black box multigrid for nonsymmetric problems 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 previous work this defect was overcome and a code written in which one needed only to specify the logically rectangular positive definite matrix problem the code did everything else necessary to set up the auxiliary coarser problems to achieve a multigrid solution in this paper the previous work is extended to a single nonsymmetric equation

smoothing analysis of multicolour pattern schemes in this note we present an analysis of the smoothing properties of three and four colour patterns for two dimensional line relaxation schemes

distributed average tracking of networked euler lagrange systems under an extended proportional integral pi control scheme a new class of distributed average tracking dat algorithms has been developed for three different kinds of references references with steady states references with bounded derivatives and references with a common derivative the class of dat algorithms are further applied to solve the dat problem for euler lagrange el systems

on the robust solution of an isogeometric discretization of bilaplacian equation by using multigrid methods abstract in this paper we propose a multigrid method for the solution of the biharmonic problem isogeometric analysis iga is considered in order to easily obtain h 2 conforming discretizations of the bilaplacian equation which are difficult to get by means of standard finite element methods typically the design of solvers for isogeometric discretizations that are robust with respect to the polynomial degree is a challenging task here we achieve such robustness by using multiplicative schwarz methods as smoothers within the multigrid algorithm the design of the proposed solver is also supported by a local fourier analysis lfa which allows us to choose appropriately the size of the block in the smoother depending on the polynomial degree of the discretization the robustness and efficiency of the proposed multigrid method is demonstrated through numerical experiments in one and two dimensional cases

approximation properties on invariant measure and oseledec splitting in non uniformly hyperbolic systems we prove that each invariant measure in a non uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits this contributes to our main result that the mean angle definition 1 10 independence number definition 1 6 and oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic periodic orbits respectively combining this result with the approximation property of lyapunov exponents by wang and sun 2005 theorem 1 9 we strengthen katok s closing lemma 1980 by presenting more extensive information not only about the state system but also its linearization in the present paper we also study an ergodic theorem and a variational principle for mean angle independence number and liao s style number definition 1 3 which are bases for discussing the approximation properties in the main result

multi staging of jacobi relaxation to improve smoothing properties of multigrid methods for steady euler equations abstract multi stage versions of jacobi relaxation are studied for use in multigrid methods for steady euler equations it is shown that these multi stage versions allow much more general and much more efficient multigrid methods than possible with classic relaxation methods

local fourier analysis of space time relaxation and multigrid schemes we consider numerical methods for generalized diffusion equations that are motivated by the transport problems arising in electron beam radiation therapy planning while monte carlo methods are typically used for simulations of the forward peaked scattering behavior of electron beams rough calculations suggest that grid based discretizations can provide more efficient simulations if the discretizations can be made sufficiently accurate and optimal solvers can be found for the resulting linear systems the multigrid method for model two dimensional transport problems presented in c borgers and s maclachlan j comput phys 229 2010 pp 2914 2931 shows the necessary optimal scaling with some dependence on the choice of scattering kernel in order to understand this behavior local fourier analysis can be applied to the two grid cycle using this approach expressions for the error propagation operators of the coarse grid correction and relaxation steps projected onto the fine grid harmonic spac

topological features of flows with the reparametrized gluing orbit property abstract the notions of shadowing specification and gluing orbit property differ substantially for discrete and continuous time dynamical systems in the present paper we continue the study of the topological and ergodic properties of continuous flows with the reparametrized periodic and nonperiodic gluing orbit properties initiated in 3 we prove these flows satisfy a weak mixing condition with respect to balls and if the flow is komuro expansive the topological entropy is a lower bound for the exponential growth rate of periodic orbits moreover we show that periodic measures are dense in the set of all invariant probability measures and that ergodic measures are generic furthermore we prove that irrational rotations and some minimal flows on tori and circle extensions over expanding maps satisfy gluing orbit properties thus emphasizing the difference of this property with respect the notion of specification

robust multigrid algorithms for the navier strokes equations abstract anisotropies occur naturally in computational fluid dynamics where the simulation of small scale physical phenomena such as boundary layers at high reynolds numbers causes the grid to be highly stretched leading to a slowdown in convergence of multigrid methods several approaches aimed at making multigrid a robust solver have been proposed and analyzed in the literature using the scalar diffusion equation however they have rarely been applied to solving more complicated models such as the incompressible navier stokes equations this paper contains the first published numerical results of the behavior of two popular robust multigrid approaches alternating plane smoothers combined with standard coarsening and plane implicit smoothers combined with semi coarsening for solving the 3 d incompressible navier stokes equations in the simulation of the driven cavity and a boundary layer over a flat plate on a stretched grid grid size grid stretching and reynolds number are the factors considered in evaluating the robustness of the multigrid methods both approaches yield large increases in convergence rates over cell implicit smoothers on stretched grids the combination of plane implicit smoothers and semi coarsening was found to be fully robust in the flat plate simulation up to reynolds numbers 10 6 and the best alternative in the driven cavity simulation for reynolds numbers above 10 3 the alternating plane approach exhibits a better behavior for lower reynolds numbers below 10 3 in the driven cavity simulation

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

accelerated multigrid convergence and high reynolds recirculating flows techniques are developed for accelerating multigrid convergence in general and for advection diffusion and incompressible flow problems with small viscosity in particular it is shown by analysis that the slowing down of convergence is due mainly to poor coarse grid correction to certain error components and means for dealing with this problem are suggested analyzed and tested by numerical experiments showing very significant improvement in convergence rates at little cost

efficient multigrid methods for solving upwind biased discretizations of the convection equation this paper studies a novel multigrid approach to the solution for a second order upwind biased discretization of the convection equation in two dimensions this approach is based on semi coarsening and well balanced explicit correction terms added to coarse grid operators to maintain on coarse grids the same cross characteristic interaction as on the target fine grid multicolor relaxation schemes are used on all the levels allowing a very efficient parallel implementation the results of the smoothing analysis and numerical tests can be summarized as follows 1 the residual asymptotic convergence rate is about 3 per cycle which far surpasses the theoretical limit 4 3 predicted for standard multigrid algorithms with full coarsening 2 the full multigrid fmg algorithm with two v 0 2 cycles on the target grid and just one v 0 2 cycle on all the coarse grids always provides an approximate solution with the algebraic error less than the discretization error 3 a novel algorithm for deriving a discrete solution approximating the true continuous solution with a relative accuracy given in advance is developed the computational complexity of this algorithm is nearly optimal comparable with the complexity of the fmg algorithm applied to solve the problem on the optimally spaced target grid

smoothing factor order of prolongation and actual multigrid convergence we consider the fourier analysis of multigrid methods of galerkin type for symmetric positive definite and semi positive definite linear systems arising from the discretization of scalar partial differential equations pdes we relate the so called smoothing factor to the actual two grid convergence rate and also to the convergence rate of the v cycle multigrid we derive a two sided bound that defines an interval containing both the two grid and v cycle convergence rate this interval is narrow and away from 1 when both the smoothing factor and an additional parameter are small enough besides the smoothing factor the convergence mainly depends on the angle between the range of the prolongation and the eigenvectors of the system matrix associated with small eigenvalues nice v cycle convergence is guaranteed if the tangent of this angle has an upper bound proportional to the eigenvalue whereas nice two grid convergence requires a bound proportional to the square root of the eigenvalue we also discuss the well known rule which relates the order of the prolongation to that of the differential operator associated to the problem we first define a frequency based order which in most cases amounts to the so called high frequency order as defined in hemker j comput appl math 32 423 429 1990 we give a firmer basis to the related order rule by showing that together with the requirement of having the smoothing factor away from one it provides necessary and sufficient conditions for having the two grid convergence rate away from 1 a stronger condition is further shown to be sufficient for optimal convergence with the v cycle the presented results apply to rigorous fourier analysis for regular discrete pdes and also to local fourier analysis via the discussion of semi positive systems as may arise from the discretization of pdes with periodic boundary conditions

efficient geometric multigrid implementation for triangular grids this paper deals with a stencil based implementation of a geometric multigrid method on semi structured triangular grids triangulations obtained by regular refinement of an irregular coarse triangulation for linear finite element methods an efficient and elegant procedure to construct these stencils using a reference stencil associated to a canonical hexagon is proposed local fourier analysis lfa is applied to obtain asymptotic convergence estimates numerical experiments are presented to illustrate the efficiency of this geometric multigrid algorithm which is based on a three color smoother

a simple and efficient segregated smoother for the discrete stokes equations we consider the multigrid solution of the generalized stokes equations with a segregated i e equationwise gauss seidel smoother based on a uzawa type iteration we analyze the smoother in the framework of local fourier analysis and obtain an analytic bound on the smoothing factor showing uniform performance for a family of stokes problems these results are confirmed by the numerical computation of the two grid convergence factor for different types of grids and discretizations numerical results also show that the actual convergence of the w cycle is approximately the same as that obtained by a vanka smoother despite this latter smoother being significantly more costly per iteration step

multigrid line smoothers for higher order upwind discretizations of convection dominated problems in this paper we present new multigrid line smoothers for the solution of higher order discretizations of convection dominated problems directly the behavior of the smoothers is analyzed theoretically with fourier smoothing and two grid analysis a parallel tri line variant is presented and evaluated the smoothers are applied to scalar convection diffusion problems discretized with limiters and systems of incompressible navier stokes and euler equations

on three grid fourier analysis for multigrid in this paper we present three grid fourier analysis for multigrid methods due to the recursive structure of a multigrid iteration this analysis can be deduced from the well known two grid fourier analysis the coarse grid correction part of multigrid algorithms can be more accurately evaluated with the three grid analysis we apply the analysis to several scalar equations and discretizations with an emphasis on problems with a multigrid coarse grid correction difficulty like upwind discretizations of the convection diffusion equation the main focus lies on possible improvements by carefully chosen galerkin operators and or by an additional acceleration with restarted gmres gmres m numerical test calculations validate the theoretical predictions

krylov subspace acceleration of nonlinear multigrid with application to recirculating flows this paper deals with the combination of two solution methods multigrid and gmres siam j sci comput 14 1993 pp 856 869 the generality and parallelizabili nonlinear pdes as the preconditioner for a nonlinear krylov subspace method we use the full approximation storage fas scheme math comp 31 1977 pp 333 390 a nonlinear multigrid method the nonlinear krylov acceleration is applied also on coarse grids so that recirculating incompressible flow problems discretized with a higher order upwind scheme can be solved efficiently

towards adaptive smoothed aggregation alpha sa for nonsymmetric problems applying smoothed aggregation sa multigrid to solve a nonsymmetric linear system a mathbf x mathbf b is often impeded by the lack of a minimization principle that can be used as a basis for the coarse grid correction process this paper proposes a petrov galerkin pg approach based on applying sa to either of two symmetric positive definite spd matrices sqrt a t a or sqrt aa t these matrices however are typically full and difficult to compute so it is not computationally efficient to use them directly to form a coarse grid correction the proposed approach approximates these coarse grid corrections by using sa to accurately approximate the right and left singular vectors of a that correspond to the lowest singular value these left and right singular vectors are used to construct the restriction and interpolation operators respectively a preliminary two level convergence theory is presented suggesting that more relaxation should be applied than for an spd problem additionally a nonsymmetric version of adaptive sa alpha sa is given that automatically constructs sa multigrid hierarchies using a stationary relaxation process on all levels numerical results are reported for convection diffusion problems in two dimensions with varying amounts of convection for constant variable and recirculating convection fields the results suggest that the proposed approach is algorithmically scalable for problems coming from these nonsymmetric scalar pdes with the exception of recirculating flow this paper serves as a first step for nonsymmetric alpha sa the long term goal of this effort is to develop nonsymmetric alpha sa for systems of pdes where the sa framework has proven to be well suited for adaptivity in spd problems

observers are unnecessary for output feedback control of lagrangian systems we address and solve some long standing yet well documented open problems on output feedback tracking control of euler lagrange systems with arbitrarily high relative degree this includes underactuated systems our main contribution is to establish a theoretical foundation for the use of so called dirty derivatives a common ad hoc replacement of unavailable state measurements such as generalized velocities whence obviating the the use of observers for the purpose of position feedback tracking control reminiscent of passivity based control for robot manipulators our control law is globally lipschitz and the controller dynamics is linear for relative degree two fully actuated lagrangian systems without dissipative forces friction and with bounded inertia matrix we establish uniform global asymptotic stability in closed loop furthermore we show that our control approach applies to lagrangian systems augmented by a chain of integrators relative degree m 2 systems the design method which is based on a recursive procedure in the spirit of backstepping control is intuitive as it exploits structural properties such as passivity and inherent input output stability as a corollary we solve an output feedback global tracking control problem for flexible joint robots but also for systems coupled with output feedback linearizable actuator dynamics in addition we discuss remaining open problems of fairly general interest in the realm of analysis and design of robust nonlinear systems

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

exponential rate of periodic points and metric entropy in nonuniformly hyperbolic systems abstract for an ergodic hyperbolic measure of a c 1 r diffeomorphism f there is an full measured set such that for every invariant measure m inv f the exponential growth rate of the number of periodic points whose atomic measures approximate equals to the metric entropy h f

distributed adaptive coordination for multiple lagrangian systems under a directed graph without using neighbors velocity information in this paper we study the distributed coordination problem for multiple lagrangian systems in the presence of parametric uncertainties under a directed graph without using neighbors velocity information in the absence of communication we consider two cases namely the distributed containment control problem with multiple stationary leaders and the leaderless synchronization problem in both cases distributed adaptive control algorithms without using neighbors velocity information are proposed the control gains in the algorithms are varying with distributed updating laws furthermore necessary and sufficient conditions on the directed graph are presented respectively such that all followers converge to the stationary convex hull spanned by the stationary leaders asymptotically in the containment control problem and the systems synchronize asymptotically in the leaderless synchronization problem finally simulation examples are provided to show the effectiveness of the proposed control algorithms

hp multigrid as smoother algorithm for higher order discontinuous galerkin discretizations of advection dominated flows part i multilevel analysis the hp multigrid as smoother algorithm hp mgs for the solution of higher order accurate space time discontinuous galerkin discretizations of advection dominated flows is presented this algorithm combines p multigrid with h multigrid at all p levels where the h multigrid acts as smoother in the p multigrid the performance of the hp mgs algorithm is further improved using semi coarsening in combination with a new semi implicit runge kutta method as smoother a detailed multilevel analysis of the hp mgs algorithm is presented to obtain more insight into the theoretical performance of the algorithm as model problem a fourth order accurate space time discontinuous galerkin discretization of the advection diffusion equation is considered the multilevel analysis shows that the hp mgs algorithm has excellent convergence rates both for steady state and time dependent problems and low and high cell reynolds numbers including highly stretched meshes

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

ergodic properties of systems with asymptotic average shadowing property abstract in this paper we explore a topological system f m m with asymptotic average shadowing property and extend sigmund s results from bowen s specification case we show that every non empty compact and connected subset v m inv f coincides with some v f y moreover we show that the set m v y m v f y v is dense in v v supp in particular if max m inv f supp coincides with m then m max y v f y m inv f is residual in m as consequences we have several corollaries one is that every invariant measure has generic points another is that the set consisting of those points for which the birkhoff ergodic average does not exist called irregular set is either dense in max residual provided that max m or empty in particular we give an uncountable division of irregular set and obtain a refined characterization which can be as a substantial generalization of 18 with a different method remark that for systems with asymptotic average shadowing property this article studies birkhoff ergodic average from topological viewpoint but it is still unknown how about the theory from the perspective of dimension theory

brief paper a globally exponentially convergent immersion and invariance speed observer for mechanical systems with non holonomic constraints the problem of velocity estimation for general n degrees of freedom mechanical systems is of great practical and theoretical interest for unconstrained systems many partial solutions have been reported in the literature however even in this case the basic question of whether it is possible to design a globally convergent speed observer remains open in this paper an affirmative answer to the question is given for general mechanical systems with knon holonomic constraints by proving the existence of a 3n 2k 1 dimensional globally exponentially convergent speed observer an observer for unconstrained mechanical systems is obtained as a particular case of this general result instrumental for the construction of the speed observer is the use of the immersion and invariance technique in which the observer design problem is recast as a problem of rendering attractive and invariant a manifold defined in the extended state space of the plant and the observer

a new petrov galerkin smoothed aggregation preconditioner for nonsymmetric linear systems we propose a new variant of smoothed aggregation sa suitable for nonsymmetric linear systems the new algorithm is based on two key generalizations of sa restriction smoothing and local damping restriction smoothing refers to the smoothing of a tentative restriction operator via a damped jacobi like iteration restriction smoothing is analogous to prolongator smoothing in standard sa and in fact has the same form as the transpose of prolongator smoothing when the matrix is symmetric local damping refers to damping parameters used in the jacobi like iteration in standard sa a single damping parameter is computed via an eigenvalue computation here local damping parameters are computed by considering the minimization of an energy like quantity for each individual grid transfer basis function numerical results are given showing how this method performs on highly nonsymmetric systems

dispersion induced by the pollution for the wave equation in this paper we give a computational method to measure the dispersion which is directly related to the pollution effect this pollution effect is observed by various researchers when the helmholtz equation is solved for high wave numbers the main effect of the pollution is that the wave number of the solution obtained via a finite element method is different from the wave number of the exact solution and this is what is called dispersion using fourier analysis we present a numerical method to measure the dispersion corresponding to a finite element method

synchronization of networks of nonidentical euler lagrange systems with uncertain parameters and communication delays this paper addresses the problem of synchronizing networks of nonidentical nonlinear dynamical systems described by euler lagrange equations which are assumed fully actuated with their states available for measurement but with unknown parameters the only assumption made on the communication graph is that it is connected moreover the communication is subject to constant time delays which are also unknown the main result of the paper is the construction of an adaptive controller that achieves global full state synchronization i e the difference between the agents positions and velocities asymptotically converges to zero if a desired trajectory for all systems is given a slight modification to the proposed scheme achieves also full state synchronization simulations using a ten robot manipulator network are used to illustrate the performance of the proposed schemes

half space analysis of the defect correction method for fromm discretization of convection a novel comprehensive discrete half space analysis for the defect correction method has been developed this analysis plays the same role for nonelliptic problem solvers as the full space fourier mode analysis plays for elliptic problem solvers numerical simulations confirm the accuracy of the half space analysis the following important findings about the defect correction method applied to the fromm discretization of the two dimensional convection equation are reported the initial convergence rate of the defect correction method is principally a function of the relative accuracy of the operators involved in the defect correction iterations the asymptotic convergence rate is about 0 5 per defect correction iteration if the driver operator is first order accurate then the initial convergence rates may be slow the number of iterations required to get into the asymptotic convergence regime or and to converge the algebraic error below the discretization error level can be proportional to h 1 3 this h dependent delay is a multidimensional phenomenon it cannot be observed in one dimensional problems and it disappears in the case of close alignment between the grid and the convection equation characteristic if the driver operator is second order accurate the defect correction solver demonstrates the asymptotic convergence rate from the very beginning only one defect correction iteration is required to converge algebraic error substantially below the discretization error level

acceleration of multigrid flow computations through dynamic adaptation of the smoothing procedure abstract the paper presents the development and investigation of an adaptive smoothing as procedure in conjunction with the full multigrid full approximation storage method the latter has been previously implemented by the authors 1 for the incompressible navier stokes equations in conjunction with the artificial compressibility method and forms the basis for investigating the current as approach the principle of adaptive smoothing is to exploit the nonuniform convergence behavior of the numerical solution during the iterations to reduce the size of the computational domain and subsequently to reduce the total computing time the implementation of the as approach is investigated in conjunction with three different adaptivity criteria for two and three dimensional incompressible flows furthermore a dynamic procedure henceforth labeled dynamic adaptivity for defining variably the as parameters during the computation is also proposed and its performance is investigated in contrast to as with constant parameters henceforth labeled static adaptivity both static and dynamic adaptivity can provide similar acceleration but the latter additionally provides more stable numerical solutions and also requires less experimentation for optimizing the performance of the method numerical experiments are presented for attached and separated flows around airfoils as well as for three dimensional flow in a curved channel for external flows the as performs better when it is applied in all grid levels of the multigrid method while for internal flows the best performance is achieved when as is applied in the finest grid only overall the results show that substantial acceleration of multigrid computations can be achieved by using adaptive smoothing

multigrid waveform relaxation on spatial finite element meshes the discrete time case the efficiency of numerically solving time dependent partial differential equations on parallel computers can be greatly improved by computing the solution on many time levels simultaneously the theoretical properties of one such method namely the discrete time multigrid waveform relaxation method are investigated for systems of ordinary differential equations obtained by spatial finite element discretisation of linear parabolic initial boundary value problems the results are compared to the corresponding continuous time results the theory is illustrated for a one dimensional and a two dimensional model problem and checked against results obtained by numerical experiments

local fourier analysis for cell centered multigrid methods on triangular grids this work is focused on the design of efficient multigrid methods for discretizations on voronoi meshes associated with regular refinements of acute triangles for this purpose a special local fourier analysis for this type of discretizations is developed this analysis helps us to obtain appropriate novel smoothers depending on the geometry of the grid a two and three grid analysis are performed to analyze the behavior of the multigrid methods numerical test calculations validate the theoretical predictions furthermore the application of this analysis for the solution of more real problems by using semi structured grids is shown

direct multi grid methods for linear systems with harmonic aliasing patterns multi level numerical methods that obtain the exact solution of a linear system are presented the methods are devised by combining ideas from the full multi grid algorithm and perfect reconstruction filters the problem is stated as whether a direct solver is possible in a full multi grid scheme by avoiding smoothing iterations and using different coarse grids at each step the coarse grids must form a partition of the fine grid and thus establishes a strong connection with domain decomposition methods an important analogy is established between the conditions for direct solution in multi grid solvers and perfect reconstruction in filter banks furthermore simple solutions of these conditions for direct multi grid solvers are found by using mirror filters as a result different configurations of direct multi grid solvers are obtained and studied

lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits lyapunov exponents of a hyperbolic ergodic measure are approximated by lyapunov exponents of hyperbolic atomic measures on periodic orbits

on multigrid solution of high reynolds incompressible entering flows an approach is presented for effectively separating the solution process of the elliptic component of high reynolds incompressible steady entering flow for which classical multigrid techniques are well suited from that of the non elliptic part for which other methods are more effective it is shown by analysis and numerical calculations that such an approach is very effective in terms of asymptotic convergence as well as reduction of errors well below discretization level in a 1 fmg algorithm

high order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems high mixed order finite difference discretization of optimality systems arising from elliptic nonlinear constrained optimal control problems are discussed for the solution of these systems an efficient and robust multigrid algorithm is presented theoretical and experimental results show the advantages of higher order discretization and demonstrate that the proposed multigrid scheme is able to solve efficiently constrained optimal control problems also in the limit case of bang bang control

fourier mode analysis of multigrid methods for partial differential equations with random coefficients partial differential equations with random coefficients appear for example in reliability problems and uncertainty propagation models various approaches exist for computing the stochastic characteristics of the solution of such a differential equation in this paper we consider the spectral expansion approach this method transforms the continuous model into a large discrete algebraic system we study the convergence properties of iterative methods for solving this discretized system we consider one level and multi level methods the classical fourier mode analysis technique is extended towards the stochastic case this is done by taking the eigenstructure into account of a certain matrix that depends on the random structure of the problem we show how the convergence properties depend on the particulars of the algorithm on the discretization parameters and on the stochastic characteristics of the model numerical results are added to illustrate some of our theoretical findings

flexible multiple semicoarsening for three dimensional singularly perturbed problems we present robust parallel multigrid based solvers for 3d scalar partial differential equations the robustness is obtained by combining multiple semicoarsening strategies matrix dependent transfer operators and a krylov subspace acceleration the basis for the 3d preconditioner is a 2d method with multiple semicoarsened grids based on the mg s method from c w oosterlee appl numer math 19 1995 pp 115 128 and t washio and c w oosterlee gmd arbeitspapier 949 gmd st augustin germany 1995 the 2d method is generalized to three dimensions with a line smoother in the third dimension the method based on semicoarsening has been parallelized with the grid partitioning technique j linden b steckel and k stuben parallel comput 7 1988 pp 461 475 o a mcbryan et al impact comput sci engrg 3 1991 pp 1 75 and is evaluated as a solver and as a preconditioner on a mimd machine the robustness of the 3d method is shown for finite volume and finite difference discretizations of 3d anisotropic diffusion equations and convection dominated convection diffusion problems

algebraic multigrid for stokes equations a method is investigated for solving stationary or time dependent discrete stokes equations it uses one of the standard flavors of algebraic multigrid for coupled partial differential equations which however is not applied directly to the linear system stemming from discretization but to an equivalent system obtained with a simple algebraic transformation which may be seen as a form of preconditioning in the literal sense a two grid analysis is provided showing that the eigenvalues of the preconditioned matrix are within a region of the complex plane that is both bounded and away from the origin independently of the mesh or grid size as well as of other main problem parameters on the other hand whereas the approach can in principle be combined with any type of algebraic multigrid scheme an investigation of the properties of the coarse grid matrices reveals that plain aggregation has to be preferred to maintain nice two grid convergence at coarser levels eventually numerical experiments are

analysis of a multigrid strokes solver model problem analysis or local mode analysis is used to investigate the two level convergence rates for multigrid strokes solvers the solvers are based on the discretization on the staggered grid and the distributive relaxation of the continuity equation the influence of the grid geometry is analyzed by also considering the poisson equation on the semistaggered grid various grid transfer operators are compared for an isotropic h x h y discretization simple tranfer operators are sufficient to obtain optimal rates but not in the anisotropic case the theoretical results are compared with observed multigrid convergence rates

algebraic multigrid solvers for complex valued matrices in the mathematical modeling of real life applications systems of equations with complex coefficients often arise while many techniques of numerical linear algebra e g krylov subspace methods extend directly to the case of complex valued matrices some of the most effective preconditioning techniques and linear solvers are limited to the real valued case here we consider the extension of the popular algebraic multigrid method to such complex valued systems the choices for this generalization are motivated by classical multigrid considerations evaluated with the tools of local fourier analysis and verified on a selection of problems related to real life applications

high accuracy multigrid solution of the 3d convection diffusion equation we present an explicit fourth order compact finite difference scheme for approximating the three dimensional 3d convection diffusion equation with variable coefficients this 19 point formula is defined on a uniform cubic grid fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1 we design a parallelization oriented multigrid method for fast solution of the resulting linear system using a four color gauss seidel relaxation technique for robustness and efficiency and a scaled residual injection operator to reduce the cost of multigrid inter grid transfer operator numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method several test problems are solved and highly accurate solutions of the 3d convection diffusion equations are obtained for small to medium values of the grid reynolds number effects of using different residual projection operators are compared on both vector and serial computers

on the order of prolongations and restrictions in multigrid procedures abstract it is well known in the world of multigrid that the order of the prolongation and the order of the restriction in a multigrid method should satisfy certain conditions a rule of thumb is that the sum of the orders of the prolongation and of the restriction should at least be equal to the order of the differential equation solved in this note we show the correctness of this rule we notice that we have to distinguish between low frequency and high frequency orders for the transfer operators for the restriction the low frequency order is related with its accuracy whereas for the interpolation operator both orders are related with the accuracy of the result of the interpolation if an interpolation rule leaves all polynomials of degree k 1 invariant then both the low and the high frequency order are equal to k it is the high frequency order that plays a role in the above mentioned rule of thumb

continuity properties of entropy on donne une borne superieure sur le defaut en semicontinuite superieure des entropies topologiques et metriques d une auto application c r d une variete c r r 1

an evaluation of parallel multigrid as a solver and a preconditioner for singularly perturbed problems in this paper we try to achieve h independent convergence with preconditioned gmres y saad and m h schultz siam j sci comput 7 1986 pp 856 869 and bicgstab h a van der vorst siam j sci comput 13 1992 pp 63 644 for two dimensional 2d singularly perturbed equations three recently developed multigrid methods are adopted as a preconditioner they are also used as solution methods in order to compare the performance of the methods as solvers and as preconditioners two of the multigrid methods differ only in the transfer operators one uses standard matrix dependent prolongation operators from j e dendy jr j comput phys 48 1982 pp 366 386 w hackbusch multi grid methods and applications springer berlin 1985 the second uses upwind prolongation operators developed in p m de zeeuw j comput appl math 33 1990 pp 1 27 both employ the galerkin coarse grid approximation and an alternating zebra line gauss seidel smoother the third method is based on the block lu decomposition of a matrix and on an approximate schur complement this multigrid variant is presented in a reusken a multigrid method based on incomplete gaussian elimination university of eindhoven the netherlands 1995 all three multigrid algorithms are algebraic methods the eigenvalue spectra of the three multigrid iteration matrices are analyzed for the equations solved in order to understand the convergence of the three algorithms furthermore the construction of the search directions for the multigrid preconditioned gmres solvers is investigated by the calculation and solution of the minimal residual polynomials for poisson and convection diffusion problems all solution methods are investigated and evaluated for finite volume discretizations on fine grids the methods have been parallelized with a grid partitioning technique and are compared on an mimd machine

technical communique global output feedback tracking control for a class of lagrangian systems the problem of global tracking control without velocity measurement of the so called euler lagrange systems has been paid a lot of attention for the past several years in this note a nice property of one degree of freedom euler lagrange systems is highlighted which in particular allows us to obtain a new solution to the problem of tracking control for this class of systems the solution is under the form of a linear like observer based controller which gives fairly good results as illustrated in the simulation the method can be generalized to any system having the same property

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

new estimates for multilevel algorithms including the v cycle the purpose of this paper is to provide new estimates for certain multilevel algorithms in particular we are concerned with the simple additive multilevel algorithm discussed recently together with j xu and the standard v cycle algorithm with one smoothing step per grid we shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity for example the theory applies to the standard multigrid vcycle on the l shaped domain or a domain with a crack and yields a uniform convergence rate we also prove uniform convergence rates for the multigrid v cycle for problems with nonuniformly refined meshes finally we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid v cycle algorithms

on generalizing the algebraic multigrid framework we present a theory for algebraic multigrid amg methods that allows for general smoothing processes and general coarsening approaches the goal of the theory is to provide guidance in the development of new more robust amg algorithms in particular we introduce several compatible relaxation methods and give theoretical justification for their use as tools for measuring the quality of coarse grids

multigrid solution to steady state two dimensional conservation laws a new genuinely two dimensional compact approach to discretizing scalar steady state conservation laws is presented it provides the possibility to separate treatment of the streamwise and cross stream directions due to this separation the artificial viscosity can be added in the streamwise direction only a high resolution mechanism is introduced in the cross stream direction the resulting schemes are shown to be second order accurate and monotonic providing a good resolution of discontinuities representing them in the numerical solution by this oscillation free transition layers due to their good stability properties the resulting difference equations can be solved efficiently by a multigrid algorithm employing a simple pointwise relaxation numerical experiments are presented

automated transformations for pde systems with application to multigrid solvers an approach for transforming systems of partial differential equations in order to obtain new formulations which are more accessible to numerical solution is studied an algorithm is developed for generating such transformations automatically using symbolic computations employing grobner bases the algorithm is implemented using freely available software this approach along with planned developments will potentially provide a powerful set of tools for handling large systems of partial differential equations

convergence analysis of iterative methods by pseudodifference operators abstract the convergence of iterative methods to solve linear partial differential equations numerically is analyzed by the theory of pseudodifference operators approximate inverses are determined from the symbol of the iterative operator so that implicit and preconditioned methods are covered the behavior of the symbol for higher wave numbers describes the convergence rate for the corresponding error modes the results are applied to krylov subspace methods stationary iterations of gauss seidel type the multigrid algorithm and time stepping methods for flow problems

on quantitative analysis methods for multigrid solutions limitations of the classical local mode fourier lmf analysis in application to multigrid solutions of variable coefficient problems are well known this paper introduces alternative very general quantitative analysis methods for multigrid solutions of partial differential equations the methods are applied to available nonperfect multigrid solvers that deal with practical problems the analysis methods considered in this paper focus on the main complementary parts of a multigrid cycle relaxation and coarse grid correction tasks for these parts e g error smoothing for relaxation and reduction of smooth errors for coarse grid correction are assumed to be assigned in advance idealized relaxation ir and idealized coarse grid icg iterations are introduced in these iterations one part of the cycle relaxation for ir iterations and coarse grid correction for icg iterations is replaced with an idealized imitation known to be efficient for the assigned task its complementary part is the actual part of a two grid cycle the analysis compares performances of the actual cycle and idealized iterations the ir and icg iterations are very general and can be directly applied in the most complicated simulations including highly variable or nonlinear coefficients complex geometries and unstructured grids the results of this analysis are not single number estimates they are rather convergence patterns of the iterations that may either confirm or refute expectations indicating what part of the actual solver is inefficient in carrying out the assigned task the generality of the analysis makes it a valuable tool for analyzing complicated large scale computational problems where no other analysis methods are currently available in this paper the ir and icg iterations are applied to model problems of progressive complexity the analysis proved sensitive to very delicate details of the actual multigrid cycle pointing clearly to each instance when a part of the algorithm was inefficient

a multigrid method for constrained optimal control problems we consider the fast and efficient numerical solution of linear quadratic optimal control problems with additional constraints on the control discretization of the first order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints the complementarity conditions are treated by a primal dual active set strategy that serves as outer iteration at each iteration step a kkt system has to be solved here we develop a multigrid method for its fast solution to this end we use a smoother which is based on an inexact constraint preconditioner we present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying elliptic pde problem furthermore when combined with a nested iteration the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the pde solution

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

distributed global output feedback control for a class of euler lagrange systems this paper investigates the distributed tracking control problem for a class of euler lagrange multiagent systems when the agents can only measure the positions in this case the lack of the separation principle and the strong nonlinearity in unmeasurable states pose severe technical challenges to global output feedback control design to overcome these difficulties a global nonsingular coordinate transformation matrix in the upper triangular form is first proposed such that the nonlinear dynamic model can be partially linearized with respect to the unmeasurable states and a new type of velocity observers is designed to estimate the unmeasurable velocities for each system then based on the outputs of the velocity observers we propose distributed control laws that enable the coordinated tracking control system to achieve uniform global exponential stability both theoretical analysis and numerical simulations are presented to validate the effectiveness of the proposed control scheme

convergence analysis of the defect correction iteration for hyperbolic problems this paper explains some of the convergence behaviour of iterative implicit and defect correction schemes for the solution of the discrete steady euler equations such equations are also commonly solved by pseudo time integration the steady solution being achieved as the limit for t to infty of the solution of a time dependent problem implicit schemes are then often chosen for their favourable stability properties permitting large timesteps for efficiency an important class of implicit schemes involving first and second order accurate upwind discretisations is considered in the limit of an infinite timestep these implicit schemes approach defect correction algorithms thus our analysis is informative for both types of construction simple scalar linear model problems are introduced for the one dimensional and two dimensional cases these model problems are analyzed in detail by both fourier and matrix analyses the convergence behaviour appears to be strongly dependent on a parameter beta

multigrid solution of an elliptic boundary value problem with integral constraints an efficient multigrid approach for solving a discretized elliptic equation whose boundary values are determined in part by integral relations is developed analyzed and tested the algorithm is motivated by a problem that is solved during integration of the 3d quasigeostrophic qg equations which model large scale rotating stratified flows where the integral constraints represent mass conservation

a v cycle multigrid for multilevel matrix algebras proof of optimality we analyze the convergence rate of a multigrid method for multilevel linear systems whose coefficient matrices are generated by a real and nonnegative multivariate polynomial f and belong to multilevel matrix algebras like circulant tau hartley or are of toeplitz type in the case of matrix algebra linear systems we prove that the convergence rate is independent of the system dimension even in presence of asymptotical ill conditioning this happens iff f takes the zero value more precisely if the d level coefficient matrix has partial dimension nr at level r with r 1 ldots d then the size of the system is n varvec n prod r 1 d n r varvec n n 1 ldots n d and o n n operations are required by the considered v cycle multigrid in order to compute the solution within a fixed accuracy since the total arithmetic cost is asymptotically equivalent to the one of a matrix vector product the proposed method is optimal some numerical experiments concerning linear systems arising in 2d and 3d applications are considered and discussed

coarse grid correction for nonelliptic and singular perturbation problems the suitability of coarse grid correction operators for multigrid solution of discretized nonelliptic partial differential equations is studied it is found that the requirements for the orders of intergrid transfers and the order of approximation of the coarse grid operators differ in an important way from the well known rules pertaining to elliptic operators both galerkin coarsening and coarsening schemes based on the differential operator are examined and new coarse grid correction strategies are suggested in light of this analysis

fully distributed flocking with a moving leader for lagrange networks with parametric uncertainties this paper addresses the leader follower flocking problem with a moving leader for networked lagrange systems with parametric uncertainties under a proximity graph here a group of followers move cohesively with the moving leader to maintain connectivity and avoid collisions for all time and also eventually achieve velocity matching in the proximity graph the neighbor relationship is defined according to the relative distance between each pair of agents each follower is able to obtain information from only the neighbors in its proximity involving only local interaction we consider two cases i the leader moves with a constant velocity and ii the leader moves with a varying velocity in the first case a distributed continuous adaptive control algorithm accounting for unknown parameters is proposed in combination with a distributed continuous estimator for each follower in the second case a distributed discontinuous adaptive control algorithm and estimator are proposed then the algorithm is extended to be fully distributed with the introduction of gain adaptation laws in all proposed algorithms only one hop neighbors information e g the relative position and velocity measurements between the neighbors and the absolute position and velocity measurements is required and flocking is achieved as long as the connectivity and collision avoidance are ensured at the initial time and the control gains are designed properly numerical simulations are presented to illustrate the theoretical results

on the numerical radius and its applications abstract we give a brief account of the numerical radius of a linear bounded operator on a hilbert space and some of its better known properties both finite and infinite dimensional aspects are discussed as well as applications to stability theory of finite difference approximations for hyperbolic initial value problems

local mode analysis of multicolor and composite relaxation schemes abstract we develop local mode analysis of multicolor or more generally multicoupling numerical processes and provide practical procedures for computing their symbols and predicting their convergence and smoothing properties in the context of multigrid methods we apply our analysis including simple numerical examples to multicolor point relaxation especially in the context of the novel composite relaxation a brandt barriers to achieving textbook multigrid efficiency tme in cfd tech report gauss center report wi gc 10 weizmann institute december 1998 o e livne multiscale eigenbasis algorithms ph d thesis the weizmann institute of science rehovot israel 2000 for products of discrete operators such as the biharmonic

rigorous quantitative analysis of multigrid i constant coefficients two level cycle with l 2 norm exact numerical convergence factors for any multigrid cycle can be predicted by local mode fourier analysis for general linear elliptic partial differential equation pde systems with piecewise smooth coefficients in general domains discretized by uniform grids it is proved that in the limit of small meshsizes these predicted factors are indeed obtained provided the cycle is supplemented with a proper processing at and near the boundaries that processing it is proved costs negligible extra computer work apart from mode analysis a coarse grid approximation cga condition is introduced which is both necessary and sufficient for the multigrid algorithm to work properly the present part studies the l 2 convergence in one cycle for equations with constant coefficients in the sequel brandt rigorous quantitative analysis of multigrid ii extensions and practical implications manuscript extensions are discussed to many cycles asymptotic convergence to more levels with arbitrary cycle ty

fourier analysis for multigrid methods on triangular grids in this paper a local fourier analysis technique for multigrid methods on triangular grids is presented the analysis is based on an expression of the fourier transform in new coordinate systems both in space variables and in frequency variables associated with reciprocal bases this tool makes it possible to study different components of the multigrid method in a very similar way to that of rectangular grids different smoothers for the discrete laplace operator obtained with linear finite elements are analyzed a new three color smoother has been studied and has proven to be the best choice for near equilateral triangles it is also shown that the block line smoothers are more appropriate for irregular triangles numerical test calculations validate the theoretical predictions

spectra and pseudospectra of waveform relaxation operators we describe the spectra and pseudospectra of continuous time and discrete time waveform relaxation operators since the spectrum of the finite interval waveform relaxation operator is only a singleton the spectrum of the infinite interval operator is typically used to describe the behavior of waveform relaxation algorithms here we show that the pseudospectrum is a more useful tool for analyzing convergence of waveform iterations on finite intervals and prove that the pseudospectrum of the infinite interval operator is in fact the limit of the pseudospectra of the finite interval operators

multigrid algorithms for solving multigroup neutron transport equations abstract this paper describes multigrid algorithms for solving multigroup neutron transport equations local mode analysis of multilevel is performed for transport problems and numerical fourier analysis is carried out to study the smoothing and convergence properties of the multigrid algorithms for neutron diffusion and general pn approximate case respectively numerical test calculations validate the theoretical predictions

fields of values and iterative methods abstract the performance of an iterative scheme to solve x t x c t c n n c c n is often judged by spectral properties of t if t is not normal it is however well known that only conclusions about the assymptotic behavior of an iterative method can be drawn from spectral information to anticipate the progress of the iteration after a finite number of steps the knowledge of the eigenvalues alone is often useless in addition the spectrum of t may be highly sensitive to perturbations if t is not normal an iterative method which on the basis of some spectral information is predicted to converge rapidly for t may well diverge if t is slightly perturbed in practice the convergence of the iteration x m t x m 1 c is therefore frequently measured by some norm verbar t verbar rather than by the spectral radius t but apart from the fact that norms lead to error estimates which are often too pessimistic they cannot be used to analyze more general schemes such as e g the chebyshev iterative methods here we discuss another tool to analyze the behavior of an iterative method namely the field of values w t the collection of all rayleigh quotients of t w t contains the eigenvalues of t and the numerical radius t max z w t z defines a norm on c n n the field of values represents therefore an intermediate concept to judge an iterative scheme by it is related to the spectral approach but has also certain norm properties

geometric multigrid with applications to computational fluid dynamics an overview is given of the development of the geometric multigrid method with emphasis on applications in computational fluid dynamics over the last ten years both compressible and incompressible flow problems and their corresponding multigrid solution methods are discussed the state of the art is described with respect to methods employed in industry as well as the multigrid efficiency obtained in academic applications

distributed coordinated tracking with a dynamic leader for multiple euler lagrange systems in this note we study a distributed coordinated tracking problem for multiple networked euler lagrange systems the objective is for a team of followers modeled by full actuated euler lagrange equations to track a dynamic leader whose vector of generalized coordinates is time varying under the constraints that the leader is a neighbor of only a subset of the followers and the followers have only local interaction we consider two cases i the leader has a constant vector of generalized coordinate derivatives and ii the leader has a varying vector of generalized coordinate derivatives in the first case we propose a distributed continuous estimator and an adaptive control law to account for parametric uncertainties in the second case we propose a model independent sliding mode control algorithm simulation results on multiple networked two link revolute joint arms are provided to show the effectiveness of the proposed control algorithms