the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

generalized aor method for solving a class of generalized saddle point problems abstract in this paper by constructing appropriate splitting a generalized aor gaor method for solving a class of generalized saddle point problems has been presented and the convergence of the method is studied the numerical experiments results show the efficiency and robustness of the new method

perturbation bound of singular linear systems a classical perturbation bound in numerical linear algebra is extended to more general cases of singular linear algebraic systems and weighted linear least squares problem componentwise perturbation analysis of linear least squares problem is also derived

an unstructured multigrid method for elliptic problems a multigrid algorithm for implementation on unstructured meshes is proposed the algorithm uses a sequence of unnested grids and requires the development of efficient inter grid interpolation procedures it is demonstrated how elliptic problems can be solved in this fashion by using jacobi smoothers

a comparison of preconditioned iterative techniques using rapid operator application against direct solution methods the p version of the finite element method which uses high order hierarchic basis functions within an element allows for ease of adaptive computation and high accuracy with relatively few degrees of freedom however the solution time and storage requirements for large 3 d problems can be very high when direct methods are used to factor global matrices iterative techniques with rapid operator application have minimal storage requirements because neither element nor global matrices are formed but may not converge rapidly enough we study an approach that attempts to combine the two methods by using direct factorization of the matrix corresponding to low level polynomials as a preconditioner to an iterative method at a higher polynomial level this resulting solution scheme converges more rapidly than an unpreconditioned scheme but has much lower storage requirements than direct methods we illustrate the effectiveness of this scheme on some large three dimensional problems in structural analysis

formulation and solution of hierarchical finite element equations the paper presents a general hierarchical formulation applicable to both elliptic and hyperbolic problems static and eigenvalue linear elastic problems as well as convection diffusion problems are studied the hierarchical formulation is well suited for adaptive procedures for the convection diffusion problem the hierarchical approximation is made in time only different hierarchical functions are proposed for different types of problems both weighted residual and least squares formulations are applied a combination of these two gives a penalty method with a constraint equation corresponding to the least squares method a whole class of time integration formulae is obtained these are all suitable for adaptive procedures owing to the hierarchical approximation in the time domain if a linear discontinuous hierarchical base function is used in the galerkin weak formulation the method so obtained corresponds to the discontinuous galerkin method in time and is especially suited for convection dominated problems the streamline diffusion method is found to be the aforementioned penalty method this paper also examines the sequence of nested equation systems that results from a hierarchical finite element formulation properties of these systems arising from static problems are investigated the paper presents some new possibilities for iterative solution of hierarchic element equations and different procedures are compared in a numerical example finally a simple id convection diffusion problem clearly shows that the proposed hierarchical formulation in time gives a stable and accurate solution even for convection dominated flow

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

efficient preconditioning for the p version finite element method in two dimensions parallel preconditioners are formulated and analyzed for systems of equations arising from the p version finite element method applied to second order self adjoint elliptic boundary value problems in two dimensions by using new theoretical results for polynomial spaces it is proved that the condition number of the preconditioned system grows as log 2 p where p is the degree of the polynomial space numerical results are presented showing that the condition number indeed grows very slowly with p

an adaptive multigrid technique for three dimensional elasticity a program for finite element analysis of 3d linear elasticity problems is described the program uses quadratic hexahedral elements the solution process starts on an initial coarse mesh here error estimators are determined by the standard babuska rheinboldt method and local refinement is performed by partitioning of indicated elements each hexahedron into eight new elements then the discrete problem is solved on the second mesh and the refinement process proceeds in the following way on the ith mesh only the elements caused by refinement on the i 1 th mesh can be refined the control of refinement is the task of the user because the dimension of the discrete problem grows very rapidly in 3d the discrete problem is being solved by the frontal solution method on the initial mesh and by a newly developed and very efficient local multigrid method on the refined meshes the program can be successfully used for solving problems with structural singularities such as re entrant corners and moving boundary conditions a numerical example shows that such problems are solved with the same efficiency as regular problems

on the augmented system approach to sparse least squares problems we study the augmented system approach for the solution of sparse linear least squares problems it is well known that this method has better numerical properties than the method based on the normal equations we use recent work by arioli et al 1988 to introduce error bounds and estimates for the components of the solution of the augmented system in particular we find that using iterative refinement we obtain a very robust algorithm and our estimates of the error are accurate and cheap to compute the final error and all our error estimates are much better than the classical or skeel s error analysis 1979 indicates moreover we prove that our error estimates are independent of the row scaling of the augmented system and we analyze the influence of the bjorck scaling 1967 on these estimates we illustrate this with runs both on large scale practical problems and contrived examples comparing the numerical behaviour of the augmented systems approach with a code using the normal equations these experiments show that while the augmented system approach with iterative refinement can sometimes be less efficient than the normal equations approach it is comparable or better when the least squares matrix has a full row and is in any case much more stable and robust

the role of the strengthened cauchy buniakowskii schwarz inequality in multilevel methods the basic theory of the strengthened cauchy buniakowskii schwarz inequality and its applications in multilevel methods for the solution of linear systems arising from finite element or finite difference discretisation of elliptic partial differential equations is surveyed proofs are given both in a finite element context and in purely algebraic form

a class of uzawa sor methods for saddle point problems in this paper we consider a class of uzawa sor methods for saddle point problems and prove the convergence of the proposed methods we solve a lower triangular system per iteration in the proposed methods instead of solving a linear equation az b actually the new methods can be considered as an inexact iteration method with the uzawa as the outer iteration and the sor as the inner iteration although the proposed methods cannot achieve the same convergence rate as the gsor methods proposed by bai et al z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 but our proposed methods have less workloads per iteration step experimental results show that our proposed methods are feasible and effective

symmetric sor method for augmented systems in this paper we develop symmetric successive overrelaxation ssor method to find solution of augmented systems these systems have appeared in many different applications of scientific computing e g the finite element approximation to solve the navier stokes equation and the constrained optimization the convergence of ssor method are studied finally we apply the new method to solve augmented systems the iteration number of this method to receive the solution is very better than sor like method

simplex algorithm for countable state discounted markov decision processes we consider discounted markov decision processes mdps with countably infinite state spaces finite action spaces and unbounded rewards typical examples of such mdps are inventory management and queueing control problems in which there is no specific limit on the size of inventory or queue existing solution methods obtain a sequence of policies that converges to optimality in value but may not improve monotonically ie a policy in the sequence may be worse than preceding policies our proposed approach considers countably infinite linear programming cilp formulations of the mdps a cilp is defined as a linear program lp with countably infinite numbers of variables and constraints under standard assumptions for analyzing mdps with countably infinite state spaces and unbounded rewards we extend the major theoretical extreme point and duality results to the resulting cilps under additional mild assumptions which are satisfied by several applications of interest we present a simplex type algor

high performance multilevel iterative aggregation solver for large finite element structural analysis problems the present study is aimed to overcome difficulties faced with industrial applications of multilevel iterative methods to arbitrary finite element fe structural analysis problems the coarse grid concept used in multigrid methods is substituted with an aggregation coarse model based on the mechanical principle on the base of this approach together with previously developed multilevel preconditioner an efficient iterative equation solver feags was developed for using in standard comprehensive finite element software systems numerical examples for analyses of three dimensional 3 d frame structures demonstrate the efficiency of the method comparison with incomplete cholesky conjugate gradient iccg method and direct methods is presented

the augmented system variant of ipms in two stage stochastic linear programming computation abstract the application of interior point methods ipm to solve the deterministic equivalent of two stage stochastic linear programming problems is a known and natural idea experiments have proved that among the interior point methods the augmented system approach gives the best performance on these problems however most of their implementations encounter numerical difficulties in certain cases which can result in loss of efficiency we present a new approach for the decomposition of the augmented system which automatically exploits the special behavior of the problems we show that the suggested approach can be implemented in a fast and numerically robust way by solving a number of large scale two stage stochastic linear programming problems the comparison of our solver with fo1aug which is considered as a state of the art augmented system implementation of interior point methods is also given

a cholesky dual method for proximal piecewise linear programming a quadratic programming method is given for minimizing a sum of piecewise linear functions and a proximal quadratic term subject to simple bounds on variables it may be used for search direction finding in nondifferentiable optimization algorithms an efficient implementation is described that updates a cholesky factorization of active constraints and provides good accuracy via iterative refinement numerical experience is reported for some large problems

on using cholesky based factorizations and regularization for solving rank deficient sparse linear least squares problems by examining the performance of modern parallel sparse direct solvers and exploiting our knowledge of the algorithms behind them we perform numerical experiments to study how they can be used to efficiently solve rank deficient sparse linear least squares problems arising from practical applications the cholesky factorization of the normal equations breaks down when the least squares problem is rank deficient while applying a symmetric indefinite solver to the augmented system can give an unacceptable level of fill in the factors to try to resolve these difficulties we consider a regularization procedure that modifies the diagonal of the unregularized matrix this leads to matrices that are easier to factorize we consider both the regularized normal equations and the regularized augmented system we employ the computed factors of the regularized systems as preconditioners with an iterative solver to obtain the solution of the original unregularized problem furthermore we look at using limited me

on generalized symmetric sor method for augmented systems in this paper we establish the generalized symmetric sor method gssor for solving the large sparse augmented systems of linear equations which is the extension of the ssor iteration method the convergence of the gssor method for augmented systems is studied numerical resume shows that this method is effective

tensor methods for large sparse nonlinear least squares problems this paper introduces tensor methods for solving large sparse nonlinear least squares problems where the jacobian either is analytically available or is computed by finite difference approximations tensor methods have been shown to have very good computational performance for small to medium sized dense nonlinear least squares problems in this paper we consider the application of tensor methods to large sparse nonlinear least squares problems where a sparse factorization of the jacobian can be stored this involves an entirely new way of solving the tensor model that is efficient for sparse problems a number of interesting linear algebraic implementation issues are addressed the test results of the tensor method applied to a set of sparse nonlinear least squares problems compared with those of the standard gauss newton method reveal that the tensor method is significantly more robust and efficient than the standard gauss newton method

a survey of condition number estimation for triangular matrices we survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix and make recommendations concerning the use of the estimates in applications each of the methods is shown to bound the condition number the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms for each bound we examine by how much at worst it can overestimate or underestimate the condition number numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators

the simplex and policy iteration methods are strongly polynomial for the markov decision problem with a fixed discount rate we prove that the classic policy iteration method howard r a 1960 dynamic programming and markov processes mit cambridge and the original simplex method with the most negative reduced cost pivoting rule of dantzig are strongly polynomial time algorithms for solving the markov decision problem mdp with a fixed discount rate furthermore the computational complexity of the policy iteration and simplex methods is superior to that of the only known strongly polynomial time interior point algorithm ye y 2005 a new complexity result on solving the markov decision problem math oper res 30 3 733 749 for solving this problem the result is surprising because the simplex method with the same pivoting rule was shown to be exponential for solving a general linear programming problem klee v g j minty 1972 how good is the simplex method technical report o shisha ed inequalities iii academic press new york the simplex method with the smallest index pivoting rule was shown to be exponential for solving an mdp regardless of discount rates melekopoglou m a condon 1994 on the complexity of the policy improvement algorithm for markov decision processes informs j comput 6 2 188 192 and the policy iteration method was recently shown to be exponential for solving undiscounted mdps under the average cost criterion we also extend the result to solving mdps with transient substochastic transition matrices whose spectral radii are uniformly below one

on mixed and componentwise condition numbers for moore penrose inverse and linear least squares problems classical condition numbers are normwise they measure the size of both input perturbations and output errors using some norms to take into account the relative of each data component and in particular a possible data sparseness componentwise condition numbers have been increasingly considered these are mostly of two kinds mixed and componentwise in this paper we give explicit expressions computable from the data for the mixed and componentwise condition numbers for the computation of the moore penrose inverse as well as for the computation of solutions and residues of linear least squares problems in both cases the data matrices have full column row rank

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

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

finite state approximations for denumerable multidimensional state discounted markov decision processes abstract a finite state iterative scheme introduced by white in recent developments in markov decision processes r hartley l c thomas and d j white eds academic press new york 1980 to approximate the value function of denumerable state markov decision processes is extended to denumerable multidimensional state space under essentially the same assumptions given in op cit a simpler proof of the convergence theorem is obtained together with convergence rates the iterative scheme is used to determine an asymptotically discount optimal policy which in turn can be used to obtain a discount optimal stationary policy

two level domain decomposition preconditioning for the p version finite element method in three dimensions this paper presents a flexible approach to domain decomposition preconditioning for problems expressed in terms of local stiffness matrices of subdomains evaluation of the preconditioner requires fully parallelizable local computations and the solution of a global auxiliary problem with few variables per subdomain convergence factors can be bounded using properties of the local stiffness matrices only the method is applied to the p version finite element method for three dimensional elasticity we treat each element as a subdomain and compute explicit convergence bounds

on the multi grid technique for solving three dimensional boundary value engineering problems the paper proposes an iterative technique with a similar multi grid operator for solving complex spatial bounctary value problems the description of the method and efficient vectorial algorithms based on a simple ctata structure applicable to solving three dimensional engineering problems are presented the discussion of the method s convergence includes theoretical grounds and practical aspects we give a general proof of convergence and consider the model cases to estimate the rate of convergence we also address the histories of convergence derived from the numerical tests and consider two examples of practical problems

shadow prices in infinite dimensional linear programming we consider the class of linear programs that can be formulated with infinitely many variables and constraints but where each constraint has only finitely many variables this class includes virtually all infinite horizon planning problems modeled as infinite stage linear programs examples include infinite horizon production planning under time varying demands and equipment replacement under technological change we provide under a regularity condition conditions that are both necessary and sufficient for strong duality to hold moreover we show that under these conditions the lagrangean function corresponding to any pair of primal and dual optimal solutions forms a linear support to the optimal value function thus extending the shadow price interpretation of an optimal dual solution to the infinite dimensional case we illustrate the theory through an application to production planning under time varying demands and costs where strong duality is established

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

warm start of the primal dual method applied in the cutting plane scheme a practical warm start procedure is described for the infeasible primal dual interior point method ipm employed to solve the restricted master problem within the cutting plane method in contrast to the theoretical developments in this field the approach presented in this paper does not make the unrealistic assumption that the new cuts are shallow moreover it treats systematically the case when a large number of cuts are added at one time the technique proposed in this paper has been implemented in the context of hopdm the state of the art yet public domain interior point code numerical results confirm a high degree of efficiency of this approach regardless of the number of cuts added at one time can be thousands in the largest examples and regardless of the depth of the new cuts reoptimizations are usually done with a few additional iterations

adaptive multivlevel methods in three space dimensions we consider the approximate solution of self adjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non uniform tetrahedral mesh which is generated adaptively the arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner here the accuracy of the iterative solution is coupled with the discretization error as the performance of hierarchical bases preconditioners deteriorates in three space dimensions the bpx preconditioner is used taking special care of an efficient implementation reliable a posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements to illustrate the theoretical results we consider a familiar model problem involving reentrant corners and a real life problem arising from hyperthermia a recent clinical method for cancer therapy

dual based methods for solving infinite horizon nonstationary deterministic dynamic programs we develop novel dual ascent and primal dual methods to solve infinite horizon nonstationary deterministic dynamic programs these methods are finitely implementable and converge in value to optimality moreover the dual ascent method produces a sequence of improving dual solutions that pointwise converge to an optimal dual solution while the primal dual algorithm provides a sequence of primal basic feasible solutions with value error bounds from optimality that converge to zero our dual based methods work on a more general class of infinite network flow problems that include the shortest path formulation of dynamic programs as a special case to our knowledge these are the first dual based methods proposed in the literature to solve infinite horizon nonstationary deterministic dynamic programs

semi convergence analysis of gmssor methods for singular saddle point problems recently some authors darvishi and hessari 2011 zhang et al 2012 discussed the generalized modified symmetric successive over relaxation gmssor method for saddle point problems in this paper we further study the gmssor method for solving singular saddle point problems we prove the semi convergence of the gmssor method without the assumption that the 1 1 block sub matrix a should be symmetric and analyze the spectral property of the corresponding preconditioned matrix numerical experiments are given to illustrate the efficiency of gmssor with appropriate parameters

convergence of a generalized mssor method for augmented systems recently wu et al s l wu t z huang x l zhao a modified ssor iterative method for augmented systems j comput appl math 228 1 2009 424 433 introduced a modified ssor mssor method for augmented systems in this paper we establish a generalized mssor gmssor method for solving the large sparse augmented systems of linear equations which is the extension of the mssor method furthermore the convergence of the gmssor method for augmented systems is analyzed and numerical experiments are carried out which show that the gmssor method with appropriate parameters has a faster convergence rate than the mssor method with optimal parameters

a multigrid method for the generalized symmetric eigenvalue problem part i algorithm and implementation a multigrid method is described that can solve the generalized eigenvalue problem encountered in structural dynamics the algorithm combines relaxation on a fine mesh with the solution of a singular equation on a coarse mesh a sequence of coarser meshes may be used to quickly solve this singular equation using another multigrid method the hierarchy of increasingly finer meshes can be further exploited using a nested iteration scheme whereby initial approximations to the fine mesh eigenvectors are computed using interpolated coarse mesh eigenvectors the solution of some simple plate problems on a convex c240 demonstrates the efficiency of a vectorized version of the multigrid algorithm

on the method of weighting for equality constrained least squares problems the generalized singular value decomposition is used to analyze the problem of minimizing ax b 2 subject to the constraint bx d a byproduct of the analysis is a new iterative procedure that can be used to improve an approximate solution obtained via the method of weights all that is required to implement the procedure is a single qr factorization these developments turn out to be of interest when a and b are sparse and for the case when systolic architectures are used to carry out the computations

variants of the uzawa method for saddle point problem in this paper we first propose three variants of the uzawa method for solving the saddle point problem and then we provide convergence results for the three proposed methods numerical experiments show that our proposed methods with three parameters perform about twice as fast as the gsor generalized sor method with two parameters since the proposed methods have less workload per iteration than the gsor

a hierarchical iterative procedure for the analysis of composite laminates a multilevel recursively defined preconditioner for use with the preconditioned conjugate gradient pcg algorithm is developed and applied to solve thin and thick laminated plate problems the preconditioner is constructed from a sequence of hierarchical vector spaces arising from the p version of the finite element fe method numerical studies have been conducted using quadrilateral elements for two dimensional problems up to p 8 and serendipity brick elements for three dimensional problems up to p 5 the efficiency of the iterative procedure is illustrated using bending response of laminated plates the effects of element span ratios convergence tolerance polynomial order material properties and plate span ratios on the number of iterations required for convergence has been investigated results indicate that the preconditioner developed herein can be used to produce an efficient iterative solver for two and three dimensional problems in structural mechanics

circumventing the slater conundrum in countably infinite linear programs abstract duality results on countably infinite linear programs are scarce subspaces that admit an interior point which is a sufficient condition for a zero duality gap yield a dual where the constraints cannot be expressed using the ordinary transpose of the primal constraint matrix subspaces that permit a dual with this transpose do not admit an interior point this difficulty has stumped researchers for a few decades it has recently been called the slater conundrum we find a way around this hurdle we propose a pair of primal dual spaces with three properties the series in the primal and dual objective functions converge the series defined by the rows and columns of the primal constraint matrix converge and the order of sums in a particular iterated series of a double sequence defined by the primal constraint matrix can be interchanged so that the dual is defined by the ordinary transpose weak duality and complementary slackness are then immediate instead of using interior point conditions to establish a zero duality gap we call upon the planning horizon method when the series in the primal and dual constraints are continuous we prove that strong duality holds if a sequence of optimal solutions to finite dimensional truncations of the primal and dual cilps has an accumulation point we show by counterexample that the requirement that such an accumulation point exist cannot be relaxed our results are illustrated using several examples and are applied to countable state markov decision processes and to a problem in robust optimization

a shadow simplex method for infinite linear programs we present a simplex type algorithm that is an algorithm that moves from one extreme point of the infinite dimensional feasible region to another not necessarily adjacent extreme point for solving a class of linear programs with countably infinite variables and constraints each iteration of this method can be implemented in finite time whereas the solution values converge to the optimal value as the number of iterations increases this simplex type algorithm moves to an adjacent extreme point and hence reduces to a true infinite dimensional simplex method for the important special cases of nonstationary infinite horizon deterministic and stochastic dynamic programs

solution techniques for the p version of the adaptive finite element method this paper investigates the performance of a class of methods based on the preconditioned conjugate gradient pcg method for the solution of large and sparse systems of linear equations which arise from the p version of the adaptive finite element method the hierarchical type of preconditioning used in the past is extended to the ssor and incomplete factorization type preconditioning and applied with a global handling of the stiffness matrix the paper also presents some new ideas on domain decomposition dd matrix handling techniques whereby the explicit formulation of the resulting schur complement is avoided by utilizing the pcg method with efficient preconditioners and by processing separately on the coarse and the fine mesh the suitability of the methods to the p version of the adaptive finite element method is demonstrated through extensive numerical tests in two dimensions

convergence of relaxation iterative methods for saddle point problem in this paper we first provide convergence results of three relaxation iterative methods for solving saddle point problem next we propose how to find near optimal parameters for which preconditioned krylov subspace method performs nearly best when the relaxation iterative methods are applied to the preconditioners of krylov subspace method lastly we provide efficient implementation for the relaxation iterative methods and efficient computation for the preconditioner solvers numerical experiments show that the miaor method and the bicgstab with maor preconditioner using near optimal parameters perform more than twice faster than the gsor method

duality in infinite dimensional linear programming we consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finitely many constraints examples include production planning and equipment replacement over an infinite horizon we form the natural dual linear programming problem and prove strong duality under a transversality condition that dual prices are asymptotically zero that is we show under this transversality condition that optimal solutions are attained in both primal and dual problems and their optimal values are equal the transversality condition and hence strong duality is established for an infinite horizon production planning problem

a linear programming approach to nonstationary infinite horizon markov decision processes nonstationary infinite horizon markov decision processes mdps generalize the most well studied class of sequential decision models in operations research namely that of stationary mdps by relaxing the restrictive assumption that problem data do not change over time linear programming lp has been very successful in obtaining structural insights and devising solution methods for stationary mdps however an lp approach for nonstationary mdps is currently missing this is because the lp formulation of a nonstationary infinite horizon mdp includes countably infinite variables and constraints and research on such infinite dimensional lps has traditionally faced several hurdles for instance duality results may not hold an extreme point may not be a basic feasible solution and in the context of a simplex algorithm a pivot operation may require infinite data and computations and a sequence of improving extreme points need not converge in value to optimal in this paper we tackle these challenges a

solving symmetric indefinite systems in an interior point method for linear programming we describe an implementation of a primal dual path following method for linear programming that solves symmetric indefinite augmented systems directly by bunch parlett factorization rather than reducing these systems to the positive definite normal equations that are solved by cholesky factorization in many existing implementations the augmented system approach is seen to avoid difficulties of numerical instability and inefficiency associated with free variables and with dense columns in the normal equations approach solving the indefinite systems does incur an extra overhead whose median is about 40 in our tests but the augmented system approach proves to be faster for a minority of cases in which the normal equations have relatively dense cholesky factors a detailed analysis shows that the augmented system factorization is reliable over a fairly large range of the parameter settings that control the tradeoff between sparsity and numerical stability

a class of new extended shift splitting preconditioners for saddle point problems abstract in this paper a class of new extended shift splitting ness preconditioners containing t 0 are proposed for saddle point problems which contain the existing shift splitting based preconditioners as special cases the convergence properties of the ness iteration and the spectral distribution of the ness preconditioned matrix are investigated we give an estimated bound for the real eigenvalues of the ness preconditioned matrix whose left end is positive and show that the non real eigenvalues of the ness preconditioned matrix are located in an intersection of two rings particularly these non real eigenvalues are located in an intersection of two rings and one circle if t 1 2 moreover we show that under appropriate conditions our proposed eigenvalue bounds are tighter than the corresponding bounds given in ren et al 2017 using fourier analysis we derive an optimized parameter t independent of the viscosity for the continuous version of the ness preconditioned gmres method for the 2d stokes equation moreover we find that the ness preconditioned gmres method with a constant multiple of the optimized parameter t is effective and robust to solve 2d stokes problems in practice numerical experiments are used to verify our theoretical results

study of interlaminar stresses in composite laminates subjected to torsional loading a computational procedure for accurate determination of interlaminar stresses in thick and thin composite laminates subjected to torsional loading is presented a multilevel recursively defined preconditioner in conjunction with the preconditioned conjugate gradient pcg algorithm is constructed from a sequence of hierarchical vector spaces arising from the p version of the finite element method the computational procedure is used to model stress fields in thick and thin laminates in torsion using three dimensional and quasi three dimensional models results indicate that the preconditioner developed here can be used to produce an efficient iterative solver for the determination of two and three dimensional stress fields in composite structures

componentwise analysis of direct factorization of real symmetric and hermitian matrices abstract we derive componentwise error bound for the factorization h gjg t where h is a real symmetric matrix g has full column rank and j is diagonal with 1 s on the diagonal we also derive a componentwise forward error bound that is we bound the difference between the exact and the computed factor g in the cases where such a bound is possible we extend these results to the hermitian case and to the well known bunch parlett factorization finally we prove bounds for the scaled condition of the matrix g and show that the factorization can have the rank revealing property

a modified symmetric successive overrelaxation method for augmented systems in this paper we establish a modified symmetric successive overrelaxation mssor method to solve augmented systems of linear equations which uses two relaxation parameters this method is an extension of the symmetric sor ssor iterative method the convergence of the mssor method for augmented systems is studied numerical examples show that the new method is an efficient method

parallel domain decomposition solver for adaptive hp finite element methods in this paper the development and implementation of highly parallelizable domain decomposition solvers for adaptive hp finite element methods is discussed two level orthogonalization is used to obtain a reduced system which is preconditioned by a coarse grid operator the condition number of the preconditioned system for poisson problems in two space dimensions is proved to be bounded by c 1 hp h 2 1 log p 2 and cp 1 hp h 2 1 log p 2 for different choices of coarse grid operators where h is the subdomain size p is the maximum spectral order h is the size of the smallest element in the subdomain and c is a constant independent of the mesh parameters the work here extends the work of bramble et al math comp 47 1986 pp 103 134 on the h version and babuska et al siam j numer anal 29 1991 pp 624 661 on the p version of the finite element method a preliminary version of this solver was first announced by oden patra and feng in domain decomposition solver for adaptive hp finite elements vii conference on domain decomposition state college pa october 1993 numerical experiments show fast convergence of the solver and good control of the condition number on a variety of discretizations

preconditioning and two level multigrid methods of arbitrary degree of approximation let h be a mesh parameter corresponding to a finite element mesh for an elliptic problem we describe preconditioning methods for two level meshes which for most prob lems solved in practice behave as methods of optimal order in both storage and computa tional complexity namely per mesh point these numbers are bounded above by relatively small constants for all h ho where ho is small enough to cover all but excessively fine meshes we note that in practice multigrid methods are actually solved on a finite often even a fixed number of grid levels in which case also these methods are not asymptotically optimal as h 0 numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems variable coefficients general domains and boundary conditions for all but excessively fine meshes furthermore most of the latter methods have been implemented only for difference schemes of second order of accuracy whereas our methods are applicable to higher order approximations we claim that our scheme could be added fairly easily to many existing finite element codes 1 introduction consider the numerical solution of elliptic boundary value prob lems discretized by finite element methods we assume that the boundary is polygonal or consists of planes we note that in practical problems one often has a fine enough grid already after the definition of the boundary and the minimal number of vertices needed for a first coarse triangulation anyhow if not so in most cases one makes only a few steps of mesh refinement hence the power of multigrid methods their optimal order of computational complexity is most often not achieved fully because optimality requires a large number of recursively defined meshes for details see e g 4 and for further references see 7 hence one might as well consider other methods perhaps simpler and more effective on a fixed mesh but which are not asymptotically optimal here we shall describe a method which uses only a fixed mesh but for which one nevertheless achieves a low order of computational complexity and of seemingly optimal order except for from a practical viewpoint excessively small meshes to be more precise the computational cost per mesh point is bounded by clog n for n no where n is the number of mesh points no is large enough to cover most applications and c is small enough that the method is competitive with multigrid methods as is well known the latter need recursion and the usual smoothing followed by corrections of the solutions on the different mesh levels we claim that the new method is more suitable for implementation in existing finite element packages in fact most packages for the multigrid methods are only for second order difference methods

two preconditioners based on the multi level splitting of finite element spaces the hierarchical basis preconditioner and the recent preconditioner of bramble pasciak and xu are derived and analyzed within a joint framework this discussion elucidates the close relationship between both methods special care is devoted to highly nonuniform meshes exclusively local properties like the shape regularity of the finite elements are utilized

adaptive solution strategy for solving large systems of p type finite element equations a solution strategy is proposed and implemented for taking advantage of the hierarchical structure of linear equation sets arising from the p type finite element method using a hierarchical basis function set the algorithm dynamically branches to either direct or iterative solution methods in the iterative solution branch the substructure of the finite element equation set is used to generate a lower order preconditioner for a preconditioned conjugate gradient pcg method the convergence rate of the pcg algorithm is monitored to improve the heuristics used in the choice of the preconditioner the robustness and efficiency of the method are demonstrated on a variety of three dimensional examples utilizing both hexahedral and tetrahedral mesh discretizations this strategy has been implemented in a p version finite element code which has been used in an industrial environment for over two years to solve mechanical design problems

variants of the uzawa method for three order block saddle point problem abstract in this paper we first propose three kinds of variants of the uzawa method for solving three order block saddle point problem and study the convergence conditions of the proposed methods second we obtain the approximate optimal relaxation factors of the three proposed uzawa type methods by using variable control method finally the experimental results show that our proposed uzawa type methods for solving three order block saddle point problem have less workload per iteration step than the corresponding uzawa type methods for solving standard saddle point problem which explains that our proposed methods are feasible and efficient

scalable algorithms for the solution of navier s equations of elasticity very large scale mathematical modelling such as accurate modelling using navier s equations of elasticity needs both massively parallel computing and scalable algorithms it is shown in this paper that efficient methods must be scalable with respect to the speedup measured as the ratio of the computing time of the best sequential algorithm on one processor and the computing time of the parallel algorithm on p processors for a class of multilevel methods for elliptic partial differential equations it is shown how to balance the coarsest mesh size to the finest and the number of processors to the size of the problem to get smallest computing time and maximal efficiency it turns out that the number of processors should grow slowly in proportion with the problem size further it should grow slightly slower by a logarithmic or a polylogarithmic function for asymptotically maximal efficiency than the number of processors required for minimal computing time