a hybridizable discontinuous galerkin method for stokes flow abstract in this paper we introduce a hybridizable discontinuous galerkin method for stokes flow the method is devised by using the discontinuous galerkin methodology to discretize a velocity pressure gradient formulation of the stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization one of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries thereby leading to a significant reduction in the size of the resulting matrix moreover by using an augmented lagrangian method the globally coupled unknowns are further reduced to the numerical trace of the velocity only another important feature is that the approximations of the velocity pressure and gradient converge with the optimal order of k 1 in the l 2 norm when polynomials of degree k 0 are used to represent the approximate variables based on the optimal convergence of the hdg method we apply an element by element postprocessing scheme to obtain a new approximate velocity which converges with order k 2 in the l 2 norm for k 1 the postprocessing performed at the element level is less expensive than the solution procedure numerical results are provided to assess the performance of the method

local discontinuous galerkin methods for the stokes system in this paper we introduce and analyze local discontinuous galerkin methods for the stokes system for a class of shape regular meshes with hanging nodes we derive a priori estimates for the l2 norm of the errors in the velocities and the pressure we show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k 1 for the pressure for any k ge1 we also consider the case in which all the unknowns are approximated with polynomials of degree k and show that although the orders of convergence remain the same the method is more efficient numerical experiments verifying these facts are displayed

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

smoothed aggregation multigrid solvers for high order discontinuous galerkin methods for elliptic problems abstract we develop a smoothed aggregation based algebraic multigrid solver for high order discontinuous galerkin discretizations of the poisson problem algebraic multigrid is a popular and effective method for solving the sparse linear systems that arise from discretizing partial differential equations however high order discontinuous galerkin discretizations have proved challenging for algebraic multigrid the increasing condition number of the matrix and loss of locality in the matrix stencil as p increases in addition to the effect of weakly enforced dirichlet boundary conditions all contribute to the challenging algebraic setting we propose a smoothed aggregation approach that addresses these difficulties in particular the approach effectively coarsens degrees of freedom centered at the same spatial location as well as degrees of freedom at the domain boundary moreover the character of the near null space particularly at the domain boundary is captured by interpolation one classic prolongation smoothing step of weighted jacobi is also shown to be ineffective at high order and a more robust energy minimization approach is used along with block relaxation that more directly utilizes the block diagonal structure of the discontinuous galerkin discretization finally we conclude by examining numerical results in support our proposed method

optimal a priori error estimates for the hp version of the local discontinuous galerkin method for convection diffusion problems we study the convergence properties of the hp version of the local discontinuous galerkin finite element method for convection diffusion problems we consider a model problem in a one dimensional space domain we allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh width h in the polynomial degree p and in the regularity of the exact solution we identify a special numerical flux for which the estimates are optimal in both h and p the theoretical results are confirmed in a series of numerical examples

correspondence authors reply to comments on state estimation for linear systems with state equality constraints automatica 43 2007 1363 1368

discrete time minmax filtering subject to a norm constrained state estimate abstract this paper considers discrete time minmax filtering when the state estimate or part of the state estimate is subjected to a norm constraint derivation and use of minmax filters are motivated by the desire to estimate systems states in a more robust manner there are various methods for enforcing constraints on the state estimate using the kalman filter however enforcing constraints on the state estimate in the minmax framework has been less explored this paper presents the derivation of a minmax filter for a discrete time system with a full or partial equality norm constraint on the state estimate this is accomplished by modifying the minmax filter s optimization problem using lagrange multipliers to enforce the constraint on the state estimates

preconditioning methods for local discontinuous galerkin discretizations a multilevel interior penalty method is used as an efficient preconditioner for the schur complement of the local discontinuous galerkin ldg discretization of a poisson problem the method is then used in a block triangular preconditioner of the ldg saddle point system the block preconditioner is of the same efficiency as the schur complement version finally the block preconditioner is extended to the discretization of the stokes problem by the ldg method again the preconditioned saddle point problem can be solved in about as many steps as the schur complement the influence of several parameters on the performance of these methods is investigated

brief paper state estimation for linear systems with state equality constraints this paper deals with the state estimation problem for linear systems with linear state equality constraints using noisy measurements which are available from the observable system we construct the optimal estimate which also satisfies linear equality constraints for this purpose after reviewing modeling problems in linear stochastic systems with state equality constraints we formulate a projected system representation by using the constrained kalman filter for the projected system and comparing its filter riccati equation with those of the unconstrained and the projected kalman filters we clearly show without using optimality that the constrained estimator outperforms the other filters for estimating the constrained system state finally a numerical example is presented which demonstrates performance differences among those filters

target tracking problems subject to kinematic constraints filtering problems with kinematic constraints which may arise in target tracking problems are considered a novel approach which treats kinematic constraints as additional fictitious or pseudomeasurements is proposed a numerical example is provided to show the technical feasibility of the proposed idea for target tracking problems this example shows that the proposed method can improve estimation accuracy significantly for velocity and acceleration states in the tracking problem however it is noted that the tracking performance may be deteriorated if the constraints do not properly represent the target characteristics and a small r sub c is chosen

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

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

extremal solutions of the two dimensionall problem of moments by using the theory of the principal function of a hyponormal operator with rank one self commutator one classifies some extremal points of the solutions of the truncatedl problem of moments in two real variables these extremal elements called degenerated solutions of thel problem are proved to be in a natural bijective correspondence with the bounded quadrature domains in the complex plane the main result of the paper characterizes these degenerated solutions from three distinct perspectives complex analysis moment problems and operator theory the interplay between the three points of view each with specific techniques constitutes the main body of the paper

algebraic multigrid for higher order finite elements two related approaches for solving linear systems that arise from a higher order finite element discretization of elliptic partial differential equations are described the first approach explores direct application of an algebraic based multigrid method amg to iteratively solve the linear systems that result from higher order discretizations while the choice of basis used on the discretization has a significant impact on the performance of the solver results indicate that amg is capable of solving operators from both poisson s equation and a first order system least squares fosls formulation of stoke s equation in a scalable manner nearly independent of basis order p for 3

high resolution viscous fingering simulation in miscible displacement using a p adaptive discontinuous galerkin method with algebraic multigrid preconditioner abstract high resolution simulation of viscous fingering can offer an accurate and detailed prediction for subsurface engineering processes involving fingering phenomena the fully implicit discontinuous galerkin dg method has been shown to be an accurate and stable method to model viscous fingering with high peclet number and mobility ratio in this paper we present two techniques to speedup large scale simulations of this kind the first technique relies on a simple p adaptive scheme in which high order basis functions are employed only in elements near the finger fronts where the concentration has a sharp change as a result the number of degrees of freedom is significantly reduced and the simulation yields almost identical results to the more expensive simulation with uniform high order elements throughout the mesh the second technique for speedup involves improving the solver efficiency we present an algebraic multigrid amg preconditioner which allows the dg matrix to leverage the robust amg preconditioner designed for the continuous galerkin cg finite element method the resulting preconditioner works effectively for fixed order dg as well as p adaptive dg problems with the improvements provided by the p adaptivity and amg preconditioning we can perform high resolution three dimensional viscous fingering simulations required for miscible displacement with high peclet number and mobility ratio in greater detail than before for well injection problems

two grid method for linear elasticity on unstructured meshes we propose an abstract two grid algorithm with convergence independent of the coarse space size the abstract algorithm is applied to problems of three dimensional linear elasticity discretized on unstructured meshes with no regularity assumptions we prove uniform convergence with respect to coarse space size domain essential boundary conditions and jumps in young modulus numerical experiments confirm the theory and show that the method works well even if some assumptions of the theory are violated

continuous time norm constrained kalman filtering this paper considers continuous time state estimation when part of the state estimate or the entire state estimate is norm constrained in the former case continuous time state estimation is considered by posing a constrained optimization problem the optimization problem can be broken up into two separate optimization problems one which solves for the optimal observer gain associated with the unconstrained state estimates while the other solves for the optimal observer gain associated with the constrained state estimates the optimal constrained state estimate is found by projecting the time derivative of an unconstrained estimate onto the tangent space associated with the norm constraint the special case where the entire state estimate is norm constrained is briefly discussed the utility of the filtering results developed are highlighted through a spacecraft attitude estimation example numerical simulation results are included

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

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

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

deflation accelerated preconditioning of the poisson neumann schur problem on long domains with a high order discontinuous element based collocation method a combination of block jacobi and deflation preconditioning is used to solve a high order discontinuous element based collocation discretization of the schur complement of the poisson neumann system as arises in the operator splitting of the incompressible navier stokes equations the preconditioners and deflation vectors are chosen to mitigate the effects of ill conditioning due to highly elongated domains typical of simulations of strongly non hydrostatic environmental flows and to achieve generalized minimum residual method gmres convergence independent of the size of the number of elements in the long direction the ill posedness of the poisson neumann system manifests as an inconsistency of the schur complement problem but it is shown that this can be accounted for with appropriate projections out of the null space of the schur complement matrix without affecting the accuracy of the solution the block jacobi preconditioner is shown to yield gmres convergence independent of the polynomial order and only weakly dependent on the number of elements within a subdomain in the decomposition the combined deflation and block jacobi preconditioning is compared with two level non overlapping block jacobi preconditioning of the schur problem and while both methods achieve convergence independent of the grid size deflation is shown to require half as many gmres iterations and 25 less wall clock time for a variety of grid sizes and domain aspect ratios the deflation methods shown to be effective for the two dimensional poisson neumann problem are extensible to the three dimensional problem assuming a fourier discretization in the third dimension a fourier discretization results in a two dimensional helmholtz problem for each fourier component that is solved using deflated block jacobi preconditioning on its schur complement here again deflation is shown to be superior to two level non overlapping block jacobi preconditioning requiring about half as many gmres iterations and 15 less time

use of a kinematic constraint in tracking constant speed maneuvering targets the use of a kinematic constraint as a pseudomeasurement in the tracking of constant speed maneuvering targets is considered the kinematic constraint provides additional information about the target motion that can be processed as a pseudomeasurement to improve tracking performances a new formulation of the constraint equation is presented and the rationale for the new formulation is discussed the filter using the kinematic constraint as a pseudomeasurement is shown to be unbiased and sufficient conditions for stochastic stability of the filter are given simulated tracking results are given to demonstrate the potential that the new formulation provides for improving tracking performance

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

adaptive aggregation based domain decomposition multigrid for the lattice wilson dirac operator in lattice quantum chromodynamics qcd computations a substantial amount of work is spent in solving discretized versions of the dirac equation conventional krylov solvers show critical slowing down for large system sizes and physically interesting parameter regions we present a domain decomposition adaptive algebraic multigrid method used as a preconditioner to solve the clover improved wilson discretization of the dirac equation this approach combines and improves two approaches namely domain decomposition and adaptive algebraic multigrid that have been used separately in lattice qcd before we show in extensive numerical tests conducted with a parallel production code implementation that considerable speedup can be achieved compared to conventional krylov subspace methods domain decomposition methods and other hierarchical approaches for realistic system sizes

a hybrid discontinuous galerkin method for elliptic problems a new family of hybrid discontinuous galerkin methods is studied for second order elliptic equations our proposed method is a generalization of the cell boundary element cbe method y jeon and e j park appl numer math 58 2008 pp 800 814 which allows high order polynomial approximations our method can be viewed as a hybridizable discontinuous galerkin method b cockburn j gopalakrishnan and r lazarov siam j numer anal 47 2009 pp 1319 1365 using a bauman oden type local solver the method conserves the mass in each element and the average flux is continuous across the interelement boundary for even degree polynomial approximations optimal order error estimates measured in the energy norm are proved numerical examples are presented to show the performance of the method

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

two level fourier analysis of a multigrid approach for discontinuous galerkin discretization in this paper we study a multigrid mg method for the solution of a linear second order elliptic equation discretized by discontinuous galerkin dg methods and we give a detailed analysis of the convergence for different block relaxation strategies we find that pointwise block partitioning gives much better results than the classical cellwise partitioning both for the baumann oden method and for the symmetric dg method with and without interior penalty ip the block relaxation methods jacobi gauss seidel and symmetric gauss seidel give excellent smoothing procedures in a classical mg setting independent of the mesh size simple mg cycles give convergence factors of 0 075 0 4 per iteration sweep for the different discretization methods studied

performance of discontinuous galerkin methods for elliptic pdes in this paper we compare the performance of several discontinuous galerkin dg methods for elliptic partial differential equations pdes on a model problem theoretical estimates of the condition number of the stiffness matrix are given for dg methods whose bilinear form is symmetric and which are shown to be numerically sharp then the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes

smoothed aggregation multigrid for the discontinuous galerkin method the aim of this paper is to investigate theoretically as well as experimentally an algebraic multilevel algorithm for the solution of the linear systems that arise from the discontinuous galerkin method the smoothed aggregation multigrid introduced by vanek for the conforming finite element method is applied to low order discretizations of convection diffusion equations for the elliptic model problem the algorithm is shown to be quasi optimal adjustments for the case of nonvanishing advection such as directionally implicit smoothing and a suitable splitting of the operator are discussed several numerical experiments are presented for two dimensional problems including a newton type linearization of the compressible navier stokes equations

an adaptive algebraic multigrid algorithm for low rank canonical tensor decomposition a new algorithm based on algebraic multigrid is presented for computing the rank r canonical decomposition of a tensor for small r standard alternating least squares als is used as the relaxation method transfer operators and coarse level tensors are constructed in an adaptive setup phase that combines multiplicative correction and bootstrap algebraic multigrid an accurate solution is computed by an additive solve phase based on the full approximation scheme numerical tests show that for certain test problems our multilevel method significantly outperforms standalone als when a high level of accuracy is required

block triangular preconditioners for saddle point problems with a penalty term block triangular preconditioners for a class of saddle point problems with a penalty term are considered an important example is the mixed formulation of the pure displacement problem in linear elasticity it is shown that the spectrum of the preconditioned system is contained in a real positive interval and that the interval bounds can be made independent of the discretization and penalty parameters this fact is used to construct bounds of the convergence rate of the gmres method with respect to an energy norm numerical results are given for gmres and bi cgstab

robust smoothers for high order discontinuous galerkin discretizations of advection diffusion problems the multigrid method for discontinuous galerkin discretizations of advection diffusion problems is presented it is based on a block gauss seidel smoother with downwind ordering honoring the advection operator the cell matrices of the dg scheme are inverted in this smoother in order to obtain robustness for higher order elements employing a set of experiments we show that this technique actually yields an efficient preconditioner and that both ingredients downwind ordering and blocking of cell matrices are crucial for robustness discretizations of partial differential equations by higher order finite element methods or hp methods gained popularity during the last years after computing power and memory became readily available these methods yield much higher accuracy with similar degrees of freedom than standard schemes based on linear or bilinear polynomials nevertheless the utility of these schemes depends very much on the availability of efficient solvers for the discrete problems in this article we continue previous work and demonstrate that multilevel methods with suitably chosen smoothers can provide such solvers even in the presence of non self adjoint advection terms and locally refined meshes in 8 we devised a multilevel preconditioner for the symmetric interior penalty method and proved it s optimality these results were extended to other adjoint

a multilevel discontinuous galerkin method a variable v cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented an analysis under a mild regularity assumption shows that the preconditioner is uniform the interior penalty method is then combined with a discontinuous galerkin scheme to arrive at a discretization scheme for an advection diffusion problem for which an error estimate is proved a multigrid algorithm for this method is presented and numerical experiments indicating its robustness with respect to diffusion coefficient are reported

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

superconvergence of the local discontinuous galerkin method for elliptic problems on cartesian grids in this paper we present a superconvergence result for the local discontinuous galerkin ldg method for a model elliptic problem on cartesian grids we identify a special numerical flux for which the l2 norm of the gradient and the l2 norm of the potential are of orders k 1 2 and k 1 respectively when tensor product polynomials of degree at most k are used for arbitrary meshes this special ldg method gives only the orders of convergence of k and k 1 2 respectively we present a series of numerical examples which establish the sharpness of our theoretical results

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

a feti dp preconditioner for a composite finite element and discontinuous galerkin method in this paper a nitsche type discretization based on a discontinuous galerkin dg method for an elliptic two dimensional problem with discontinuous coefficients is considered the problem is posed on a polygonal region omega which is a union of n disjoint polygonal subdomains omega i of diameter o h i the discontinuities of the coefficients possibly very large are assumed to occur only across the subdomain interfaces partial omega i inside each subdomain a conforming finite element space on a quasi uniform triangulation with mesh size o h i is introduced to handle the nonmatching meshes across the subdomain interfaces a dg discretization is applied only on the interfaces for solving the resulting discrete system a feti dp type method is proposed and analyzed it is established that the condition number of the preconditioned linear system is estimated by c 1 max i log h i h i 2 with a constant c independent of h i h i h j h i and the jumps of coefficients the

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 multigrid method with guaranteed convergence rate we consider the iterative solution of large sparse symmetric positive definite linear systems we present an algebraic multigrid method which has a guaranteed convergence rate for the class of nonsingular symmetric m matrices with nonnegative row sum the coarsening is based on the aggregation of the unknowns a key ingredient is an algorithm that builds the aggregates while ensuring that the corresponding two grid convergence rate is bounded by a user defined parameter for a sensible choice of this parameter it is shown that the recursive use of the two grid procedure yields a convergence independent of the number of levels provided that one uses a proper amli cycle on the other hand the computational cost per iteration step is of optimal order if the mean aggregate size is large enough this cannot be guaranteed in all cases but is analytically shown to hold for the model poisson problem for more general problems a wide range of experiments suggests that there are no complexity issues and further demonstrates the robustness of the method the experiments are performed on systems obtained from low order finite difference or finite element discretizations of second order elliptic partial differential equations pdes the set includes two and three dimensional problems with both structured and unstructured grids some of them with local refinement and or reentering corner and possible jumps or anisotropies in the pde coefficients

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

a characterization of k hyponormality via weak subnormality abstract an operator t acting on a hilbert space h is said to be weakly subnormal if there exists an extension t acting on k h such that t t f t t f for all f h when such partially normal extensions exist we denote by m p n e t the minimal one on the other hand for k 1 t is said to be k hyponormal if the operator matrix t j t i i j 1 k is positive we prove that a 2 hyponormal operator t always satisfies the inequality t t t t t 2 t t and as a result t is automatically weakly subnormal thus a hyponormal operator t is 2 hyponormal if and only if there exists b such that ba a t and t a 0 b is hyponormal where a t t 1 2 more generally we prove that t is k 1 hyponormal if and and only if t is weakly subnormal and m p n e t is k hyponormal as an application we obtain a matricial representation of the minimal normal extension of a subnormal operator as a block staircase matrix

adaptive algebraic multigrid methods our ability to simulate physical processes numerically 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 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 the principles which guide the adaptivity are highlighted as well as their application to algebraic multigrid solution of certain symmetric positive definite linear 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

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

hyponormal operators with rank two self commutators abstract in this paper it is shown that if t l h satisfies i t is a pure hyponormal operator ii t t is of rank two and iii ker t t is invariant for t then t is either a subnormal operator or the putinar s matricial model of rank two more precisely if t ker t t has a rank one self commutator then t is subnormal and if instead t ker t t has a rank two self commutator then t is either a subnormal operator or the kth minimal partially normal extension t k k of a k 1 hyponormal operator t k which has a rank two self commutator for any k z hence in particular every weakly subnormal or 2 hyponormal operator with a rank two self commutator is either a subnormal operator or a finite rank perturbation of a k hyponormal operator for any k z

operator theory and algebraic geometry on presente des resultats en theorie des operateurs multivariable dont les demonstrations se relient a des techniques issues de la geometrie algebrique

a new method for the nonlinear transformation of means and covariances in filters and estimators this paper describes a new approach for generalizing the kalman filter to nonlinear systems a set of samples are used to parametrize the mean and covariance of a not necessarily gaussian probability distribution the method yields a filter that is more accurate than an extended kalman filter ekf and easier to implement than an ekf or a gauss second order filter its effectiveness is demonstrated using an example

on kalman filtering with nonlinear equality constraints the state space description of some physical systems possess nonlinear equality constraints between some state variables in this paper we consider the problem of applying a kalman filter type estimator in the presence of such constraints we categorize previous approaches into pseudo observation and projection methods and identify two types of constraints those that act on the entire distribution and those that act on the mean of the distribution we argue that the pseudo observation approach enforces neither type of constraint and that the projection method enforces the first type of constraint only we propose a new method that utilizes the projection method twice once to constrain the entire distribution and once to constrain the statistics of the distribution we illustrate these algorithms in a tracking system that uses unit quaternions to encode orientation

norm preconditioners for discontinuous galerkin hp finite element methods we consider a norm preconditioning approach for the solution of discontinuous galerkin finite element discretizations of second order partial differential equations with a nonnegative characteristic form our solution method is a norm preconditioned three term gmres routine we find that for symmetric positive definite diffusivity tensors the convergence of our solver is independent of discretization while for the semidefinite case both theory and experiment indicate dependence on both h and p numerical results are included to illustrate performance on several test cases

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

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

correspondence comments on state estimation for linear systems with state equality constraints automatica 43 2007 1363 1368 the state estimation problem for linear systems with linear state equality constraints was dealt with in ko bitmead ko s bitmead r 2007 state estimation for linear systems with state equality constraints automatica 43 1363 1368 in this correspondence it is first shown that a necessary assumption on the covariance of the process noise is missing in the main result of the paper it is then shown that the main result of the paper can be achieved in a convenient and more general way without any additional assumptions on the covariance of the process noise except positive definiteness

linear analysis of quadrature domains iii the linear data of a quadrature domain introduced in the first paper in this series is further investigated the matricial structure of all hyponormal operators with finite rank self commutators which is inductively constructed from such linear data is discussed in particular a conservation law of a determinantal function in the iterative process of defining these matrices is exploited the spectrum of the determining left upper corner of some of these operators is analyzed in connection with the existence of generalized quadrature formulas on planar domains

a numerical study of a semi algebraic multilevel preconditioner for the local discontinuous galerkin method in this work we consider the local discontinuous galerkin ldg method applied to second order elliptic problems arising in the modeling of single phase flows in porous media it has been recently proven that the spectral condition number of the stiffness matrix exhibits an asymptotic behavior of o h 2 on structured and unstructured meshes where h is the mesh size thus efficient preconditioners are mandatory we present a semi algebraic multilevel preconditioner for the ldg method using local lagrange type interpolatory basis functions we show numerically that its performance does not degrade or at least the number of iterations increases very slowly as the number of unknowns augments the preconditioner is tested on problems with high jumps in the coefficients which is the typical scenario of problems arising in porous media

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

stencil reduction algorithms for the local discontinuous galerkin method the problem of reducing the stencil of the local discontinuous galerkin method applied to second order differential operator is discussed heuristic algorithms to minimize the total number of non zero blocks of the reduced stiffness matrix are presented and tested on a wide variety of unstructured and structured grids in 2d and 3d copyright 2009 john wiley sons ltd

truly monolithic algebraic multigrid for fluid structure interaction the coupling of flexible structures to incompressible fluids draws a lot attention during the last decade many different solution schemes have been proposed in this contribution we concentrate on strong coupling fluid structure interaction by means of monolithic solution schemes therein a newton krylov method is applied to the monolithic set of nonlinear equations such schemes require good preconditioning to be efficient we propose two preconditioners that apply algebraic multigrid techniques to the entire fluid structure interaction system of equations the first is based on a standard block gauss seidel approach where approximate inverses of the individual field blocks are based on a algebraic multigrid hierarchy tailored for the type of the underlying physical problem the second is based on a monolithic coarsening scheme for the coupled system that makes use of prolongation and restriction projections constructed for the individual fields the resulting nonsymmetric monolithic algebraic multigrid method therefore involves coupling of the fields on coarse approximations to the problem yielding significantly enhanced performance

a multiscale restriction smoothed basis method for high contrast porous media represented on unstructured grids a wide variety of multiscale methods have been proposed in the literature to reduce runtime and provide better scaling for the solution of poisson type equations modeling flow in porous media we present a new multiscale restricted smoothed basis msrsb method that is designed to be applicable to both rectilinear grids and unstructured grids like many other multiscale methods msrsb relies on a coarse partition of the underlying fine grid and a set of local prolongation operators multiscale basis functions that map unknowns associated with the fine grid cells to unknowns associated with blocks in the coarse partition these mappings are constructed by restricted smoothing starting from a constant a localized iterative scheme is applied directly to the fine scale discretization to compute prolongation operators that are consistent with the local properties of the differential operators the resulting method has three main advantages first of all both the coarse and the fine grid can have general polyhedral geometry and unstructured topology this means that partitions and good prolongation operators can easily be constructed for complex models involving high media contrasts and unstructured cell connections introduced by faults pinch outs erosion local grid refinement etc in particular the coarse partition can be adapted to geological or flow field properties represented on cells or faces to improve accuracy secondly the method is accurate and robust when compared to existing multiscale methods and does not need expensive recomputation of local basis functions to account for transient behavior dynamic mobility changes are incorporated by continuing to iterate a few extra steps on existing basis functions this way the cost of updating the prolongation operators becomes proportional to the amount of change in fluid mobility and one reduces the need for expensive tolerance based updates finally since the msrsb method is formulated on top of a cell centered conservative finite volume method it is applicable to any flow model in which one can isolate a pressure equation herein we only discuss single and two phase incompressible models compressible flow e g as modeled by the black oil equations is discussed in a separate paper

the local discontinuous galerkin method for the oseen equations we introduce and analyze the local discontinuous galerkin method for the oseen equations of incompressible fluid flow for a class of shape regular meshes with hanging nodes we derive optimal a priori estimates for the errors in the velocity and the pressure in l 2 and negative order norms numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of reynolds numbers

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

robust multigrid for high order discontinuous galerkin methods we present a polynomial multigrid method for nodal interior penalty and local discontinuous galerkin formulations of the poisson equation on cartesian grids for smoothing we propose two classes of overlapping schwarz methods the first class comprises element centered and the second face centered methods within both classes we identify methods that achieve superior convergence rates prove robust with respect to the mesh spacing and the polynomial order at least up to p 32 consequent structure exploitation yields a computational complexity of o pn where n is the number of unknowns further we demonstrate the suitability of the face centered method for element aspect ratios up to 32 p multigrid for discontinuous formulations of poisson problems on cartesian grids robust with respect to the mesh size and polynomial order at least up to p 32 convergence rates between 0 02 and 0 003 improving with increasing p computational complexity of pn for n unknowns linear complexity in runtime excellent performance up to aspect ratios of 1 16

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

a nearly optimal multigrid method for general unstructured grids in this paper we develop a multigrid method on unstructured shape regular grids for a general shape regular unstructured grid of mathcal o n o n elements we present a construction of an auxiliary coarse grid hierarchy on which a geometric multigrid method can be applied together with a smoothing on the original grid by using the auxiliary space preconditioning technique such a construction is realized by a cluster tree which can be obtained in mathcal o n log n o nlogn operations for a grid of n elements this tree structure in turn is used for the definition of the grid hierarchy from coarse to fine for the constructed grid hierarchy we prove that the convergence rate of the multigrid preconditioned cg for an elliptic pde is 1 mathcal o 1 log n 1 o 1 logn numerical experiments confirm the theoretical bounds and show that the total complexity is in mathcal o n log n o nlogn

algebraic multigrid preconditioning of high order spectral elements for elliptic problems on a simplicial mesh algebraic multigrid is investigated as a solver for linear systems that arise from high order spectral element discretizations an algorithm is introduced that utilizes the efficiency of low order finite elements to precondition the high order method in a multilevel setting in particular the efficacy of this approach is highlighted on simplexes in two and three dimensions with nodal spectral elements up to order n 11 additionally a hybrid preconditioner is also developed for use with discontinuous spectral element methods the latter approach is verified for the discontinuous galerkin method on elliptic problems

adaptive smoothed aggregation alpha sa multigrid 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 multigrid methods applied to discretizations of elliptic problems these assumptions typically consist of knowledge of the near kernel or 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 kernel components is unavailable this extension is accomplished in an adaptive process that uses the method itself to determine near kernel components and adjusts the coarsening processes accordingly