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

inexact schwarz algebraic multigrid preconditioners for crack problems modeled by extended finite element methods summary traditional algebraic multigrid amg preconditioners are not well suited for crack problems modeled by extended finite element methods xfem this is mainly because of the unique xfem formulations which embed discontinuous fields in the linear system by addition of special degrees of freedom these degrees of freedom are not properly handled by the amg coarsening process and lead to slow convergence in this paper we proposed a simple domain decomposition approach that retains the amg advantages on well behaved domains by avoiding the coarsening of enriched degrees of freedom the idea was to employ a multiplicative schwarz preconditioner where the physical domain was partitioned into healthy or unfractured and cracked subdomains first the healthy subdomain containing only standard degrees of freedom was solved approximately by one amg v cycle followed by concurrent direct solves of cracked subdomains this strategy alleviated the need to redesign special amg coarsening strategies that can handle xfem discretizations numerical examples on various crack problems clearly illustrated the superior performance of this approach over a brute force amg preconditioner applied to the linear system copyright 2011 john wiley sons ltd

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

structure of basic lie superalgebras and of their affine extensions we generalize to the case of superalgebras several properties of simple lie algebras involving the use of dynkin diagrams if to a simple lie algebra can be associated one dynkin diagram it is a finite set of nonequivalent ones which can be constructed for a basic superalgebra or b s a the knowledge of these diagrams which can be obtained for each b s a in a systematic way allows us to deduce the regular subsuperalgebras of a b s a the symmetries of the dynkin diagrams are related to outer automorphisms of b s a and lead to some singular subsuperalgebras finally we consider the extended dynkin diagrams in order to classify the affine b s a and use their symmetries to construct the twisted basic superalgebras

element free amge general algorithms for computing interpolation weights in amg we propose a new general algorithm for constructing interpolation weights in algebraic multigrid amg it exploits a proper extension mapping outside a neighborhood about a fine degree of freedom dof to be interpolated the extension mapping provides boundary values based on the coarse dofs used to perform the interpolation at the boundary of the neighborhood the interpolation value is then obtained by matrix dependent harmonic extension of the boundary values into the interior of the neighborhood we describe the method present examples of useful extension operators provide a two grid analysis of model problems and by way of numerical experiments demonstrate the successful application of the method to discretized elliptic problems

multiple galerkin adaptive algebraic multigrid algorithm for the helmholtz equations introduced in this paper is a multiple galerkin adaptive algebraic multigrid algorithm for solving linear discrete equations arising from discretization of partial differential operators with a rich and diverse near kernel standard multigrid methods struggle with solving such equations due to their inability to produce accurate coarse correction to all near kernel components the motivating model problem here is the indefinite helmholtz operators for which the character of near kernel components is at least approximately known in the algorithm some of these components are used to create multiple prolongation operators and consecutive coarse descriptions which collectively approximate the entire near kernel the algorithm consists of two parts with multiple galerkin corrections accelerating a standard multigrid v cycle both parts largely employ standard multigrid techniques the resulting algorithm is then used to precondition the gmres numerical experiments in one and two dimensions are presented

a fully supersymmetric akns theory we construct a fully supersymmetric bihamiltonian theory in four superfields admitting zero curvature and lax formulation this theory is an extension of the classical akns which can be recovered as a reduction other supersymmetric theories are obtained as reductions of the susy akns namely a nonlinear schrodinger a modified kdv and the manin radul kdv the susy nonlinear schrodinger hierarchy is related to the one of roelofs and kersten we determine its bihamiltonian and lax formulation finally we show that the susy kdv s mentioned before are related through a susy miura map

multilevel preconditioning methods for discrete models of lattice block materials in this paper we construct optimal preconditioners for the discrete mathematical models arising in modeling the elastic responses of lattice block materials we present extensive numerical experiments to show that the preconditioned system has a uniformly bounded condition number with respect to the size of problem and with respect to the parameter relating the stretching and bending of the beams in a lattice using the limiting system of partial differential equations we show theoretically that for square lattices the proposed preconditioners are efficient by proving a uniform bound on the condition number of the preconditioned system

adaptive smoothed aggregation alpha sa substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice these systems often arise from discretizing partial differential equations over unstructured meshes and the particular parameters or geometry of the physical problem being discretized may be unavailable to the solver algebraic multigrid amg and multilevel domain decomposition methods of algebraic type have been of particular interest in this context because of their promises of optimal performance without the need for explicit knowledge of the problem geometry these methods construct a hierarchy of coarse problems based on the linear system itself and on certain assumptions about the smooth components of the error for smoothed aggregation sa methods applied to discretizations of elliptic problems these assumptions typically consist of knowledge of the near nullspace of the weak form this paper introduces an extension of the sa method in which good convergence properties are achieved in situations where explicit knowledge of the near nullspace components is unavailable this extension is accomplished by using the method itself to determine near nullspace components and adjusting the coarsening processes accordingly

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

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

parallel multigrid smoothing polynomial versus gauss seidel gauss seidel is often the smoother of choice within multigrid applications in the context of unstructured meshes however maintaining good parallel efficiency is difficult with multiplicative iterative methods such as gauss seidel this leads us to consider alternative smoothers we discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems two particular polynomials are considered chebyshev and a multilevel specific polynomial the advantages of polynomial smoothing over traditional smoothers such as gauss seidel are illustrated on several applications poisson s equation thin body elasticity and eddy current approximations to maxwell s equations while parallelizing the gauss seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates we show that although parallel computers are the main motivation polynomial smoothers are often surprisingly competitive with gauss seidel smoothers on serial machines

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

weight diagram construction of lax operators it is demonstrated and proven that cyclic weight diagrams corresponding to representations of affine lie algebras allow one to construct the associated lax operator the resultant lax operator is in the miura like form and generates the modified kdv equations the algorithm is extended to the supersymmetric case

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

integrable system constructed out of two interacting superconformal fields we consider an interaction between superconformal fields of the same gradation this entails the construction of a supersymmetric poisson tensor for these fields generating a new class of hamiltonian systems the lax representation is found for one of them by supersymmetrizing the lax operator for the hirota satsuma equation the supersymmetric equation is not reducible to the classical hirota satsuma case we show that our generalized system can be reduced to the the supersymmetric kdv equation a 4 surprisingly the integrals of motion are not reduced to integrals of motion of the supersymmetric kdv equation

on the bihamiltonian structure of the supersymmetric kdv hierarchies a lie superalgebraic approach we give a lie superalgebraic interpretation of the bihamiltonian structure of known supersymmetric kdv equations we show that the loop algebra of a lie superalgebra carries a natural poisson pencil and we subsequently deduce the bihamiltonian structure of the supersymmetric kdv hierarchies by applying to loop superalgebras an appropriate reduction technique this construction can be regarded as a superextension of the drinfeld sokolov method for building a kdv type hierarchy from a simple lie algebra

algebraic multigrid for markov chains an algebraic multigrid amg method is presented for the calculation of the stationary probability vector of an irreducible markov chain the method is based on standard amg for nonsingular linear systems but in a multiplicative adaptive setting a modified amg interpolation formula is proposed that produces a nonnegative interpolation operator with unit row sums we show how the adoption of a previously described lumping technique maintains the irreducible singular m matrix character of the coarse level operators on all levels together these properties are sufficient to guarantee the well posedness of the algorithm numerical results show how it leads to nearly optimal multigrid efficiency for a representative set of test problems

a self learning algebraic multigrid method for extremal singular triplets and eigenpairs a self learning algebraic multigrid method for dominant and minimal singular triplets and eigenpairs is described the method consists of two multilevel phases in the first multiplicative phase setup phase tentative singular triplets are calculated along with a multigrid hierarchy of interpolation operators that approximately fit the tentative singular vectors in a collective and self learning manner using multiplicative update formulas in the second additive phase solve phase the tentative singular triplets are improved up to the desired accuracy by using an additive correction scheme with fixed interpolation operators combined with a ritz update a suitable generalization of the singular value decomposition is formulated that applies to the coarse levels of the multilevel cycles the proposed algorithm combines and extends two existing multigrid approaches for symmetric positive definite eigenvalue problems to the case of dominant and minimal singular triplets numerical tests on model probl

local rigidity of discrete groups acting on complex hyperbolic space the superrigidity theorem of margulis see zimmer 17 classifies finite dimensional representations of lattices in semi simple lie groups of real rank strictly larger than 1 it is a fundamental problem to obtain the classification of finite dimensional representations of lattices in rank i semi simple lie groups that this problem will be considerably harder than the previous one is suggested by the existence of continuous families of inequivalent representations or in other words the existence of non trivial deformations in johnson millson 6 and kourouniotis 8 such deformations were constructed for certain representations of lattices in so n 1 based on a construction of thurston called bending the deformation space of the representation of a lattice f so n 1 obtained by restricting an inclusion so n 1 so n l 1 to f is of particular interest if n 2 the space of infinitesimal deformations is hi f r 1 where f acts on r 1 by the restriction of the standard action of so n 1 the space of infinitesimal deformations is non zero for the standard arithmetic examples and the main point of the papers cited above was to establish that some of these infinitesimal deformations are integrable in 6 it is also shown that some are not in this paper we study the complex analogue of the above example we let f be a cocompact torsion free lattice in su n 1 and consider the deformation space of the representation of f obtained by restricting an inclusion su n 1 su n i 1 if n l the space of infinitesimal deformations is h f r h f c 1 in the first summand f acts trivially and the infinitesimal deformations are tangent to the obvious deformations obtained by deforming f in u n 1 by a curve of homomorphisms into the center of u n 1 observe that the above inclusion factors as su n 1 u n 1 su n i 1 in the second summand f acts by the restriction of the standard action of su n 1 this summand is non zero for the standard arithmetic examples

a class of spectral two level preconditioners it is well known that the convergence of krylov methods for solving the linear system often depends to a large extent on the eigenvalue distribution in many cases it is observed that removing the smallest eigenvalues can greatly improve the convergence several techniques have been proposed in the past few years that attempt to tackle this problem the proposed approaches can be split into two main families depending on whether the scheme enlarges the generated krylov space or adaptively updates the preconditioner in this paper we follow the second approach and propose a class of preconditioners both for unsymmetric and for symmetric linear systems that can also be adapted for symmetric positive definite problems we effectively solve the preconditioned system exactly in the low dimensional space associated with the smallest eigenvalues and use this to update the preconditioned residual this update results in shifting eigenvalues from close to the origin to near to one for the new preconditioner this is ideal when there are only a few eigenvalues near the origin while all the others are close to one because the updated preconditioned system becomes close to the identity we illustrate the performance of our method through extensive numerical experiments on a set of general linear systems finally we show the advantages of the preconditioners for solving dense linear systems arising in electromagnetism applications which were the main motivation for this work

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

parallel algebraic multigrids for structural mechanics this paper presents the results of a comparison of three parallel algebraic multigrid amg preconditioners for structural mechanics applications in particular we are interested in investigating both the scalability and robustness of the preconditioners numerical results are given for a range of structural mechanics problems with various degrees of difficulty and they show that algebraic multigrid methods can be applied successfully even to difficult problems in structural mechanics

algebraic multigrid based on computational molecules 2 linear elasticity problems this paper deals with a new approach in algebraic multigrid amg for self adjoint and elliptic problems arising from finite element discretization of linear elasticity problems generalizing our approach for scalar problems j k kraus and j schicho computing 77 2006 pp 57 75 we propose an edge matrix concept for point block systems of linear algebraic equations this gives a simple and reliable method for the evaluation of the strength of nodal dependence that can be applied to symmetric positive definite non m matrices in this paper the edge matrix concept is developed for and applied to two as well as three dimensional linear elasticity problems we consider approximate splittings of the problem related element matrices into symmetric positive semidefinite edge matrices of rank one the reproduction of edge matrices on coarse levels offers the opportunity to combine classical coarsening techniques with robust energy minimizing interpolation schemes the computational molecules involved in this process are assembled from edge matrices this yields a flexible new variant of amg that allows also for an efficient solution of problems with discontinuous coefficients e g for composite materials

surface group representations with maximal toledo invariant we develop the theory of maximal representations of the fundamental group 1 of a compact connected oriented surface possibly with boundary into lie groups g of hermitian type for any homomorphism p 1 g we define the toledo invariant t p a numerical invariant which has both topological and analytical interpretations we establish important properties of t among which continuity uniform boundedness on the representation variety additivity under connected sum of surfaces and congruence relations mod we thus obtain information about the representation variety as well as striking geometric properties of maximal representations that is representations whose toledo invariant achieves the maximum value moreover we establish properties of boundary maps associated to maximal representations which generalize naturally monotonicity properties of semiconjugations of the circle we define a rotation number function for general locally compact groups and study it in detail for groups of hermitian type properties of the rotation number together with the existence of boundary maps lead to additional invariants for maximal representations and show that the subset of maximal representations is always real semialgebraic

algebraic multigrid for finite element elasticity equations determination of nodal dependence via edge matrices and two level convergence we study an algebraic multigrid amg method for solving elliptic finite element equations of linear elasticity problems in this method which has been proposed in kraus siam j sci comput 2008 30 505 524 the coarsening is based on the so called edge matrices which allows to generalize the concept of strong and weak connections as used in the classical amg to algebraic vertices that accumulate the nodal degrees of freedom in case of vector field problems the major contribution of this work is related to the investigation of a measure for the nodal dependence and on the generation of the edge matrices which are the basic building blocks of this method a natural measure is the cosine of the abstract angle between the two subspaces spanned by the basis functions corresponding to the respective algebraic vertices another original contribution of this work is a two level convergence analysis of the method the presented numerical results cover also problems with jumps in young s modulus of elasticity and orthotropic materials copyright 2010 john wiley sons ltd

n 2 super w algebras and generalized n 2 super kdv heirarchies based on lie superalgebras the authors propose a hierarchy of super lax equations based on an affine lie superalgebra a n n 1 as the generalized n 2 super kdv hierarchy and show that the associated super gel fand dikii bracket defines an n 2 super w algebra for a 2 2 1 they obtain the n 2 super boussiensque equation and the n 2 super w2 algebra containing additional currents of spins 2 5 2 3

robustness and scalability of algebraic multigrid algebraic multigrid amg is currently undergoing a resurgence in popularity due in part to the dramatic increase in the need to solve physical problems posed on very large unstructured grids while amg has proved its usefulness on various problem types it is not commonly understood how wide a range of applicability the method has in this study we demonstrate that range of applicability while describing some of the recent advances in amg technology moreover in light of the imperatives of modern computer environments we also examine amg in terms of algorithmic scalability finally we show some of the situations in which standard amg does not work well and indicate the current directions taken by amg researchers to alleviate these difficulties

amg e based on element agglomeration this paper contains the main ideas for an amge algebraic multigrid for finite elements method based on element agglomeration in the method coarse grid elements are formed by agglomerating fine grid elements compatible interpolation operators are constructed which yield coarse grid basis functions with a minimal energy property heuristics based on interpolation quality measures are used to guide the agglomeration procedure the performance of the resulting method is demonstrated in two level numerical experiments

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

greedy coarsening strategies for nonsymmetric problems the solution of large scale linear systems in computational science and engineering requires efficient solvers and preconditioners often the most effective such techniques are those based on multilevel splittings of the problem in this paper we consider the problem of partitioning both symmetric and nonsymmetric matrices based solely on algebraic criteria a new algorithm is proposed that combines attractive features of two previous techniques proposed by the authors it offers rigorous guarantees of certain properties of the partitioning yet is naturally compatible with the threshold based dropping known to be effective for incomplete factorizations numerical results show that the new partitioning scheme leads to improved results for a variety of problems the effects of further matrix reordering within the fine scale block are also considered

accelerated multigrid for graph laplacian operators we consider multigrid type techniques for the numerical solution of large linear systems whose coefficient matrices show the structure of weighted graph laplacian operators we combine ad hoc coarser grid operators with iterative techniques used as smoothers empirical tests suggest that the most effective smoothers have to be of krylov type with subgraph preconditioners while the projectors which define the coarser grid operators have to be designed for maintaining as much as possible the graph structure of the projected matrix at the inner levels the main theoretical contribution of the paper is the characterization of necessary and sufficient conditions for preserving the graph structure in this framework it is possible to explain why the classical projectors inherited from differential equations are good in the differential context and why they may behave unsatisfactorily for unstructured graphs furthermore we report and discuss several numerical experiments showing that our approach is effective even in very difficult cases where the known approaches are rather slow as a conclusion the main advantage of the proposed approach is the robustness since our multigrid type technique behaves uniformly well in all cases without requiring either the setting or the knowledge of critical parameters as it happens when using the best known preconditioned krylov methods

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

lie superalgebraic approach to super toda lattice and generalized super kdv equations we propose a super lax type equation based on a certain class of lie superalgebra as a supersymmetric extension of generalized modified kdv hierarchy we are able to construct an infinite set of conservation laws and the consistent time evolution generators for generalized modified super kdv equations thefirst few of the conserved currents the modified super kdv equation and the super miura transformation are worked out explicitly in the case of twisted affine lie superalgebraosp 2 2 2

the extended supersymmetrization of the multicomponent kadomtsev petviashvilli hierarchy we describe three different approaches to the extended n 2 supersymmetrization of the multicomponent kp hierarchy in the first one we utilize only superfermions while in the second only superbosons and in the third superbosons as well as superfermions it is shown that many soliton equations can be embedded in the supersymmetry theory by using the first approach even if we do not change these equations in the bosonic limit of the supersymmetry in the second or third approach we obtain a generalization of the soliton equations in the bosonic limit which remains in the class of the usual commuting functions as a byproduct of our analysis we prove that for the first procedure the bosonic part of the one component supersymmetric kp hierarchy coincides with the usual classical two component kp hierarchy

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

spectral amge rho amge we introduce spectral element based algebraic multigrid rho amge a new algebraic multigrid method for solving systems of algebraic equations that arise in ritz type finite element discretizations of partial differential equations the method requires access to the element stiffness matrices which enables accurate approximation of algebraically smooth vectors i e error components that relaxation cannot effectively eliminate most other algebraic multigrid methods are based in some manner on predefined concepts of smoothness coarse grid selection and prolongation for example are often defined assuming that smooth errors vary slowly in the direction of strong connections relatively large coefficients in the operator matrix one aim of rho amge is to broaden the range of problems to which the method can be successfully applied by avoiding any implicit premise about the nature of the smooth error rho amge uses the spectral decomposition of small collections of element stiffness matrices to determine local representations of algebraically smooth error components this provides a foundation for generating the coarse level and for defining effective interpolation this paper presents a theoretical foundation for rho amge along with numerical experiments demonstrating its robustness

convergence analysis of multigrid methods with residual scaling techniques in this paper multigrid methods with residual scaling techniques for symmetric positive definite linear systems are considered the idea of perturbed two grid methods proposed in 7 is used to estimate the convergence factor of multigrid methods with residual scaled by positive constant scaling factors we will show that if the convergence factors of the two grid methods are uniformly bounded by s s 0 5 then the convergence factors of the w cycle multigrid methods are uniformly bounded by s email protected whether the residuals are scaled at some or all levels this result extends notay s theorem 3 1 in 7 to more general cases the result also confirms the viewpoint that the w cycle multigrid method will converge sufficiently well as long as the convergence factor of the two grid method is small enough in the case where the convergence factor of the two grid method is not small enough by appropriate choice of the cycle index c we can guarantee that the convergence factor of the multigrid methods with residual scaling techniques still has a uniform bound less than s email protected numerical experiments are provided to show that the performance of multigrid methods can be improved by scaling the residual with a constant factor the convergence rates of the two grid methods and the multigrid methods show that the w cycle multigrid methods perform better if the convergence rate of the two grid method becomes smaller these numerical experiments support the proposed theoretical results in this paper

supersymmetric two dimensional toda lattice the two dimensional toda lattice connected with contragradient lie superalgebras is studied the systems of linear equations associated with the models for which the inverse scattering method is applicable are written down the reduction group is calculated

solvability of the super kp equation and a generalization of the birkhoff decomposition the unique solvability of the initial value problem for the total hierarchy of the super kadomtsev petviashvili system is established to prove the existence we use a generalization of the birkhoff decomposition which is obtained by replacing the loop variable and loop groups in the original setting by a super derivation operator and groups of infinite order super micro i e pseudo differential operators to show the uniqueness we generalize the fact that every flat connection admits horizontal sections to the case of an infinite dimensional super algebra bundle defined over an infinite dimensional super space the usual kp system with non commutative coefficients is also studied the kp system is obtained from the super kp system by reduction modulo odd variables on the other hand the first modified kp equation can be obtained from the super kp system by elimination of odd variables thus the super kp system is a natural unification of the kp system and the modified kp systems

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

global basis two level method for indefinite systems part 1 convergence studies a robust two level solver for high indefinite system of equations arising from the finite element discretization is developed it is shown that the optimal coarse model is spanned by the spectrum of the highest eigenmodes of the smoothing iteration matrix convergence studies conducted on a model prolongation operator reveal pathological sensitivity to any deviation from the optimal coarse model copyright 2000 john wiley sons ltd

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

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

density of zariski density for surface groups we show that a surface group contained in a reductive real algebraic group can be deformed to become zariski dense unless its zariski closure acts transitively on a hermitian symmetric space of tube type this is a kind of converse to a rigidity result of burger iozzi and wienhard

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

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

an approach to generate superextensions of integrable systems an algorithm to generate integrable systems is extended to the super case some new examples of superextensions of integrable systems are illustrated we also generalize the trace identity due to tu to the super case and use it to establish hamiltonian structures of superextensions of integrable systems under consideration

uniform convergence of multigrid v cycle iterations for indefinite and nonsymmetric problems in this paper an analysis of a multigrid method for nonsymmetric and or indefinite elliptic problems is presented in this multigrid method various types of smoothers may be used one type of smoother considered is defined in terms of an associated symmetric problem and includes point and line jacobi and gauss seidel iterations smoothers based entirely on the original operator are also considered one smoother is based on the normal form that is the product of the operator and its transpose other smoothers studied include point and line jacobi and gauss seidel it is shown that the uniform estimates of j h bramble and j e pasciak math comp 60 1993 pp 447 471 for symmetric positive definite problems carry over to these algorithms more precisely the multigrid iteration for the nonsymmetric and or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine but not dependent on the number of multigrid levels

division algebras and extended n 2 4 8 superkdvs a no go result for integrable minimal n 8 supersymmetric extensions of kdv is found however allowing for non associative realizations of the extended supersymmetries the first example of an n 8 supersymmetric kdv equation is explicitly constructed it involves eight bosonic and eight fermionic fields and corresponds to the unique n 8 solution based on a generalized hamiltonian dynamics with generalized poisson brackets given by the non associative n 8 superconformal algebra the complete list of inequivalent classes of parametric dependent n 3 and n 4 superkdvs obtained from the non associative n 8 sca is also furnished furthermore a fundamental domain characterizing the class of inequivalent n 4 superkdvs based on the minimal n 4 sca is given

the gromov norm of the kaehler class of symmetric domains let d be bounded symmetric domain of rank p and let o9 be the krhler form of its bergmann metric where the metric is normalized so that the minimum holomorphic sectional curvature is 1 i fx is a compact manifold whose universal cover is d to defines a cohomology class to e h 2 x r the purpose of this paper is to compute its sup norm in the sense of gromov 5 we show ii to ii co pn strictly speaking we only prove this for three of the four classes of classical domains cf sect 2 this theorem has the following topological corollary let s be a riemann surface of genus 0 1 and f s x a continuous map then f t o 4 p 1 n

a parallel block multi level preconditioner for the 3d incompressible navier stokes equations the development of robust and efficient algorithms for both steady state simulations and fully implicit time integration of the navier stokes equations is an active research topic to be effective the linear subproblems generated by these methods require solution techniques that exhibit robust and rapid convergence in particular they should be insensitive to parameters in the problem such as mesh size time step and reynolds number in this context we explore a parallel preconditioner based on a block factorization of the coefficient matrix generated in an oseen nonlinear iteration for the primitive variable formulation of the system the key to this preconditioner is the approximation of a certain schur complement operator by a technique first proposed by kay loghin and wathen siam j sci comput 2002 and silvester elman kay and wathen j comput appl math 128 2001 261 the resulting operator entails subsidiary computations solutions of pressure poisson and convection diffusion subproblems that are similar to those required for decoupled solution methods however in this case these solutions are applied as preconditioners to the coupled oseen system one important aspect of this approach is that the convection diffusion and poisson subproblems are significantly easier to solve than the entire coupled system and a solver can be built using tools developed for the subproblems in this paper we apply smoothed aggregation algebraic multigrid to both subproblems previous work has focused on demonstrating the optimality of these preconditioners with respect to mesh size on serial two dimensional steady state computations employing geometric multi grid methods we focus on extending these methods to large scale parallel three dimensional transient and steady state simulations employing algebraic multigrid amg methods our results display nearly optimal convergence rates for steady state solutions as well as for transient solutions over a wide range of cfl numbers on the two dimensional and three dimensional lid driven cavity problem

multigrid methods for variational problems this paper develops a very simple but powerful theory for multigrid methods which applies directly to variationally posed operator equations

surface group representations and u p q higgs bundles using the l2 norm of the higgs field as a morse function we study the moduli spaces of u p q higgs bundles over a riemann surface we require that the genus of the surface be at least two but place no constraints on p q a key step is the identification of the function s local minima as moduli spaces of holomorphic triples in a companion paper 7 we prove that these moduli spaces of triples are nonempty and irreducible because of the relation between flat bundles and fundamental group representations we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple pu p q representations the topological invariants of the flat bundles are used to label subspaces these invariants are bounded by a milnor wood type inequality for each allowed value of the invariants satisfying a certain coprimality condition we prove that the corresponding subspace is nonempty and connected if the coprimality condition does not hold our results apply to the closure of the moduli space of irreducible representations 1 members of vbac vector bundles on algebraic curves which is partially supported by eager ec fp5 contract no hprn ct 2000 00099 and by edge ec fp5 contract no hprn ct 2000 00101 2 partially supported by the national science foundation under grant dms0072073 3 partially supported by the ministerio de ciencia y tecnolo ia spain under grant bfm2000 0024 4 partially supported by the fundacao para a ciencia e a tecnologia portugal through the centro de matematica da universidade do porto and through grant no sfrh bpd 1606 2000 5 partially supported by the portugal spain bilateral programme acciones integradas grant nos hp2000 0015 and ai 01 24 6 partially supported by a british epsrc grant october december 2001 received 11 27 2002

acceleration of the generalized global basis ggb method for nonlinear problems two heuristic strategies intended to enhance the performance of the generalized global basis ggb method h waisman j fish r s tuminaro j shadid the generalized global basis ggb method international journal for numerical methods in engineering 61 8 1243 1269 applied to nonlinear systems are presented the standard ggb accelerates a multigrid scheme by an additional coarse grid correction that filters out slowly converging modes this correction requires a potentially costly eigen calculation this paper considers reusing previously computed eigenspace information the ggb a scheme enriches the prolongation operator with new eigenvectors while the modified method mggb selectively reuses the same prolongation both methods use the criteria of principal angles between subspaces spanned between the previous and current prolongation operators numerical examples clearly indicate significant time savings in particular for the mggb scheme

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

an improved algebraic multigrid method for solving maxwell s equations we propose two improvements to the reitzinger and schoberl algebraic multigrid amg method for solving the eddy current approximations to maxwell s equations the main focus in the reitzinger schoberl method is to maintain null space properties of the weak nabla times nabla times operator on coarse grids while these null space properties are critical they are not enough to guarantee h independent convergence of the overall multigrid method we illustrate how the reitzinger schoberl amg method loses h independence due to the somewhat limited approximation property of the grid transfer operators we present two improvements to these operators that not only maintain the important null space properties on coarse grids but also yield significantly improved multigrid convergence rates the first improvement is based on smoothing the reitzinger schoberl grid transfer operators the second improvement is obtained by using higher order nodal interpolation to derive the corresponding amg interpolation operators while not completely h independent the resulting amg cg method demonstrates improved convergence behavior while maintaining low operator complexity

supersymmetry for integrable hierarchies on loop superalgebras an algebraic approach is employed to formulate n 2 supersymmetry transformations in the context of integrable systems based on loop superalgebras with homogeneous gradation we work with extended integrable hierarchies which contain supersymmetric akns and lund regge sectors we derive the one soliton solution for p 1 which solves positive and negative evolution equations of the n 2 supersymmetric model

recursively accelerated multilevel aggregation for markov chains a recursive acceleration method is proposed for multiplicative multilevel aggregation algorithms that calculate the stationary probability vector of large sparse and irreducible markov chains pairs of consecutive iterates at all branches and levels of a multigrid w cycle with simple nonoverlapping aggregation are recombined to produce improved iterates at those levels this is achieved by solving quadratic programming problems with inequality constraints the linear combination of the two iterates is sought that has a minimal two norm residual under the constraint that all vector components are nonnegative it is shown how the two dimensional quadratic programming problems can be solved explicitly in an efficient way the method is further enhanced by windowed top level acceleration of the w cycles using the same constrained quadratic programming approach recursive acceleration is an attractive alternative to smoothing the restriction and interpolation operators since the operator complexity is better controlled and the probabilistic interpretation of coarse level operators is maintained on all levels numerical results are presented showing that the resulting recursively accelerated multilevel aggregation cycles for markov chains combined with top level acceleration converge significantly faster than w cycles and lead to close to linear computational complexity for challenging test problems

on the fly adaptive smoothed aggregation multigrid for markov chains a new adaptive algebraic multigrid scheme is developed for the solution of markov chains where the hierarchy of operators is adapted on the fly in a setup process that is interlaced with the solution process the setup process feeds the solution process with improved operators while the solution process provides the adaptive setup process with better approximations on which to base further improved operators the approach is demonstrated using petrov galerkin smoothed aggregation where only the prolongation operator is smoothed while the restriction remains of low order results show that the on the fly adaptive scheme can improve the performance of multigrid solvers that require extensive setup computations in both serial and parallel environments