an algebraic multigrid method for a class of elliptic differential systems the development numerical investigation and analysis of an algebraic multigrid amg method for the numerical solution of a representative class of elliptic differential systems are presented our amg scheme is based on the amg strategy of brandt ruge and stuben however the present approach may be implemented within other amg schemes by means of numerical experiments with model problems arising from some application areas it is demonstrated that the computational performance of the proposed amg scheme is comparable to that of amg applied to a single scalar equation an amg convergence theory is presented that extends convergence results for amg methods for scalar problems to the case of amg applied to elliptic systems some details of our amg implementation are given

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

mechanisms for the hard bubbling transition in symmetrically coupled chaotic systems we investigate mechanisms for the hard bubbling transition in symmetrically coupled one dimensional maps a transition from strong to weak synchronization occurs via a first period doubling or pitchfork transverse bifurcation of a periodic saddle embedded in the synchronous chaotic attractor the consequence of such transverse bifurcations depends on the existence of an absorbing area controlling the global dynamics in the presence of an absorbing area a subcritical transverse bifurcation is found to give rise to abrupt appearance of transient intermittent bursts with large amplitude for this case any small parameter mismatch that is unavoidable in real systems induces a hard bubbling transition via an interior crisis through a detailed numerical analysis we present explicit examples for the subcritical period doubling and pitchfork transverse bifurcations leading to hard bubbling

a new preprocessing algorithm for the computation of the generalized singular value decomposition in this note a new algorithm is proposed for the preprocessing phase of paige s algorithm for computing the generalized singular value decomposition gsvd this new algorithm substantially reduces the complexity of paige s algorithm and makes it much easier to implement it is also proved that the preprocessing phase is backward stable and a numerical example is demonstrated

on interval estimates of perturbations of generalized eigenvalues for diagonalizable pairs abstract on generalized eigenvalue perturbation bounds for diagonalizable matrix pairs upper bounds are discussed all the time in this paper we mainly consider not only upper bounds but also lower bounds of perturbation of generalized eigenvalues for diagonalizable matrix pairs sharper upper bounds and sharp lower bounds on perturbation of generalized eigenvalues for diagonalizable matrix pairs are obtained in terms of the distances of two points in grassmann manifold the main results improve the existing results of chen 2007 1 and numerical examples illustrate the efficiency of the theoretical results

a comparison of algebraic multigrid and geometric immersed interface multigrid methods for interface problems in l adams and z li siam j sci comput 24 2002 pp 463 479 a multigrid method was designed specifically for interface problems that have been discretized using the methods described in l adams and z li siam j sci comput 24 2002 pp 463 479 and in z li and k ito siam j sci comput 23 2001 pp 339 361 for elliptic interface problems using the maximum principle preserving schemes in l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 a new method was introduced that utilizes a new interpolator for grid points near the immersed interface and a new restrictor that guarantees the coarse grid matrices are m matrices this paper compares the immersed interface multigrid methods introduced in l adams and z li siam j sci comput 24 2002 pp 463 479 and l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 with algebraic multigrid which uses no geometric information to set up the multigrid components for coarse grid correction we show that algebraic multigrid is a robust solver for our test problems it outperforms the method in l adams and z li siam j sci comput 24 2002 pp 463 479 and performs nearly as well as the method in l adams and t p chartier siam j sci comput 25 2002 pp 1516 1533 which is shown to be the most efficient for all problem parameters and sizes

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

an application of multigrid methods for a discrete elastic model for epitaxial systems we apply an efficient and fast algorithm to simulate the atomistic strain model for epitaxial systems recently introduced by schindler et al phys rev b 67 075316 2003 the discrete effects in this lattice statics model are crucial for proper simulation of the influence of strain for thin film epitaxial growth but the size of the atomistic systems of interest is in general quite large and hence the solution of the discrete elastic equations is a considerable numerical challenge in this paper we construct an algebraic multigrid method suitable for efficient solution of the large scale discrete strain model using this method simulations are performed for several representative physical problems including an infinite periodic step train a layered nanocrystal and a system of quantum dots the results demonstrate the effectiveness and robustness of the method and show that the method attains optimal convergence properties regardless of the problem size the geometry and the physical parameters the effects of substrate depth and of invariance due to traction free boundary conditions are assessed for a system of quantum dots the simulated strain energy density supports the observations that trench formation near the dots provides strain relief

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

perturbation analysis for the generalized singular value problem this paper discusses perturbation bounds for generalized singular values and for subspaces associated with the generalized singular value decomposition of two matrices having the same number of columns by a limiting procedure from our results we can derive those for the ordinary singular value problem which were proved by l mirsky quart j math oxford 11 1960 pp 50 59 and p a wedin bit 12 1972 pp 99 111

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

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

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

an extension of analytic solutions of a class of constrained matrix minimization problems abstract in this paper we consider an extension of analytic solutions of a class of constrained matrix minimization problems and propose analytic solutions of the following matrix minimization problem min s k s k h i n l k l k h i m det c i m k 1 s a k s k b k l k min s k s k h i n l k l k h i m t r c i m k 1 s a k s k b k l k where a k c m n b k c n m c is a complex number i m denotes m order identity matrix and det t r denote matrix determinant and trace function the proposed results improve the corresponding existing ones in xu et al 2019 numerical examples are provided to illustrate the efficiency of the proposed theoretical results

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

robust norm equivalencies for diffusion problems additive multilevel methods offer an efficient way for the fast solution of large sparse linear systems which arise from a finite element discretization of an elliptic boundary value problem these solution methods are based on multilevel norm equivalencies for the associated bilinear form using a suitable subspace decomposition to obtain a robust iterative scheme it is crucial that the constants in the norm equivalence do not depend or depend only weakly on the ellipticity constants of the problem in this paper we present such a robust norm equivalence for the model problem w u with a scalar diffusion coefficient in c r 2 our estimates involve only very weak information about w and the results are applicable for a large class of diffusion coefficients namely we require to be in the muckenhoupt class a 1 a function class well studied in harmonic analysis the presented multilevel norm equivalencies are a main step towards the realization of an optimal and robust multilevel preconditioner for scalar diffusion problems

studying basin bifurcations in nonlinear triopoly games by using 3d visualization we consider three dimensional discrete dynamical system obtained by the iteration of a noninvertible map of which simulates the time evolution of an oligopoly game with three competing firms the model is characterized by the presence of several coexisting stable equilibria each with its own basin of attraction in this paper we face the question of the delimitation of the basins and the detection of the global bifurcations that cause the creation of non connected basins this requires a study of the global properties of the 3dimensional noninvertible map by the method of critical sets based on the determination of the contact bifurcations through a systematic computer assisted study this requires the visualization of surfaces the critical surfaces and the basins boundaries which sometimes are nested one inside the other enhanced graphical methods based on two level volume rendering are employed in order to modulate the opacity of outer objects so that the contacts between the basins boundaries and critical surfaces can be visualized this is obtained through the realization of ad hoc routines which allow interactive 3d visualization

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

on riddling and weak attractors abstract we propose general definitions for riddling and partial riddling of a subset v of r m with non zero lebesgue measure and show that these properties are invariant for a large class of dynamical systems we introduce the concept of a weak attractor a weaker notion than a milnor attractor and use this to re examine and classify riddled basins of attractors we find that basins of attraction can be partially riddled but if this is the case then any partial riddling must be evident near the attractor we use these concepts to aid classification of bifurcations of attractors from invariant subspaces in particular our weak attractor is a generalisation of the absorbing area investigated by other authors and we suggest that a transition of a basin to riddling is usually associated with loss of stability of a weak attractor

on perturbations of matrix pencils with real spectra a revisit this paper continues earlier studies by bhatia and li on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectra a unifying framework for creating crucial perturbation equations is developed with the help of a recent result on generalized commutators involving unitary matrices new and much sharper bounds are obtained

symmetry breaking bifurcation for coupled chaotic attractors the authors consider transitions from synchronous to asynchronous chaotic motion in two identical dissipatively coupled one dimensional mappings they show that the probability density of the asymmetric component satisfies a scaling law the exponent in this scaling law varies continuously with the distance from the bifurcation point and is determined by the spectrum of local lyapunov exponents of the uncoupled map finally they show that the topology of the invariant set is rather unusual though the attractor for supercritical coupling is a line it is surrounded by a strange invariant set which is dense in a two dimensional neighbourhood of the attractor

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

a new algebraic multigrid approach for stokes problems standard discretizations of stokes problems lead to linear systems of equations in saddle point form making difficult the application of algebraic multigrid methods in this paper a new approach is proposed it consists in first transforming the system by pre and post multiplication with simple algebraic sparse block triangular matrices this is a form of pre conditioning in the literal sense designed to make sure that the transformed matrix is well adapted to multigrid in particular after transformation all the diagonal blocks are symmetric and positive definite and correspond to or resemble a discrete laplace operator then to each of these diagonal blocks is associated a prolongation that works well for it using any relevant algebraic or geometric multigrid method next a multigrid scheme for the global system is naturally set up by combining these partial prolongations with a galerkin coarse grid matrix for this approach combined with damped jacobi smoothing a uniform two grid convergence bound is derived for the global system under the assumption that the two grid schemes for the different diagonal blocks are themselves uniformly convergent this result is illustrated by a few examples showing further that time dependent problems and variable viscosity can be handled in a natural way without requiring parameter adjustment a numerical comparison also shows that the new approach can be more effective than state of the art block preconditioning techniques

first order system least squares for incompressible resistive magnetohydrodynamics magnetohydrodynamics mhd is a fluid theory that describes plasma physics by treating the plasma as a fluid of charged particles hence the equations that describe the plasma form a nonlinear system that couples navier stokes equations with maxwell s equations this paper shows that the first order system least squares fosls finite element method is a viable discretization for these large mhd systems to solve this system a nested iteration newton fosls amg approach is taken most of the work is done on the coarse grid including most of the linearizations we show that at most one newton step and a few v cycles are all that are needed on the finest grid here we describe how the fosls method can be applied to incompressible resistive mhd and how it can be used to solve these mhd problems efficiently a 3d steady state and a reduced 2d time dependent test problem are studied the latter equations can simulate a large aspect ratio tokamak the goal is to resolve as much physics from the test problems with the least amount of computational work we show that this is achieved in a few dozen work units or fine grid residual evaluations

accurate gradient approximation for complex interface problems in 3d by an improved coupling interface method most elliptic interface solvers become complicated for complex interface problems at those exceptional points where there are not enough neighboring interior points for high order interpolation such complication increases especially in three dimensions usually the solvers are thus reduced to low order accuracy in this paper we classify these exceptional points and propose two recipes to maintain order of accuracy there aiming at improving the previous coupling interface method 26 yet the idea is also applicable to other interface solvers the main idea is to have at least first order approximations for second order derivatives at those exceptional points recipe 1 is to use the finite difference approximation for the second order derivatives at a nearby interior grid point whenever this is possible recipe 2 is to flip domain signatures and introduce a ghost state so that a second order method can be applied this ghost state is a smooth extension of the solution at the exceptional point from the other side of the interface the original state is recovered by a post processing using nearby states and jump conditions the choice of recipes is determined by a classification scheme of the exceptional points the method renders the solution and its gradient uniformly second order accurate in the entire computed domain numerical examples are provided to illustrate the second order accuracy of the presently proposed method in approximating the gradients of the original states for some complex interfaces which we had tested previous in two and three dimensions and a real molecule 1d63 which is double helix shape and composed of hundreds of atoms

compatible relaxation and coarsening in algebraic multigrid we introduce a coarsening algorithm for algebraic multigrid amg based on the concept of compatible relaxation cr the algorithm is significantly different from standard methods most notably because it does not rely on any notion of strength of connection we study its behavior on a number of model problems and evaluate the performance of an amg algorithm that incorporates the coarsening approach finally we introduce a variant of cr that provides a sharper metric of coarse grid quality and demonstrate its potential with two simple examples

role of invariant and minimal absorbing areas in chaos synchronization in this paper the method of critical curves a tool for the study of the global dynamical properties of two dimensional noninvertible maps is applied to the study of chaos synchronization and related phenomena of riddling blowout and on off intermittency a general procedure is suggested in order to obtain the boundary of a particular two dimensional compact trapping region called absorbing area containing the onedimensional chaotic set on which synchronized dynamics occur the main purpose of the paper is to show that only invariant and minimal absorbing areas are useful to characterize the global dynamical behavior of the dynamical system when a milnor attractor with locally riddled basin or a chaotic saddle exists and may strongly influence the effects of riddling and blowout bifurcations some examples are given for a system of two coupled logistic maps and some practical methods and numerical tricks are suggested in order to ascertain the properties of invariance and minimality of an absorbing area some general considerations are given concerning the transition from locally riddled to globally riddled basins and the role of the absorbing area in the occurrence of such transition is discussed s1063 651x 98 11611 2

first order system least squares and the energetic variational approach for two phase flow this paper develops a first order system least squares fosls formulation for equations of two phase flow the main goal is to show that this discretization along with numerical techniques such as nested iteration algebraic multigrid and adaptive local refinement can be used to solve these types of complex fluid flow problems in addition from an energetic variational approach it can be shown that an important quantity to preserve in a given simulation is the energy law we discuss the energy law and inherent structure for two phase flow using the allen cahn interface model and indicate how it is related to other complex fluid models such as magnetohydrodynamics finally we show that using the fosls framework one can still satisfy the appropriate energy law globally while using well known numerical techniques

an optimal linear solver for the jacobian system of the extreme type ii ginzburg landau problem this paper considers the extreme type ii ginzburg landau equations a nonlinear pde model for describing the states of a wide range of superconductors based on properties of the jacobian operator and an amg strategy a preconditioned newton krylov method is constructed after a finite volume type discretization numerical experiments are done for representative two and three dimensional domains strong numerical evidence is provided that the number of krylov iterations is independent of the dimension n of the solution space yielding an overall solver complexity of o n

the deflated conjugate gradient method convergence perturbation and accuracy abstract deflation techniques for krylov subspace methods have seen a lot of attention in recent years they provide means to improve the convergence speed of these methods by enriching the krylov subspace with a deflation subspace the most common approach for the construction of deflation subspaces is to use approximate eigenvectors but also more general subspaces are applicable in this paper we discuss two results concerning the accuracy requirements within the deflated cg method first we show that the effective condition number which bounds the convergence rate of the deflated conjugate gradient method depends asymptotically linearly on the size of the perturbations in the deflation subspace second we discuss the accuracy required in calculating the deflating projection this is crucial concerning the overall convergence of the method and also allows to save some computational work to show these results we use the fact that as a projection approach deflation has many similarities to multigrid methods in particular recent results relate the spectra of the deflated matrix to the spectra of the error propagator of twogrid methods in the spirit of these results we show that the effective condition number can be bounded by the constant of a weak approximation property

a perturbation bound for definite pencils we attempt to generalize a well known result on spectral variations of a hermitian matrix due to mirsky to the definite generalized eigenvalue problem we also point out that some results on perturbations of definite pencils due to stewart can be slightly improved

a multigrid method for three dimensional elasticity and algebraic convergence estimates a two level iterative method for 3d elasticity problems discretized by quadratic elements is proposed the coarse level consists of linear elements we develop a sharpened algebraic convergence theory for two level methods with smoothing by gauss seidel which applies to more general situations as well the convergence estimates do not deteriorate for strongly discontinuous coefficients observed convergence rates are compared with theoretical estimates the method is competitive with a standard direct solver

sensor array signal processing via a procrustes rotations based eigenanalysis of the esprit data pencil the authors propose the procrustes rotation based esprit algorithm pro esprit for estimating characteristic parameters associated with a scene of radiating sources given data derived from two identical sensor arrays the x and y arrays which are displaced relative to one another pro esprit uses the inherent redundancy built into the esprit array system to compensate for deviations from the ideal array model this redundancy is manifested by two fundamental properties of the noiseless x and y data matrices comprising the esprit data pencil a they have the same row space the signal subspace and 2 they have the same column space the source subspace these properties are utilized by invoking two primary tools 1 the solution to the procrustes problem for optimally approximating an invariant subspace rotation and 2 the total least squares method for perturbing each of two estimates of a common subspace in a minimal fashion until the two perturbed spaces are the same

multilevel adaptive aggregation for markov chains with application to web ranking a multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described the method is a special case of the adaptive smoothed aggregation and adaptive algebraic multigrid methods for sparse linear systems and is also closely related to certain extensively studied iterative aggregation disaggregation methods for markov chains in contrast to most existing approaches our aggregation process does not employ any explicit advance knowledge of the topology of the markov chain instead adaptive agglomeration is proposed that is based on the strength of connection in a scaled problem matrix in which the columns of the original problem matrix at each recursive fine level are scaled with the current probability vector iterate at that level the strength of connection is determined as in the algebraic multigrid method and the aggregation process is fully adaptive with optimized aggregates chosen in each step of the iteration and at all recursive levels the multilevel method is applied to a set of stochastic matrices that provide models for web page ranking numerical tests serve to illustrate for which types of stochastic matrices the multilevel adaptive method may provide significant speedup compared to standard iterative methods the tests also provide more insight into why google s pagerank model is a successful model for determining a ranking of web pages

an algebraic two level preconditioner for asymmetric positive definite systems a two level linear algebraic solver for asymmetric positive definite systems is developed using matrices arising from stabilized finite element formulations to motivate the approach supported by an analysis of a representative smoother the parent space is divided into oscillatory and smooth subspaces according to the eigenvectors of the associated normal system using a mesh based aggregation technique which relies only on information contained in the matrix a restriction prolongation operator is constructed various numerical examples on both structured and unstructured meshes are performed using the two level cycle as the basis for a preconditioner results demonstrate the complementarity between the smoother and the coarse level correction as well as convergence rates that are nearly independent of the problem size copyright 2001 john wiley sons ltd

algebraic multigrid for moderate order finite elements we investigate the use of algebraic multigrid amg methods for the solution of large sparse linear systems arising from the discretization of scalar elliptic partial differential equations with lagrangian finite elements of order at most 4 the resulting system matrices do not have the m matrix property that is required by standard analyses of classical amg and aggregation based amg methods a unified approach is presented that allows us to extend these analyses it uses an intermediate m matrix and highlights the role of the spectral equivalence constant that relates this matrix to the original system matrix this constant is shown to be bounded independently of the problem size and jumps in the coefficients of the partial differential equations provided that jumps are located at elements boundaries for two dimensional problems it is further shown to be uniformly bounded if the angles in the triangulation also satisfy a uniform bound this analysis validates the application of the amg methods to the

periodic orbit theory of the blowout bifurcation this paper presents a theory for characterization of the blowout bifurcation by periodic orbits blowout bifurcation in chaotic systems occurs when a chaotic attractor lying in some symmetric invariant subspace becomes transversely unstable we present an analysis and numerical results that indicate that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor there are two distinct groups of periodic orbits one transversely stable and another transversely unstable the bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced our results thus categorize the blowout bifurcation as a unique type of bifurcation that involves an infinite number of periodic orbits in contrast to most previously known bifurcations that are mediated by only a finite number of periodic orbits

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

smoothed aggregation multigrid for markov chains a smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse markov chain it is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for markov chains that have been proposed in the literature the proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems supplemented with a new lumping technique that assures well posedness of the coarse level problems the coarse level operators are singular m matrices on all levels resulting in strictly positive coarse level corrections on all levels numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems both when geometric and algebraic aggregation strategies are used

an algebraic study of a local multigrid method for variational problems a two level iterative method for solving systems of equations arising from local refinement of finite element meshes is proposed the smoothing steps consists of gauss seidel relaxations applied only to the local mesh a direct solver is required on the global coarse mesh we analyze this scheme by modifying the standard algebraic convergence theory the estimates of the convergence factor do not require any regularity assumption the estimates can be evaluated element by element the evaluated estimates are compared with the actual convergence factor and with another bound in several numerical examples the theory also applies to a method with a direct solver on the local fine mesh in this case we have obtained very sharp and in fact optimal estimates of the convergence factors

convergence analysis of perturbed two grid and multigrid methods we consider multigrid methods for symmetric positive definite linear systems we present a new algebraic convergence analysis of two grid schemes with inexact solution of the coarse grid system this analysis allows us to bound the convergence factor of such perturbed two grid schemes assuming only a certain bound on the convergence factor for the unperturbed scheme with exact solution of the coarse grid system applied to multigrid methods with the standard w cycle this analysis shows that if the convergence factor of the unperturbed two grid method is uniformly bounded by sigma 1 2 then the convergence factor of the multigrid method is uniformly bounded by sigma 1 sigma the analysis is purely algebraic and requires only that pre and postsmoothing are applied in a symmetric way it covers both geometric and algebraic multigrid methods and the coarse grid matrix may be of any type not necessarily galerkin

aggregation based algebraic multigrid for convection diffusion equations we consider the iterative solution of large sparse linear systems arising from the upwind finite difference discretization of convection diffusion equations the system matrix is then an m matrix with nonnegative row sum and further when the convective flow has zero divergence the column sum is also nonnegative possibly up to a small correction term we investigate aggregation based algebraic multigrid methods for this class of matrices a theoretical analysis is developed for a simplified two grid scheme with one damped jacobi postsmoothing step an uncommon feature of this analysis is that it applies directly to problems with variable coefficients e g to problems with recirculating convective flow on the basis of this theory we develop an approach in which a guarantee is given on the convergence rate thanks to an aggregation algorithm that allows an explicit control of the location of the eigenvalues of the preconditioned matrix some issues that remain beyond the analysis are discussed in the

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

computing and deflating eigenvalues while solving multiple right hand side linear systems with an application to quantum chromodynamics we present a new algorithm that computes eigenvalues and eigenvectors of a hermitian positive definite matrix while solving a linear system of equations with conjugate gradient cg traditionally all the cg iteration vectors could be saved and recombined through the eigenvectors of the tridiagonal projection matrix which is equivalent theoretically to unrestarted lanczos our algorithm capitalizes on the iteration vectors produced by cg to update only a small window of vectors that approximate the eigenvectors while this window is restarted in a locally optimal way the cg algorithm for the linear system is unaffected yet in all our experiments if the window has more than a properly chosen but small number of vectors it converges to the required eigenvectors at a rate identical to unrestarted lanczos after the solution of the linear system eigenvectors that have not accurately converged can be improved in an incremental fashion by solving additional linear systems in this case eigenvectors identified in earlier systems can be used to deflate and thus accelerate the convergence of subsequent systems we have used this algorithm with excellent results in lattice quantum chromodynamics applications where hundreds of right hand sides may be needed specifically about 70 eigenvectors are obtained to full accuracy after solving 24 right hand sides deflating these from the large number of subsequent right hand sides removes the dreaded critical slowdown where the conditioning of the matrix increases as the quark mass reaches a critical value our experiments show almost a constant number of iterations for our method regardless of quark mass and speedups of 8 over original cg for light quark masses

on perturbations of matrix pencils with real spectra perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil a ab with real spectrum are developed it is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices the applications of these bounds to the spectral variations of definite pencils are conducted in such a way that extra attention is paid to their peculiarities so as to derive more sophisticated perturbation bounds our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of hermitian matrices such as the weyl lidskii theorem and the hoffman wielandt theorem and those for generalized eigenspaces are counterparts of the celebrated davis kahan sin 0 sin 20 theorems for the eigenspace variations of hermitian matrices the paper consists of two parts part i is for generalized eigenvalue perturbations while part ii deals with generalized eigenspace perturbations

convergence of algebraic multigrid based on smoothed aggregation we prove aconvergence estimate for the algebraic multigrid method with prolongations defined by aggregation using zero energy modes followed by a smoothing the method input is the problem matrix and a matrix of the zero energy modes the estimate depends only polylogarithmically on the mesh size and requires only a weak approximation property for the aggregates which can be a priori verified computationally construction of the prolongator in the case of a general second order system is described and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements

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

transverse instability and riddled basins in a system of two coupled logistic maps riddled basins denote a characteristic type of fractal domain of attraction that can arise when a chaotic motion is restricted to an invariant subspace of total phase space an example is the synchronized motion of two identical chaotic oscillators the paper examines the conditions for the appearance of such basins for a system of two symmetrically coupled logistic maps we determine the regions in parameter plane where the transverse lyapunov exponent is negative the bifurcation curves for the transverse destabilization of low periodic orbits embedded in the chaotic attractor are obtained and we follow the changes in the attractor and its basin of attraction when scanning across the riddling and blowout bifurcations it is shown that the appearance of transversely unstable orbits does not necessarily lead to an observable basin riddling and that the loss of weak stability when the transverse lyapunov exponent becomes positive does not necessarily destroy the basin of attraction instead the symmetry of the synchronized state may break and the attractor may spread into two dimensional phase space

robust multigrid methods for nonsmooth coefficient elliptic linear systems we survey the literature on robust multigrid methods which have been developed in recent years for solving second order elliptic pdes with nonsmooth coefficients we highlight the key ideas of designing robust multigrid methods which are able to recover the usual multigrid efficiency for nonsmooth coefficient pdes on structured or unstructured grids in particular we shall describe various approaches for constructing the interpolation and the smoothing operators and the coarse grid points selections

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

a general interpolation strategy for algebraic multigrid using energy minimization algebraic multigrid methods solve sparse linear systems ax b by automatic construction of a multilevel hierarchy this hierarchy is defined by grid transfer operators that must accurately capture algebraically smooth error relative to the relaxation method we propose a methodology to improve grid transfers through energy minimization the proposed strategy is applicable to hermitian non hermitian definite and indefinite problems each column of the grid transfer operator p is minimized in an energy based norm while enforcing two types of constraints a defined sparsity pattern and preservation of specified modes in the range of p a krylov based strategy is used to minimize energy which is equivalent to solving a p j boldsymbol 0 for each column j of p with the constraints ensuring a nontrivial solution for the hermitian positive definite case a conjugate gradient cg based method is utilized to construct grid transfers while methods based on generalized minimum residual gmres and cg on the normal equations cgnr are explored for the general case the approach is flexible allowing for arbitrary coarsenings unrestricted sparsity patterns straightforward long distance interpolation and general use of constraints either user defined or auto generated we conclude with numerical evidence in support of the proposed framework

note on analytic solutions of a class of extended constrained matrix maximization problems abstract this note is devoted to dealing with flaws that existed in a recent paper xu et al 2020 we give corrections of the original 3 9 and 3 12 in theorem 3 2 of xu et al 2020

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

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

a ghost fluid level set methodology for simulating multiphase electrohydrodynamic flows with application to liquid fuel injection in this paper we present the development of a sharp numerical scheme for multiphase electrohydrodynamic ehd flows for a high electric reynolds number regime the electric potential poisson equation contains ehd interface boundary conditions which are implemented using the ghost fluid method gfm the gfm is also used to solve the pressure poisson equation the methods detailed here are integrated with state of the art interface transport techniques and coupled to a robust high order fully conservative finite difference navier stokes solver test cases with exact or approximate analytic solutions are used to assess the robustness and accuracy of the ehd numerical scheme the method is then applied to simulate a charged liquid kerosene jet

singular values doubly stochastic matrices and applications abstract the hadamard square of any square matrix a is bounded above and below by some doubly stochastic matrices times the square of the largest and the smallest singular values of a applications to graphs permanents and eigenvalue perturbations are discussed

hsl_mi20 an efficient amg preconditioner for finite element problems in 3d algebraic multigrid amg is one of the most effective iterative methods for the solution of large sparse linear systems obtained from the discretization of second order scalar elliptic self adjoint partial differential equations it can also be used as a preconditioner for krylov subspace methods in this communication we report on the design and development of a robust effective and portable fortran 95 implementation of the classical ruge stuben amg which is available as package hsl mi20 within the hsl mathematical software library the routine can be used as a black box preconditioner but it also offers the user a range of options and parameters proper tuning of these parameters for a particular application can significantly enhance the performance of an amg preconditioned krylov solver this is illustrated using a number of examples arising in the unstructured finite element discretization of the diffusion the convection diffusion and the stokes equations as well as transient thermal convection problems associated with the boussinesq approximation of the navier stokes equations in 3d copyright 2009 john wiley sons ltd

on perturbation bounds of generalized eigenvalues for diagonalizable pairs the purpose of this paper is to study the perturbation of generalized eigenvalues two perturbation bounds of the diagonalizable pairs are given these results extend the corresponding ones given by sun math numer sinica 4 23 29 1982