inertia controlling methods for general quadratic programming active set quadratic programming qp methods use a working set to define the search direction and multiplier estimates in the method proposed by fletcher in 1971 and in several subsequent mathematically equivalent methods the working set is chosen to control the inertia of the reduced hessian which is never permitted to have more than one nonpositive eigenvalue such methods will be called inertia controlling this paper presents an overview of a generic inertia controlling qp method including the equations satisfied by the search direction when the reduced hessian is positive definite singular and indefinite recurrence relations are derived that define the search direction and lagrange multiplier vector through equations related to the karush kuhn tucker system discussion is included of connections with inertia controlling methods that maintain an explicit factorization of the reduced hessian matrix

introduction of oblique incidence plane wave to stratified lossy dispersive media for fdtd analysis in the application of finite difference time domain fdtd method to the wideband plane wave scattering analysis of objects embedded in stratified lossy dispersive media it is a challenge to solve the obliquely incident wave in this work a novel configuration of auxiliary one dimensional 1 d fdtd grids is proposed to provide the obliquely incident plane wave at the total field scattered field tfsf boundary in stratified lossy dispersive media thus a closed tfsf boundary is formed where the incident waves are directly obtained from the auxiliary simulations without any interpolation the formulations of the auxiliary simulations are presented by the modified 1 d maxwell s equations given the dispersion characteristic perfectly matched layer pml is implemented to the 1 d simulations examples demonstrate the effectiveness of the proposed technique of obliquely incident wave introduction

deflated decomposition of solutions of nearly singular systems when solving the linear system ax b where a may be nearly singular and b is not consistent with a one is often interested in computing a deflated solution i e a unique solution to a nearby singular but consistent system a s x d tilde b kelley 14 15 has considered deflated solutions with a s corresponding to setting a small pivot of the lu factorization of a to zero stewart 22 proposed an iterative algorithm for computing a deflated solution with a s corresponding to setting the smallest singular value of a to zero kelley s approach explicitly uses submatrices of the lu factors whereas stewart s approach is implicit in that it only involves solving systems with a in this paper we generalize the concept of a deflated solution to that of a deflated decomposition which expresses the solution x in terms of x d and the null vectors of a s we treat such decompositions in a uniform framework that includes the approaches of kelley and stewart and introduce some new deflated

a note on the group inverse of some 2 2 block matrices over skew fields abstract let k be a skew field and km n be the set of all m n matrices over k in this paper we give the necessary and sufficient conditions to the existence and the representations of the group inverse for block matrix ax yb a b 0 under some conditions where a y km n x b kn m

block triangular preconditioners based on symmetric triangular decomposition for generalized saddle point problems abstract in this paper the symmetric triangular decomposition is further studied to construct a class of block triangular preconditioners for generalized saddle point problems such that the preconditioned generalized saddle point matrices are symmetric and positive definite then the preconditioned conjugate gradient iterative method can be used three specific preconditioners are studied in detail eigen properties of the corresponding preconditioned generalized saddle point matrices are studied in particular upper bounds on the condition number of the preconditioned matrices are analyzed finally numerical experiments of a model stokes equation are given to illustrate the efficiency of the new proposed preconditioners

on types of fuzzy numbers and extension principles when using the given crisp functional relationships in a fuzzy environment with fuzzy number input always the problem arises how to specify the fuzzy numbers and to choose the t norm within the extension principle in this paper the question is considered whether different pairs of reference function and t norm can lead to the same fuzzy value for the relationship the question is answered in the negative however for more general combination functions inserted into the extension principle this case will happen as a rule

the numerical solution of equality constrained quadratic programming problems this paper proves that a large class of iterative schemes can be used to solve a certain constrained minimization problem the constrained minimization problem considered involves the minimization of a quadratic functional subject to linear equality constraints among this class of convergent iterative schemes are generalizations of the relaxed jacobi gauss seidel and symmetric gauss seidel schemes

on practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem we present practical conditions under which the existence and uniqueness of a finite solution to a given equality quadratic program may be examined different sets of conditions allow this examination to take place when null space range space or lagrangian methods are used to find stationary points for the quadratic program

parameterized approximate block lu preconditioners for generalized saddle point problems abstract in this paper we are concerned with the iteration solution of generalized saddle point problems based on the exact block lu factorization of the coefficient matrix we construct a class of parameterized approximate block lu factorization preconditioners which rely on suitable approximations of the schur complement of the 1 1 block of the coefficient matrix convergence of the corresponding iteration methods is analyzed and the optimal iteration parameters minimizing the spectral radii are deduced algebraic characteristics of the related preconditioned matrices are discussed including eigenvalue and eigenvector distributions and upper bounds for degree of the minimal polynomial the established results extend those of the approximate factorization and variants of the hermitian and skew hermitian splitting and positive and skew hermitian splitting preconditioners for saddle point problems numerical experiments are demonstrated to illustrate the efficiency of the new preconditioners

piecewise linear recursive convolution for dispersive media using fdtd electromagnetic propagation through linear dispersive media can be analyzed using the finite difference time domain fdtd method by employing the recursive convolution rc approach to evaluate the discrete time convolution of the electric field and the dielectric susceptibility function the rc approach results in a fast and computationally efficient algorithm however the accuracy achieved is not generally as good as that obtained with other methods a new piecewise linear recursive convolution plrc method is described here that has greatly improved accuracy over the original rc approach but retains its speed and efficiency advantages

parametric representation of fuzzy numbers and application to fuzzy calculus we present several models to obtain simple parametric representations of the fuzzy numbers or intervals based on the use of piecewise monotonic functions of different forms the representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers we use the standard cuts setting but also the membership functions are obtained immediately and can be used directly to obtain error controlled approximations of the fuzzy calculus in terms of a finite set of parameters the general setting is the hermite type interpolation where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces and of the parameters by which the results are obtained the representations are designed to model the lower and the upper extremal values of each cut compact intervals of the fuzzy numbers and are able to produce almost any possible configuration differentiable continuous or with a finite number of discontinuity points by using parametric monotonic functions of different types we show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems including the numerical solution of fuzzy differential equations the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics the proposed model is called the lower upper representation and we denote the associated fuzzy numbers or intervals as lu fuzzy

vertex method for computing functions of fuzzy variables abstract this paper is to introduce a new approach the vertex method for computing functions of fuzzy variables the method is based on the cut concept and interval analysis the vertex method can avoid abnormality due to the discretization technique on the variables domain and the widening of the resulting function value set due to multi occurrence of variables in the functional expression by conventional interval analysis methods the algorithm is very easy to implement and can be applied to many practical problems some examples are given to illustrate the applications

determining optimal points of membership with dependent variables consider sets x and y each with associated membership functions further suppose that there is a map f x y a membership may be induced ony from the membership ofx calculated using the extension principle and the two memberships ony may be combined to give an overall membership the point with the highest membership in x can be determined directly or by back mapping the value with the highest membership in y t hese two manuscript prepared for submission to fuzzy sets and systems

parallel solvers for virtual element discretizations of elliptic equations in mixed form abstract the aim of this paper is twofold on the one hand we numerically test the performance of mixed virtual elements in three dimensions to solve the mixed formulation of three dimensional elliptic equations on polyhedral meshes on the other hand we focus on the parallel solution of the linear system arising from such discretization considering both direct and iterative parallel solvers in the latter case we develop two block preconditioners one based on the approximate schur complement and one on a regularization technique both these topics are numerically validated by several parallel tests performed on a linux cluster more specifically we show that the proposed virtual element discretization recovers the expected theoretical convergence rates and we analyze the performance of the direct and iterative parallel solvers taken into account

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

field of values convergence analysis of augmented lagrangian preconditioners for the linearized navier stokes problem we study a block triangular preconditioner for finite element approximations of the linearized navier stokes equations the preconditioner is based on the augmented lagrangian formulation of the problem and was introduced by the authors in siam j sci comput 28 2006 pp 2095 2113 in this paper we prove field of values type estimates for the preconditioned system which lead to optimal convergence bounds for the gmres algorithm applied to solve the system two variants of the preconditioner are considered an ideal one based on exact solves for the velocity submatrix and a more practical variant based on block triangular approximations of the velocity submatrix

linear algebra methods in a mixed approximation of magnetostatic problems in the solution of magnetostatic problems the use of a mixed formulation based on both the magnetic and magnetic displacement fields is particularly appropriate as it allows us to impose the physical conditions exactly and to maintain the continuity properties of the two fields together with an efficient treatment of boundary conditions the discretization by means of a proper finite element method yields a strongly structured algebraic linear system this paper is concerned with the solution of this large very sparse indefinite linear system in particular we present the implementation of some known direct and preconditioned iterative methods and discuss their performance on two dimensional 2 d and three dimensional 3 d specific models we show that the 2 d system can be efficiently handled by appropriate variants of these schemes while preliminary tests on the 3 d system give some insight in the understanding of the analysis that needs to be done

fuzzy weighted averages and implementation of the extension principle abstract this paper addresses the computational aspect of the extension principle when the principle is applied to algebraic mappings and in particular to weighted average operations in risk and decision analysis a computational algorithm based on the cut representation of fuzzy sets and interval analysis is described the method provides a discrete but exact solution to extended algebraic operations in a very efficient and simple manner examples are given to illustrate the method and its relation to other discrete methods and the exact approach by non linear programming the algorithm has been implemented in a computer program which can handle very general extended algebraic operations on fuzzy numbers

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

on using a cuts to evaluate fuzzy equations abstract consider an equation y f x1 xn where the extension principle is used to find y f x 1 x n when we subtitute fuzzy numbers x i for the xi 1 i n if x i denotes an cut of x i then we argue that for only certain simple equations will one obtain y equal to f x 1 x n where f is evaluated using interval arithmetic

plane waves in fdtd simulations and a nearly perfect total field scattered field boundary the total field scattered field tfsf boundary has been successfully used for a number of years to introduce energy into finite difference time domain fdtd grids if the propagation of the incident field is grid aligned a perfect tfsf implementation can be realized by using an auxiliary one dimensional fdtd simulation which models propagation of the incident field here perfect implies the incident field propagation exactly matches the way in which the field propagates in the fdtd grid however for propagation which is not grid aligned no similarly perfect implementation has previously been presented this work provides a framework for a perfect tfsf boundary for pulsed plane waves which do not propagate in a grid aligned fashion to achieve this homogeneous plane wave propagation is rigorously quantified using this knowledge and a specification of the desired incident field the dispersion relation is used to ascertain the incident field at any point in the grid it is required to account for unlike in the continuous world the electric field the magnetic field and the wavenumber vector not forming a mutually orthogonal set group velocity is also considered because of its relevance to the implementation

an iteration for indefinite systems and its application to the navier stokes equations for large sparse systems of linear equations iterative solution techniques are attractive in this paper we propose and examine the convergence of an iterative method for an important class of nonsymmetric and indefinite coefficient matrices based on the use of an indefinite and symmetric preconditioner we apply our technique to the linearized navier stokes equations the oseen equations

representations of the drazin inverse for a class of block matrices abstract in this paper we give a formula for the drazin inverse of a block matrix f i i e 0 where e is square and i is the identity matrix further we get representations of the drazin inverse for matrices of the form m a b c 0 where a is square under some conditions expressed in terms of the individual blocks

an optimal preconditioner for a class of saddle point problems with a penalty term iterative methods are considered for a class of saddle point problems with a penalty term arising from finite element discretizations of certain elliptic problems an optimal preconditioner which is independent of the discretization and the penalty parameter is constructed this approach is then used to design an iterative method with a convergence rate bounded independently of the lame parameters occurring in the equations of linear elasticity

incorporating the g tfsf concept into the analytic field propagation tfsf method previously anantha and taflove reported a generalized total field scattered field g tfsf formulation that was developed to facilitate the study of infinite scatterers such as wedges ieee trans antennas and propag vol 50 no 10 1337 1349 oct 2002 the g tfsf formulation relied upon having the tfsf boundary embedded within a perfectly matched layer pml to account for the pml the incident field terms that appear in the update equations for nodes adjacent to the tfsf boundary were scaled by a constant in accordance with the amount of attenuation produced by the pml in this work we describe how the analytic field propagation tfsf afp tfsf formulation can be used in a g tfsf like way this new approach possesses various advantages over the previously presented work for example owing to the dispersion inherent in pml s the spectral components of pulsed excitation propagate at the different speeds within the pml this dispersive behavior can be accommodated in the afp based formulation but not in the original g tfsf implementation additionally the afp based technique can directly model the infinite nature of objects such as wedges so that corners need not be embedded within the pml

preconditioning for the steady state navier stokes equations with low viscosity we introduce a preconditioner for the linearized navier stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero for constant coefficient problems with periodic boundary conditions we show that the preconditioning yields a system with a single eigenvalue equal to 1 so that performance is independent of both viscosity and mesh size for other boundary conditions we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon we also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration with virtually no degradation of performance

neumann neumann domain decomposition preconditioners for linear quadratic elliptic optimal control problems we present a class of domain decomposition dd preconditioners for the solution of elliptic linear quadratic optimal control problems our dd preconditioners are extensions of neumann neumann dd preconditioners which have been successfully applied to the solution of single pdes the dd preconditioners are based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through neumann interface conditions for the states and the adjoints this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain elliptic linear quadratic optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with neumann interface conditions for the state the subdomain schur complement operators are analyzed in the variational setting of the problem as well as the algebraic setting obtained after a finite element discretization of the problem definiteness properties of the algebraic form of the subdomain schur complement operator s are studied numerical tests show that the dependence of these preconditioners on mesh size and subdomain size is comparable to its counterpart applied to elliptic equations only these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

combination of augmented lagrangian technique and st preconditioner for saddle point problems abstract for symmetric indefinite linear systems we introduce a new triangular preconditioner based on symmetric and triangular st decomposition a new 1 1 block is obtained by augmented lagrangian technique the new st preconditioner is introduced by the combination of the new 1 1 block and symmetric and triangular st decomposition then a preconditioned system can be obtained by preconditioning technique which is superior to the original system in terms of condition number we study the spectral properties of preconditioned system such as eigenvalues the estimation of condition number and then give the quasi optimal parameter numerical examples are given to indicate that the new preconditioner has obvious efficiency advantages finally we conclude that the new st preconditioner is a better option to deal with large and sparse problems

an improved method for fuzzy interval uncertainty analysis and its application in brake instability study abstract most of the existing methods of brake squeal instability analysis are merely available to handle single type of uncertain case in this study an improved unified method is developed for uncertainty quantification which is capable of handling two types of fuzzy interval cases in the first fuzzy interval case uncertain parameters of engineering structures are assumed as either fuzzy variables or interval variables which exist in structures simultaneously and independently in the second fuzzy interval case all uncertain parameters are represented by interval variables but their lower and upper bounds just can be expressed as fuzzy variables instead of deterministic values in the proposed method fuzzy boundary interval variables are introduced to handle fuzzy interval uncertainties and based on which an improved response analysis model is established in the improved model the fuzzy boundary interval variables are firstly converted into interval boundary variables by level cut technique then by temporarily neglecting boundary uncertainties the initial interval responses can be approximated via conducting once taylor series expansion and subinterval analysis next taylor series expansion and central difference method are combined twice to deal with boundary uncertainties and the interval responses of the structures with interval boundary variables are yielded finally the fuzzy interval responses of the structures are derived on the basis of interval union operation and fuzzy decomposition theorem the improved method is subsequently extended to quantify the uncertainties in brake squeal instability analysis involving two types of fuzzy interval uncertainties the effectiveness of the proposed method on tackling fuzzy interval problems is demonstrated by numerical examples

analytic field propagation tfsf boundary for fdtd problems involving planar interfaces pecs te and tm a total field scattered field tfsf boundary can be used to introduce incident plane waves into finite difference time domain fdtd simulations for fields which are traveling obliquely to the grid axes there is no simple way to account fully for the effects of the inherent numerical artifacts associated with plane wave propagation in the fdtd grid failure to account for these artifacts causes erroneous fields to leak across the tfsf boundary recent publications have proposed ways to use the dispersion relation to describe precisely plane wave propagation in the fdtd grid thus permitting the realization of a nearly perfect tfsf boundary however these publications did not cover certain implementations details such as the type of fourier transform which is needed or their scope was so broad as to mask the relative simplicity with which the approach can be applied to problems involving planar interfaces this work considers the fourier transforms needed in order for the implementation to be exact reflection and transmission coefficients for a planar interface are derived implementation for planar perfect electric conductors is also presented

a framework for equilibrium equations the equations of equilibrium in a wide class of applications share a common structure the discrete case leads to a positive definite symmetric matrix a t ca in which a is rectangular and c is often diagonal the continuous case linear or nonlinear repeats that pattern there is a minimum principle for the potentials and a dual principle for the flow variables we try to clarify these different expressions of the same law

numerical methods for generalized least squares problems usually generalized least squares problems are solved by transforming them into regular least squares problems which can then be solved by well known numerical methods however this approach is not very effective in some cases and besides is very expensive for large scale problems in 1979 paige suggested another approach which consists of solving an equivalent equality constrained least squares problem by the orthogonal decomposition the bnp algorithm or the james implicit nullspace iterative methods in this paper we present some new developments of the numerical methods for example 2 cycle sor method and preconditioned conjugate gradient method for generalized least squares problems some numerical comparisons are included as well

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

fast nonsymmetric iterations and preconditioning for navier stokes equations discretization and linearization of the steady state navier stokes equations gives rise to a nonsymmetric indefinite linear system of equations in this paper we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization we confirm and supplement these analytic results with a series of numerical experiments indicating that krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter in addition we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner

a survey of non probabilistic uncertainty treatment in finite element analysis the objective of this paper is to critically review the emerging non probabilistic approaches for uncertainty treatment in finite element analysis the paper discusses general theoretical and practical aspects of both the interval and fuzzy finite element analysis first the applicability of the non probabilistic concepts for numerical uncertainty analysis is discussed from a theoretical viewpoint the necessary conditions for a useful application of the non probabilistic concepts are determined and are proven to be complementary rather than competitive to the classical probabilistic approach the second part of the paper focuses on numerical aspects of the interval finite element method it describes two principal strategies for the implementation i e the anti optimisation and the interval arithmetic approach and gives a state of the art of the interval finite element algorithms available from literature it is shown how the application of the interval arithmetic approach to the classical finite element procedure can result in a severe overestimation of the uncertainty on the output and the main sources of this conservatism are identified a numerical example in the final part of the paper illustrates the capabilities of the different strategies on an eigenfrequency analysis of a built up benchmark structure

convergence properties of preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite matrices for the non hermitian and positive semidefinite systems of linear equations we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned hermitian and skew hermitian splitting iteration methods we then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned hermitian and skew hermitian splitting iteration methods

on conditioning of saddle point matrices with lagrangian augmentation for saddle point matrix and its lagrangian augmentation we derive bounds on eigenvalues and on euclidean condition numbers and discuss the asymptotic behavior of the euclidean condition numbers with respect to the parameter in the augmentation

zadeh s extension principle for continuous functions of non interactive variables a parallel optimization approach there is a growing interest in the use of fuzzy intervals in many engineering applications however a direct implementation of zadeh s extension principle which forms the basis for computing with fuzzy intervals is still computationally too demanding for practical use in the case of a continuous function and fuzzy intervals that describe non interactive variables as inputs the output is a fuzzy interval as well and can be determined for each cut separately the problem thus reduces to finding the endpoints of these cuts which amounts to a number of interwoven optimization problems in the case of a non monotone continuous function however these optimization problems are non trivial in this paper different optimization algorithms are applied for that purpose gradient descent based on sequential quadratic programming simplex simulated annealing particle swarm optimization and particle swarm optimization combined with gradient descent in addition two approaches are followed to determine a suitable number of cuts either a fixed predetermined number is used or an initially very small number is chosen that is subsequently increased according to a linearity criterion both a non parallel and a parallel implementation are designed the parallel version is restricted to work with particle swarm optimization and employs communication to optimize its internal performance by exploiting the dependence between the various optimization problems different configurations are evaluated on a set of benchmark functions in terms of the mean area under the output fuzzy interval and the number of function evaluations particle swarm optimization combined with gradient descent starting from a small number of cuts leads to the most accurate fuzzy intervals at the cost of a relatively large number of function evaluations

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

a domain decomposition method with lagrange multipliers and inexact solvers for linear elasticity a new domain decomposition method with lagrange multipliers for elliptic problems is introduced it is based on a reformulation of the well known finite element tearing and interconnecting feti method as a saddle point problem with both primal and dual variables as unknowns the resulting linear system is solved with block structured preconditioners combined with a suitable krylov subspace method this approach allows the use of inexact subdomain solvers for the positive definite subproblems it is shown that the condition number of the preconditioned saddle point problem is bounded independently of the number of subregions and depends only polylogarithmically on the number of degrees of freedom of individual local subproblems numerical results are presented for a plane stress cantilever membrane problem

threshold incomplete factorization constraint preconditioners for saddle point matrices abstract this paper presents a drop threshold incomplete ld 1lt factorization constraint preconditioner for saddle point systems using a threshold parameter a transformed saddle point matrix is partitioned into a block structure with blocks of order 1 and 2 constituting a priori pivots based on these pivots an incomplete ld 1lt factorization constraint preconditioner is computed that approaches an exact form as approaches zero we prove that both the exact and incomplete factorizations exist such that the entries of the constraint block remain unaltered in the triangular factors numerical results are presented for validation

optimized analytic field propagator o afp for plane wave injection in fdtd simulations optimizations are proposed for the original frequency domain analytic field propagator afp to source plane waves in a 1d 2d 3d total field scattered field tf sf fdtd formulation the proposed technique essentially produces no field leakage error limited by 300 db machine precision error with finite sampling points dictated by temporal aliasing constraints by taking advantage of the inherent 1d nature of a plane wave the memory requirements for the original afp are reduced significantly the proposed optimized solution o afp greatly reduces the computational complexity by calculating a number of spatial fields of o n rather than o n 2 which then makes the technique practical enough to include fully 3d simulations

evaluation of st preconditioners for saddle point problems the general block st decomposition of the saddle point problem is used as a preconditioner to transform the saddle point problem into an equivalent symmetric and positive definite system such a decomposition is called a block st preconditioner two general block st preconditioners are proposed for saddle point problems with symmetric and positive definite 1 1 block some estimations of the condition number of the preconditioned system are given the same study is done for singular 1 1 block

approximating a cuts with the vertex method if f rn r is continuous and monotonic in eachvariable and if i is a fuzzy number on the ith coordinate then the membership on r induced by and by the membership onfrn given by x min 1 x1 n xn can be evaluated by determining the membership at the endpoints of the level cuts of each i here more general conditions are given for both the function and the manner in which the fuzzy numbers i are combined so that this simple method for computing induced membership may be used in particular a geometric condition is given so that the cuts computed when the fuzzy numbers are combined using min is an upper bound for the actual induced membership

block sor methods for rank deficient least squares problems many papers have discussed preconditioned block iterative methods for solving full rank least squares problems however very few papers studied iterative methods for solving rank deficient least squares problems miller and neumann 1987 proposed the 4 block sor method for solving the rank deficient problem here a 2 block sor method and a 3 block sor method are proposed to solve such problem the convergence of the block sor methods is studied the optimal parameters are determined comparison between the 2 block sor method and the 3 block sor method is given also

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

specifying pml conductivities by considering numerical reflection dependencies berenger s perfectly matched layer pml absorbing boundary condition abc has greatly enhanced finite difference time domain fdtd scattering analysis in a discretized domain however performance is signal dependent and large angle performance is poor due to a rapid reduction in layer decay rate increasing the conductivity to offset this reduction increases the discretization errors especially at near normal incidence angles however by carefully specifying the conductivity in each of the pml sublayers it is possible to balance the small and large angle performance the signal dependence of reflections may be described in terms of the number of spatial points per wavelength this lends itself to an overall strategy for which to search for pml profiles that provide superior performance for waves incident on a pml at angles between 0 75 spl deg and signals that have at least 15 spatial points per wavelength sampling a one dimensional 1 d projection method may be employed to allow an exhaustive search to become a viable alternative to optimization such a search provides profile parameters that while not necessarily optimal give excellent wide angle wide band reflection performance

preconditioned conjugate gradient method for generalized least squares problems abstract a variant of the preconditioned conjugate gradient method to solve generalized least squares problems is presented if the problem is min ax b t w 1 ax b with a r m n and w r m m symmetric and positive definite the method needs only a preconditioner a 1 r n n but not the inverse of matrix w or of any of its submatrices freund s comparison result for regular least squares problems is extended to generalized least squares problems an error bound is also given

on block preconditioners for nonsymmetric saddle point problems we discuss a class of preconditioning methods for an iterative solution of algebraic nonsymmetric saddle point problems arising from a mixed finite element discretization of partial differential equations in particular the navier stokes equation we prove that block diagonal and block triangular preconditioners based on symmetric positive definite blocks guarantee that the convergence rate of the method is independent of the mesh parameter h

on the eigenvalues of a class of saddle point matrices we study spectral properties of a class of block 2 2 matrices that arise in the solution of saddle point problems these matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system we give conditions for having a positive real spectrum and for ensuring diagonalizability of the matrix in particular we show that these properties hold for the discrete stokes operator and we discuss the implications of our characterization for augmented lagrangian formulations for krylov subspace solvers and for certain types of preconditioners

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

engineering design calculations with fuzzy parameters uncertainty in engineering analysis usually pertains to stochastic uncertainty i e v ariance in product or process parameters characterized by probability uncertainty in truth methods for calculating under stochastic uncertainty are well documented it has been proposed by the authors that other forms of uncertainty exist in engineering design imprecision or the concept of uncertainty in choice is one such form this paper considers real time techniques for calculating with imprecise parameters these techniques utilize interval mathematics and the notion of cuts from the fuzzy calculus the extremes or anomalies of the techniques are also investigated particularly the evaluation of singular or multi valued functions it will be shown that realistic engineering functions can be used in imprecision calculations with reasonable computational performance

uniform preconditioners for the time dependent stokes problem implicit time stepping procedures for the time dependent stokes problem lead to stationary singular perturbation problems at each time step these singular perturbation problems are systems of saddle point type which formally approach a mixed formulation of the poisson equation as the time step tends to zero preconditioners for discrete analogous of these systems are discussed the preconditioners uses standard positive definite elliptic preconditioners as building blocks and lead to condition numbers which are bounded uniformly with respect to the time step and the spatial discretization the construction of the discrete preconditioners is related to the mapping properties of the corresponding continuous system

the transformation method for the simulation and analysis of systems with uncertain parameters in this paper the transformation method is introduced as a powerful approach for both the simulation and the analysis of systems with uncertain model parameters based on the concept of cuts the method represents a special implementation of fuzzy arithmetic that avoids the well known effect of overestimation which usually arises when fuzzy arithmetic is reduced to interval computation systems with uncertain model parameters can thus be simulated without any artificial widening of the simulation results as a by product of the implementation scheme the transformation method also provides a measure of influence to quantitatively analyze the uncertain system with respect to the effect of each uncertain model parameter on the overall uncertainty of the model output by this a special kind of sensitivity analysis can be defined on the basis of fuzzy arithmetic finally to show the efficiency of the transformation method the method is applied to the simulation and analysis of a model for the friction interface between the sliding surfaces of a bolted joint connection

balanced truncation model reduction for semidiscretized stokes equation we discuss model reduction of linear continuous time descriptor systems that arise in the control of semidiscretized stokes equations balanced truncation model reduction methods for descriptor systems are presented these methods are closely related to the proper and improper controllability and observability gramians and hankel singular values of descriptor systems the gramians can be computed by solving projected generalized lyapunov equations important properties of the balanced truncation approach are that the asymptotic stability is preserved in the reduced order system and there is an a priori bound on the approximation error we demonstrate the application of balanced truncation model reduction to the semidiscretized stokes equation

two block triangular preconditioners for asymmetric saddle point problems in this paper two block triangular preconditioners for the asymmetric saddle point problems with singular 1 1 block are presented the spectral characteristics of the preconditioned matrices are discussed in detail theoretical analysis shows that all the eigenvalues of the preconditioned matrices are strongly clustered numerical experiments are reported to the efficiency of the proposed preconditioners