a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

the structured sensitivity of vandermonde like systems we consider a general class of structured matrices that includes possibly confluent vandermonde and vandermonde like matrices here the entries in the matrix depend nonlinearly upon a vector of parameters we define condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right hand side convenient expressions are derived for the infinity norm based condition numbers and order of magnitude estimates are given for condition numbers defined in terms of a general vector norm we then discuss the computation of the corresponding backward errors after linearising the constraints we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity norm solution of an underdetermined linear system exact componentwise condition numbers are also derived for matrix inversion and the least squares problem and the linearised least squares backward error is characterised

structured condition numbers and small sample condition estimation of symmetric algebraic riccati equations abstract this paper is devoted to a structured perturbation analysis of the symmetric algebraic riccati equations by exploiting the symmetry structure based on the analysis the upper bounds for the structured normwise mixed and componentwise condition numbers are derived due to the exploitation of the symmetry structure our results are improvements of the previous work on the perturbation analysis and condition numbers of the symmetric algebraic riccati equations our preliminary numerical experiments demonstrate that our condition numbers provide accurate estimates for the change in the solution caused by the perturbations on the data moreover by applying the small sample condition estimation method we propose a statistical algorithm for practically estimating the condition numbers of the symmetric algebraic riccati equations

mixed and componentwise condition numbers for a linear function of the solution of the total least squares problem abstract in this paper we consider the mixed and componentwise condition numbers for a linear function lx of the solution to the total least squares tls problem we derive the explicit expressions of the mixed and componentwise condition numbers through the dual techniques under both unstructured and structured componentwise perturbations the sharp upper bounds for condition numbers are obtained an efficient condition estimation algorithm is proposed which can be integrated into the iterative method for solving large scale tls problems moreover the new derived condition number expressions can recover the previous results on the condition analysis for the tls problem when l i n numerical experiments show the effectiveness of the introduced condition numbers and condition estimation algorithm

a boundary integral method for the three dimensional heat equation subject to specification of energy a boundary integral method is proposed for the numerical solution of the three dimensional heat equation subject to specification of energy a specific test problem is solved using the method

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

fully implicit finite differences methods for two dimensional diffusion with a non local boundary condition three new fully implicit methods which are based on the 5 5 crank nicolson method the 5 5 n h noye hayman implicit method and the 9 9 n h implicit method are developed for solving the heat equation in two dimensional space with non local boundary conditions the latter is fourth order while the others are second order while the implicit methods developed here like the scheme based on the standard implicit backward time centered space btcs method use a large amount of central processor cpu time the high accuracy of the new fourth order fully implicit scheme is significant like the btcs method the new methods are also unconditionally stable

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

a new approach for numerical modal analysis using the element free method a novel approach in analysing the vibration of elastic body systems is presented and discussed in this paper this method uses a meshless spatial approximation based only on nodes which constitutes an advantage over the finite element method as it is applicable to an elastic body with arbitrary shape it also has advantages over the classical rayleigh ritz method and its extensions the paper is organized as follows in section 2 hamilton s principle is used for obtaining the equations of motion for a three dimensonal simply connected elastic body in section 3 the development of weighted base functions for the moving least squares interpolant is described in section 4 the mass and stiffness matrices are derived using the previous results in section 5 a system synthesis description is given in which we also develop the approach used for integrating the boundary conditions finally in section 6 the theoretical results are validated by case studies chosen to highlight various features of the approach including optimization of the shape function parameters the particle method has already been used for structural dynamics liu et al international journal for numerical methods in engineering 1995 38 1655 1679 but for free vibration of beams and plates to the authors knowledge this paper is the first application of the element free method to modal analysis copyright c 1999 john wiley sons ltd

parametric study for an efficient meshless method in vibration analysis based on the moving least squares mls approach an efficient meshless method is employed to generate the displacement functions for vibration analysis of elastic bodies the equation of motion is established by following the standard procedure and the boundary conditions are imposed by applying penalty functions as the displacement functions are expressed in terms of weight functions the accuracy will depend on the parameters of the weight functions therefore a parametric study is carried out to determine the best values for these parameters to demonstrate the accuracy modal analyses of the beams and plates with different boundaries have been carried out in addition the responses of these structures under dynamic excitation have been analyzed the examples include simply supported beams subjected to sudden excitations and simply supported plates subjected to initial displacements

a three dimensional element free galerkin elastic and elastoplastic formulation a small strain three dimensional elastic and elastoplastic element free galerkin efg formulation is developed singular weight functions are utilized in the moving least squares mls determination of shape functions and shape function derivatives allowing accurate direct nodal imposition of essential boundary conditions a variable domain of influence efg method is introduced leading to increased efficiency in computing the mls shape functions and their derivatives the elastoplastic formulations are based on the consistent tangent operator approach and closely follow the incremental formulations for non linear analysis using finite elements several linear elastic and small strain elastoplastic numerical examples are presented to verify the accuracy of the numerical formulations copyright 1999 john wiley sons ltd

schur type methods for solving least squares problems with toeplitz structure we give an overview of fast algorithms for solving least squares problems with toeplitz structure based on generalization of the classical schur algorithm and discuss their stability properties in order to obtain more accurate triangular factors of a toeplitz matrix as well as accurate solutions for the least squares problems methods based on corrected seminormal equations csne can be used we show that the applicability of the generalized schur algorithm is considerably enhanced when the algorithm is used in conjunction with csne several numerical tests are reported where different variants of the generalized schur algorithm and csne are compared for their accuracy and speed

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

fft based preconditioners for toeplitz block least squares problems discretized two dimensional deconvolution problems arising e g in image restoration and seismic tomography can be formulated as least squares computations min b tx 2 where t is often a large scale rectangular toeplitz block matrix the authors consider solving such block least squares problems by the preconditioned conjugate gradient algorithm using square nonsingular circulant block and related preconditioners constructed from the blocks of the rectangular matrix t preconditioning with such matrices allows efficient implementation using the one dimensional or two dimensional fast fourier transform fft two block preconditioners related to those proposed by t chan and j olkin for square nonsingular toeplitz block systems are derived and analyzed it is shown that for important classes of t the singular values of the preconditioned matrix are clustered around one this extends the authors earlier work on preconditioners for toeplitz least squares iterations for one dimensiona

structured condition numbers of structured tikhonov regularization problem and their estimations both structured componentwise and structured normwise perturbation analysis of the tikhonov regularization are presented the structured matrices under consideration include toeplitz hankel vandermonde and cauchy matrices structured normwise mixed and componentwise condition numbers for the tikhonov regularization are introduced and their explicit expressions are derived for the general linear structure based on the derived expressions we prove structured condition numbers are smaller than their corresponding unstructured counterparts by means of the power method and small sample statistical condition estimation sce fast condition estimation algorithms are proposed our estimation methods can be integrated into tikhonov regularization algorithms that use the generalized singular value decomposition gsvd for large scale linear structured tikhonov regularization problems we show how to incorporate the sce into the preconditioned conjugate gradient pcg method to get the posterior error estimations the structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones and the proposed condition estimation algorithms are reliable also an image restoration example is tested to show the effectiveness of the sce for large scale linear structured tikhonov regularization problems

analysis of thin shells by the element free galerkin method a meshless approach to the analysis of arbitrary kirchhoff shells by the element free galerkin efg method is presented the shell theory used is geometrically exact and can be applied to deep shells the method is based on moving least squares approximant the method is meshless which means that the discretization is independent of the geometric subdivision into finite elements the satisfaction of the c1 continuity requirements is easily met by efg since it requires only c1 weights therefore it is not necessary to resort to mindlin reissner theory or to devices such as discrete kirchhoff theory the requirements of consistency are met by the use of a polynomial basis of quadratic or higher order a subdivision similar to finite elements is used to provide a background mesh for numerical integration the essential boundary conditions are enforced by lagrange multipliers membrane locking which is due to different approximation order for transverse and membrane displacements is removed by using larger domains of influence with the quadratic basis and by using quartic polynomial basis which can prevent membrane locking completely it is shown on the obstacle course for shells that the present technique performs well

bttb preconditioners for bttb least squares problems abstract in this paper we consider solving the least squares problem min x b t x 2 by using preconditioned conjugate gradient pcg methods where t is a large rectangular matrix which consists of several square block toeplitz toeplitz block bttb matrices and b is a column vector we propose a bttb preconditioner to speed up the pcg method and prove that the bttb preconditioner is a good preconditioner we then discuss the construction of the bttb preconditioner numerical examples including image restoration problems are given to illustrate the efficiency of our bttb preconditioner numerical results show that our bttb preconditioner is more efficient than the well known level 1 and level 2 circulant preconditioners

statistical condition estimation for the roots of polynomials this paper presents fast and reliable condition estimates for the roots of a real polynomial based on the method of statistical condition estimation sce by kenney and laub using relative perturbations we provide both a basic implementation of sce and an advanced one via the condition of the eigenvalues of the corresponding companion matrix new results for estimating the condition of eigenvalues are given fast structured calculations of the eigenvalues of the companion matrix are used and fast structured manipulations of the schur decomposition and the invariant subspaces of the companion matrix are presented the overall process is based on fast operations for sequentially semiseparable structures with small off diagonal ranks the cost of obtaining the condition estimates for all of the roots is o n 2 where n is the degree of the polynomial numerical examples are used to demonstrate the accuracy and efficiency of the proposed methods

stability analysis and fast algorithms for triangularization of toeplitz matrices a new o mn algorithm for triangularizing an m times n toeplitz matrix is presented the algorithm is based on the previously developed recursive algorithms that exploit the toeplitz structure and compute each row of the triangular factor via updating and downdating steps a forward error analysis for this existing recursive algorithm is presented which allows us to monitor the conditioning of the problem and use the method of corrected semi normal equations to obtain higher accuracy for certain ill conditioned matrices numerical experiments show that the new algorithm improves the accuracy significantly while the computational complexity stays in o mn

small sample statistical condition estimates for general matrix functions a new condition estimation procedure for general matrix functions is presented that accurately gauges sensitivity by measuring the effect of random perturbations at the point of evaluation in this procedure the number of extra function evaluations used to evaluate the condition estimate determines the order of the estimate that is the probability that the estimate is off by a given factor is inversely proportional to the factor raised to the order of the method the transpose free nature of this new method allows it to be applied to a broad range of problems in which the function maps between spaces of different dimensions this is in sharp contrast to the more common power method condition estimation procedure that is limited in the usual case where the frechet derivative is known only implicitly to maps between spaces of equal dimension a group of examples illustrates the flexibility of the new estimation procedure in handling a variety of problems and types of sensitivity estimates such as mix

on condition numbers for least squares with quadric inequality constraint abstract in this paper we will study normwise mixed and componentwise condition numbers for the linear mapping of the solution for general least squares with quadric inequality constraint glsqi and its standard form lsqi we will introduce the mappings from the data space to the interested data space and the frechet derivative of the introduced mapping can deduced through matrix differential techniques based on condition number theory we derive the explicit expressions of normwise mixed and componentwise condition numbers for the linear function of the solution for glsqi and lsqi also easier computable upper bounds for mixed and componentwise condition numbers are given numerical example shows that the mixed and componentwise condition numbers can tell us the true conditioning of the problem when its data is sparse or badly scaled compared with normwise condition numbers the mixed and componentwise condition number can give sharp perturbation bounds

statistical condition estimation for linear systems the standard approach to measuring the condition of a linear system compresses all sensitivity information into one number thus a loss of information can occur in situations in which the standard condition number with respect to inversion does not accurately reflect the actual sensitivity of a solution or particular entries of a solution it is shown that a new method for estimating the sensitivity of linear systems addresses these difficulties the new procedure measures the effects on the solution of small random changes in the input data and by properly scaling the results obtains reliable condition estimates for each entry of the computed solution moreover this approach which is referred to as small sample statistical condition estimation is no more costly than the standard 1 norm or power method 2 norm condition estimates and it has the advantage of considerable flexibility for example it easily accommodates restrictions on or structure associated with allowable perturbations the method also has a rigorous statistical theory available for the probability of accuracy of the condition estimates however it gives no estimate of an approximate null vector for nearly singular systems the theory of this approach is discussed along with several illustrative examples

a time stepping dual reciprocity boundary element method for anisotropic heat diffusion subject to specification of energy a two dimensional problem which requires determining the non steady temperature distribution in a thermally anisotropic body and a temperature control function on a certain part of the boundary of the body given that the total amount of heat energy in the body is known at all time is considered it is solved numerically using a time stepping dual reciprocity boundary element method numerical results are obtained for specific test problems

structured mixed and componentwise condition numbers of some structured matrices structured matrices such as cauchy vandermonde toeplitz hankel and circulant matrices are considered in this paper we apply a kronecker product based technique to deduce the structured mixed and componentwise condition numbers for the matrix inversion and for the corresponding linear systems