strategy for coarse grid selection efficient solution of the very large linear systems that arise in numerical modeling of real world applications is often only possible through the use of multilevel techniques while highly optimized algorithms may be developed using knowledge about the origins of the matrix problem to be considered much recent interest has been in the development of purely algebraic approaches that may be applied in many situations without problem specific tuning here we consider an algebraic approach to finding the fine coarse partitions needed in multilevel approaches the algorithm is motivated by recent theoretical analysis of the performance of two common multilevel algorithms multilevel block factorization and algebraic multigrid while no guarantee on the rate of coarsening is given the splitting is shown to always yield an effective preconditioner in the two level sense numerical performance of two level and multilevel variants of this approach is demonstrated in combination with both algebraic multigrid and multilevel block factorizations and the advantages of each of these two algorithmic settings are explored

on two ways of stabilizing the hierarchical basis multilevel methods a survey of two approaches for stabilizing the hierarchical basis hb multilevel preconditioners both additive and multiplicative is presented the first approach is based on the algebraic extension of the two level methods exploiting recursive calls to coarser discretization levels these recursive calls can be viewed as inner iterations at a given discretization level exploiting the already defined preconditioner at coarser levels in a polynomially based inner iteration method the latter gives rise to hybrid type multilevel cycles this is the so called hybrid algebraic multilevel iteration amli method the second approach is based on a different direct multilevel splitting of the finite element discretization space this gives rise to the so called wavelet multilevel decomposition based on l 2 projections which in practice must be approximated both approaches the amli one and the one based on approximate wavelet decompositions lead to optimal relative condition numbers of the multilevel preconditioners

multilevel block ilu preconditioner for sparse nonsymmetric m matrices we develop in this paper a multilevel block ilu preconditioner for solving sparse nonsymmetric m matrices arising from convection diffusion equations discretized by finite difference scheme this preconditioner is based on recursive block ilu factorization where the schur complement matrices are successively approached by some sparse matrices satisfying some hypothesis we study the preconditioned gmres iterative method and multigrid method for solving linear systems arising from this scheme

a complete set of linear phase basis functions for scatterers with flat faces and for planar apertures entire domain basis functions are introduced for the analysis of scattering from bodies with flat portions and of radiation from planar apertures they are defined via the application of the sampling theorem of shannon these basis functions are particularly convenient for polygonal contours however arbitrary contours can be treated in the same framework with a very minor pre processing time a non redundant set of basis functions is determined a priori by explicit approximate formulas either for currents and for radiated field numerical results are presented to shown the accuracy of the method

ordering strategies for modified block incomplete factorizations in this study the aim is to first investigate how the rate of convergence of unperturbed modified block incomplete factorization methods depends on the discrete partial differential equation pde to solve and next on this basis to provide simple practical rules for easily selecting orderings that result in rapid convergence it emerges from the analysis here that for discrete pdes ordering schemes that optimize the rate of convergence strongly depend on the variation of both the pde coefficients and the mesh sizes the arguments made which bring to light the appreciable potentialities of modified methods also display insight into why block versions are more robust than pointwise ones

the empirical mode decomposition and the hilbert spectrum for nonlinear and non stationary time series analysis a new method for analysing nonlinear and non stationary data has been developed the key part of the method is the empirical mode decomposition method with which any complicated data set can be dec

least squares solutions of the matrix equation axb cyd e with the least norm for symmetric arrowhead matrices in this paper the least squares solution of the matrix equation axb cyd e for symmetric arrowhead matrices with the least norm is discussed by using moore penrose inverse and the kronecker product the general expression of the solution to this problem is derived a corresponding numerical algorithm and an example are also given

preconditioning toeplitz plus diagonal linear systems using the sherman morrison woodbury formula in order to solve the toeplitz plus diagonal linear systems arising from image restorations efficiently we propose a sparse approximate inverse preconditioner based on the sherman morrison woodbury formula the preconditioner can be constructed through an incomplete factorization combined with some dropping strategies when the preconditioner is applied to the conjugate gradient method for solving the toeplitz plus diagonal linear systems numerical results show that our preconditioning method is more effective than other existing ones

fourier analysis of a robust multigrid method for convection diffusion equations we consider a two grid method for solving 2d convection diffusion problems the coarse grid correction is based on approximation of the schur complement as a preconditioner of the schur complement we use the exact schur complement of modified fine grid equations we assume constant coefficients and periodic boundary conditions and apply fourier analysis we prove an upper bound for the spectral radius of the two grid iteration matrix that is smaller than one and independent of the mesh size the convection diffusion ratio and the flow direction i e we have a strong robustness result numerical results illustrating the robustness of the corresponding multigrid w cycle are given

gaussian ring basis functions for the analysis of modulated metasurface antennas this paper presents a novel type of basis functions whose spectral and space domain properties can be exploited for the efficient method of moments mom analysis of planar metasurface mts antennas the effect of the homogenized mts is introduced in the integral equation as an impedance boundary condition ibc the proposed basis functions are shaped as gaussian type rings with small width and linear azimuthal phase the analytical form of the spectrum of the gaussian ring basis allows for a closed form evaluation of the mom impedance matrix s entries moreover these basis functions account for the global evolution of the surface current density in an effective manner reducing the size of the mom system of equations with respect to the case of subdomain basis functions these features allow one to carry out a direct solution for problems with a diameter of up to 15 wavelengths in less than 1 min using a conventional laptop the applicability on practical antennas has been tested through the full wave analysis of mts antennas implemented with small printed elements

complex conical beams for aperture field representations a method is presented for computing aperture radiated fields by means of new conical beams with azimuth phase variation these beams are generated in a natural way starting from the spectral domain radiation integral by expanding the electric field spectrum in the aperture plane in a fourier series and by approximating the obtained fourier series coefficients by a sum of complex exponentials using the generalized pencil of function method this transforms the radiation integral to a simpler form which can be evaluated analytically two types of wave objects are derived both of them arising from the same spectral gpof process that possess different properties aperture fields obtained via the new approach are successfully compared to those calculated via direct near field integration or asymptotic evaluation

upscaling methods for a class of convection diffusion equations with highly oscillating coefficients this paper investigates the upscaling method to the following parabolic equation tc uc d c f x t which stems from the application of solute transport in porous media because of the highly oscillating permeability of the porous media the darcy velocity u hence the dispersion tensor d has many scales with high contrasts thus how to calculate the macro scale equivalent coefficients of the above equation becomes the target of this paper a new upscaling method is proposed and studied via comparing with another upscaling method which was proposed in z chen w deng h ye discrete contin dyn syst 13 2005 941 960 the two different equivalent coefficients computing formulations are based on the solutions of two different cell local problems which one utilizes the elliptic operator with terms of all orders while the other only uses the second order term error estimates between the equivalent coefficients and the homogenized coefficients are given under the assumption that the oscillating coefficients are periodic which is not required by the method numerical experiments are carried out for the periodic coefficients to demonstrate the accuracy of the proposed method moreover the upscaling method is applied to solve the solute transport in a porous medium with a random log normal relative permeability the results show the efficiency and accuracy of the proposed method

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

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

sampling method using prefiltered band limited green s functions for the solution of electromagnetic integral equations in the context of far field scattering the spatial spectrum of fields or currents on a scatterer may be approximated as band limited on this premise one can greatly simplify the computation of radiation integrals by eliminating in advance those spectral components of the green s function not expected to be present in the fields this approach enables one to dispense with the notion of basis functions and instead represent fields and currents as well as the impedance matrix representing the green s function entirely in terms of a sparse set of samples thereby obviating the need for integration moreover by taking full advantage of the band limited character of the solution this sampling theoretical approach yields an efficient representation with a minimal number of degrees of freedom the method was explored for scattering from two dimensional perfect electric conductors including ellipsoidal cylinders and flat strips in both polarizations using uniform as well as nonuniform sampling and yielded surprisingly good results

on the computation of inverses and determinants of a kind of special matrices abstract in this paper the inverse and determinant of a special kind of centrosymmetric matrices are investigated based on the partition property of a matrix with centrosymmetric structure and algorithms for the inverse and determinant proposed in chen and yu 2011 a computation algorithm for the inverse and determinant of a centrosymmetric matrix is finally developed

radial angular singularity cancellation transformations derived by variable separation the variable separation method is utilized to construct new radial angular r n coordinate transformation schemes for the singularity cancellation technique derived from the concept of achieving an ideal jacobian of the transformations and with different parameter settings in the variable separation method a new augmented radial angular r 1 transformation new radial angular r 2 and radial angular r 2 cosh transformations as well as a new radial angular r 3 an augmented radial angular r 3 and an arcsinh r 3 transformation are introduced the proposed r 2 and r 3 transformations are efficient and applicable to all types of near singular kernels occurring in the method of moments mom solution of electromagnetic surface integral equations whereas the r 1 transformation is only suitable for the first order singular integrals the newly proposed transformation schemes are verified and they are also applied to deformed triangles where good performances are still achieved

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

parallel computation of flow in heterogeneous media modelled by mixed finite elements abstract in this paper we describe a fast parallel method for solving highly ill conditioned saddle point systems arising from mixed finite element simulations of stochastic partial differential equations pdes modelling flow in heterogeneous media each realisation of these stochastic pdes requires the solution of the linear first order velocity pressure system comprising darcy s law coupled with an incompressibility constraint the chief difficulty is that the permeability may be highly variable especially when the statistical model has a large variance and a small correlation length for reasonable accuracy the discretisation has to be extremely fine we solve these problems by first reducing the saddle point formulation to a symmetric positive definite spd problem using a suitable basis for the space of divergence free velocities the reduced problem is solved using parallel conjugate gradients preconditioned with an algebraically determined additive schwarz domain decomposition preconditioner the result is a solver which exhibits a good degree of robustness with respect to the mesh size as well as to the variance and to physically relevant values of the correlation length of the underlying permeability field numerical experiments exhibit almost optimal levels of parallel efficiency the domain decomposition solver doug http www maths bath ac uk parsoft used here not only is applicable to this problem but can be used to solve general unstructured finite element systems on a wide range of parallel architectures

schur complement systems in the mixed hybrid finite element approximation of the potential fluid flow problem the mixed hybrid finite element discretization of darcy s law and continuity equation describing the potential fluid flow problem in porous media leads to a symmetric indefinite linear system for the pressure and velocity vector components as a method of solution the reduction to three schur complement systems based on successive block elimination is considered the first and second schur complement matrices are formed eliminating the velocity and pressure variables respectively and the third schur complement matrix is obtained by elimination of a part of lagrange multipliers that come from the hybridization of a mixed method the structural properties of these consecutive schur complement matrices in terms of the discretization parameters are studied in detail based on these results the computational complexity of a direct solution method is estimated and compared to the computational cost of the iterative conjugate gradient method applied to schur complement systems it is shown that due to special block structure the spectral properties of successive schur complement matrices do not deteriorate and the approach based on the block elimination and subsequent iterative solution is well justified theoretical results are illustrated by numerical experiments

nested grids ilu decomposition ngilu a preconditioning technique is described which shows in many cases grid independent convergence this technique only requires an ordering of the unknowns based on the different levels of multigrid and an incomplete lu decomposition based on a drop tolerance the method is demonstrated on a variety of well known elliptic test problems including strongly varying coefficients advective terms and grid refinement

additive schur complement approximation and application to multilevel preconditioning this paper introduces an algorithm for additive schur complement approximation asca which can be applied in various iterative methods for solving systems of linear algebraic equations arising from finite element fe discretization of partial differential equations pde it is shown how the asca can be used to set up a nonlinear algebraic multilevel iteration amli method this requires the construction of a linear multiplicative two level preconditioner at each level the latter is computed in the course of a simultaneous exact two by two block factorization of local stiffness matrices associated with a covering of the entire domain by overlapping subdomains unlike in schwarz type domain decomposition methods this method does not require a global coarse problem but instead uses local coarse problems to provide global communication a robust condition number bound is proved for a particular partitioning of the nodes of a uniform grid the presented numerical tests demonstrate that the asca whe

the convergence rate of a multigrid method with gauss seidel relaxation for the poisson equation the numerical solution of the poisson equation is treated by a multigrid method for a uniform grid the convergence rate can be estimated even for the iteration with a v cycle independently of the shape of the domain as long as it is convex and polygonal the smoothing effect of the gaus seidel relaxation is described by a discrete seminorm which is weaker than the energy norm

an algebraic multilevel multigraph algorithm we describe an algebraic multilevel multigraph algorithm many of the multilevel components are generalizations of algorithms originally applied to general sparse gaussian elimination indeed general sparse gaussian elimination with minimum degree ordering is a limiting case of our algorithm our goal is to develop a procedure which has the robustness and simplicity of use of sparse direct methods yet offers the opportunity to obtain the optimal or near optimal complexity typical of classical multigrid methods

on the nystrom solutions for electromagnetic scattering by thin conducting objects solving electromagnetic em problems with thin conducting objects by integral equation approach could encounter some unfavorable factors the individual electric field integral equation efie and magnetic field integral equation mfie may deteriorate the conditioning of matrix equations due to their degeneration when the thickness reduces also there are more evaluations of near singular integrals in filling the impedance matrix because the near interactions between observation points and source patches are common in addition many triangular meshes could have a high aspect ratio owing to the small thickness and this will present a difficulty for the accurate evaluation of self and near interaction elements aiming to these factors we use the combined field integral equation cfie as a governing equation and develop an efficient nystrom solver for the problems based on a singularity treatment technique without subdividing triangular patches typical numerical examples are presented to demonstrate its robustness

a high order locally corrected nystrom scheme for charge current integral equations a high order nystrom scheme is developed for electromagnetic em scattering problems of smooth objects the implementation is based on different well known charge current formulations this study aims to develop a novel quadrature scheme to compute different singular integrals that appear on integral equation methods in ems the proposed quadrature scheme is used on a locally corrected nystrom method and applied to charge current integral equations to obtain a fast accurate and a low frequency stable method to solve em scattering problems the proposed method allows obtaining accuracy in the range 4 10 digits in the near electric and magnetic scattered field for certain simple geometries and a wide range of frequencies we also do a preliminary research on nonsmooth geometries obtaining a certain degree of accuracy depending on the particular charge current integral equation used

numerical evaluation of singular and near singular potential integrals a simple and efficient numerical procedure using a singularity cancellation scheme is presented for evaluating singular and near singular potential integrals with 1 r singularities the procedure not only has several advantages over singularity subtraction methods but also improves on some aspects of other singularity cancellation methods such as polar and duffy transformations a theoretical analysis is presented for triangles quadrilaterals tetrahedrons bricks and prisms and numerical results are presented for triangles and prisms

stabilizing the hierarchical basis by approximate wavelets ii implementation and numerical results this paper is the second part of a work on stabilizing the classical hierarchical basis hb by using wavelet like basis functions implementation techniques are of major concern for the multilevel preconditioners proposed by the authors in the first part of the work which deals with algorithms and their mathematical theory numerical results are presented to confirm the theory established there a comparison of the performance of a number of multilevel methods is conducted for elliptic problems of three space variables

potential integrals for uniform and linear source distributions on polygonal and polyhedral domains formulas for the potentials due to uniform and linearly varying source distributions defined on simply shaped domains are systematically developed and presented domains considered are infinite planar strips infinite cylinders of polygonal cross sections planar surfaces with polygonal boundaries and volumetric regions with polyhedral boundaries the expressions obtained are compact in form and their application in the numerical solution of electromagnetics problems by the method of moments is illustrated

improved schur complement preconditioners for block toeplitz systems with small size blocks in this paper we employ the preconditioned conjugate gradient method with the improved schur complement preconditioners for hermitian positive definite block toeplitz systems with small size blocks schur complement preconditioners have been proved to be an effective method for such block toeplitz systems ching et al 2007 the modification is based on taylor expansion approximation we prove that the matrices preconditioned by improved schur preconditioners have more clustered spectra compared to that of the schur complement preconditioners hence preconditioned conjugate gradient type methods will converge faster numerical examples are given to demonstrate the efficiency of the proposed method

on two types of convergence in the method of auxiliary sources the method of auxiliary sources mas is often applied to problems involving an externally illuminated smooth perfect electric conductor pec one seeks to approximately satisfy the boundary condition on the pec surface using auxiliary sources located inside the pec surface usually the underlying auxiliary surface is also smooth and closed the currents on the auxiliary sources mas currents are the initial unknowns once they have been found one can easily determine the field due to them mas field at all points external to the auxiliary surface and in particular at all points external to the pec scatterer we show that in the limit n it is possible to have a convergent mas field together with divergent mas currents and that this phenomenon is accompanied by an abrupt behavior of the limiting value of the mas field we show this possibility through an analytical study of a two dimensional scattering problem involving a circular cylinder in which mas fields and currents can be determined explicitly the analytical study proceeds from first principles it involves fundamental electromagnetics and relatively simple mathematical manipulations numerical results supplement the analytical study and demonstrate the nature of the aforementioned divergence our study sheds light on other aspects of mas in particular it establishes interesting similarities and differences between mas and its continuous version and reveals many similarities between mas currents and numerical solutions of hallen s integral equation with the approximate kernel

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

some studies on nonpolynomial interpolation and error analysis abstract in this paper we propose nonpolynomial and hermite nonpolynomial interpolation with multiple parameters and present method to determine optimal value of parameters which generate minimum error in approximation the generalized error analysis results for nonpolynomial and hermite nonpolynomial interpolations are derived we establish theoretical relationship among nonpolynomial polynomial interpolation and the fourier series and propose solution to runge s phenomenon through nonpolynomial interpolation the hermite nonpolynomial cubic spline nonpolynomial cubic spline interpolation methods and their error analysis are presented numerical simulations are carried out for the analysis of error in cubic spline interpolations proposed method is applied to the analysis of various time series to show comparison in errors between polynomial and nonpolynomial spline interpolations and to empirical mode decomposition emd to illustrate practical usefulness of the results

schur complement multigrid a schur complement multigrid method for the solution of convection diffusion problems with strongly discontinuous coefficients is the focus of this paper this algorithm turns out to be robust and efficient for our test problems its convergence rate is in all cases superior to the standard multigrid method the schur complement multigrid method is applied to the problem of the decontamination of a heterogeneous porous medium by flushing we determine the typical decontamination time for several configurations and its functional dependence on the statistical properties of the hydraulic conductivity distribution of the medium

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

singular higher order complete vector bases for finite methods this paper presents new singular curl and divergence conforming vector bases that incorporate the edge conditions singular bases complete to arbitrarily high order are described in a unified and consistent manner for curved triangular and quadrilateral elements the higher order basis functions are obtained as the product of lowest order functions and silvester lagrange interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties are discussed and these bases are proved to be fully compatible with the standard high order regular vector bases used in adjacent elements the curl divergence conforming singular bases guarantee tangential normal continuity along the edges of the elements allowing for the discontinuity of normal tangential components adequate modeling of the curl divergence and removal of spurious modes solutions these singular high order bases should provide more accurate and efficient numerical solutions of both surface integral and differential problems sample numerical results confirm the faster convergence of these bases on wedge problems

hybrid v cycle algebraic multilevel preconditioners we consider an algebraic derivation of multilevel preconditioners which are based on a sequence of finite element stiffness matrices they cor respond to a sequence of triangulations obtained by successive refinement and the associated finite element discretizations of second order selfadjoint elliptic boundary value problems the stiffness matrix at a given discretization level is partitioned into a natural hierarchical two level two by two block form then it is factored into block triangular factors the resulting schur complement is then replaced approximated by the stiffness matrix on the preceding coarser level this process is repeated successively for a fixed number ko 1 of steps after each cq steps the preconditioner so derived is corrected by a certain polynomial approximation a properly scaled and shifted chebyshev matrix polynomial which involves the preconditioner and the stiffness matrix at the considered level the hybrid f cycle preconditioner thus derived is shown to be of optimal order of complexity for 2 d and 3 d problem domains the rela tive condition number of the preconditioner is bounded uniformly with respect to the number of levels and with respect to possible jumps of the coefficients of the considered elliptic bilinear form as long as they occur only across edges faces in 3 d of elements from the coarsest triangulation in addition an adap tive implementation of our hybrid k cycle preconditioners is proposed and its practical behavior is demonstrated on a number of test problems

an improved empirical mode decomposition method based on the cubic trigonometric b spline interpolation algorithm abstract empirical mode decomposition emd is a new method presented recently for analyzing nonlinear and non stationary signals its basic idea is to decompose the signal into a series of complete orthogonal intrinsic mode functions imfs based on the local characteristics of the signal in time domain the key step of emd is to use the cubic spline interpolation to connect the maximum and minimum values of the signals into upper and lower envelopes respectively and then calculate the mean values of upper and lower envelopes based on the cubic trigonometric b spline interpolation algorithm a new improved method for emd is proposed named ctb emd in this paper in this method the interpolation curve is more flexible because of the adjustability of shape of the cubic trigonometric b splines curve thus the overshoot and undershoot problems in the cubic spline interpolation curve can be avoided and then the decomposition of the signal is more accurate and effect through numerical experiments we compare the effect of this method with other methods on decomposing simulation signals and real signals experimental results show that this method can decompose signals more effectively and accurately

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

using approximate inverses in algebraic multilevel methods this paper deals with the iterative solution of large sparse symmetric positive definite systems we investigate preconditioning techniques of the two level type that are based on a block factorization of the system matrix whereas the basic scheme assumes an exact inversion of the submatrix related to the first block of unknowns we analyze the effect of using an approximate inverse instead we derive condition number estimates that are valid for any type of approximation of the schur complement and that do not assume the use of the hierarchical basis they show that the two level methods are stable when using approximate inverses based on modified ilu techniques or explicit inverses that meet some row sum criterion on the other hand we bring to the light that the use of standard approximate inverses based on convergent splittings can have a dramatic effect on the convergence rate these conclusions are numerically illustrated on some examples

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