deterministic brownian motion generated from differential delay equations this paper addresses the question of how brownian like motion can arise from the solution of a deterministic differential delay equation to study this we analytically study the bifurcation properties of an apparently simple differential delay equation and then numerically investigate the probabilistic properties of chaotic solutions of the same equation our results show that solutions of the deterministic equation with randomly selected initial conditions display a gaussian like density for long time but the densities are supported on an interval of finite measure using these chaotic solutions as velocities we are able to produce brownian like motions which show statistical properties akin to those of a classical brownian motion over both short and long time scales several conjectures are formulated for the probabilistic properties of the solution of the differential delay equation numerical studies suggest that these conjectures could be universal for similar types of chaotic dynamics but we have been unable to prove this

variants of the deteriorated pss preconditioner for saddle point problems two new preconditioners which can be viewed as variants of the deteriorated positive definite and skew hermitian splitting preconditioner are proposed for solving saddle point problems the corresponding iteration methods are proved to be convergent unconditionally for cases with positive definite leading blocks the choice strategies of optimal parameters for the two iteration methods are discussed based on two recent optimization results for extrapolated cayley transform which result in faster convergence rate and more clustered spectrum compared with some preconditioners of similar structures the new preconditioners have better convergence properties and spectrum distributions in addition more practical preconditioning variants of the new preconditioners are considered numerical experiments are presented to illustrate the advantages of the new preconditioners over some similar preconditioners to accelerate gmres c 2018 elsevier ltd all rights reserved

a general class of shift splitting preconditioners for non hermitian saddle point problems with applications to time harmonic eddy current models abstract based on a general splitting of the 1 1 leading block matrix we first construct a general class of shift splitting gcss preconditioners for non hermitian saddle point problems convergence conditions of the corresponding matrix splitting iteration methods and preconditioning properties of the gcss preconditioned saddle point matrices are analyzed then the gcss preconditioner is specifically applied to the non hermitian saddle point problems arising from the finite element discretizations of the hybrid formulations of the time harmonic eddy current models with suitable choices of the splittings the new gcss preconditioners are easier to implement and have faster convergence rates than the existing shift splitting preconditioner and its modified variant two numerical examples are presented to verify the theoretical results and show effectiveness of the new proposed preconditioners

explicit recursive and adaptive filtering in reproducing kernel hilbert spaces this brief presents a methodology to develop recursive filters in reproducing kernel hilbert spaces unlike previous approaches that exploit the kernel trick on filtered and then mapped samples we explicitly define the model recursivity in the hilbert space for that we exploit some properties of functional analysis and recursive computation of dot products without the need of preimaging or a training dataset we illustrate the feasibility of the methodology in the particular case of the gamma filter which is an infinite impulse response filter with controlled stability and memory depth different algorithmic formulations emerge from the signal model experiments in chaotic and electroencephalographic time series prediction complex nonlinear system identification and adaptive antenna array processing demonstrate the potential of the approach for scenarios where recursivity and nonlinearity have to be readily combined

an efficient robust adaptive filtering algorithm based on parallel subgradient projection techniques this paper presents a novel robust adaptive filtering scheme based on the interactive use of statistical noise information and the ideas developed originally for efficient algorithmic solutions to the convex feasibility problems the statistical noise information is quantitatively formulated as stochastic property closed convex sets by the simple design formulae developed in this paper a simple set theoretic inspection also leads to an important statistical reason for the sensitivity to noise of the affine projection algorithm apa the proposed adaptive algorithm is computationally efficient and robust to noise because it requires only an iterative parallel projection onto a series of closed half spaces that are highly expected to contain the unknown system to be identified and is free from the computational load of solving a system of linear equations the numerical examples show that the proposed adaptive filtering scheme realizes dramatically fast and stable convergence for highly colored excited speech like input signals in severe noise situations

adaptive learning in complex reproducing kernel hilbert spaces employing wirtinger s subgradients this paper presents a wide framework for non linear online supervised learning tasks in the context of complex valued signal processing the complex input data are mapped into a complex reproducing kernel hilbert space rkhs where the learning phase is taking place both pure complex kernels and real kernels via the complexification trick can be employed moreover any convex continuous and not necessarily differentiable function can be used to measure the loss between the output of the specific system and the desired response the only requirement is the subgradient of the adopted loss function to be available in an analytic form in order to derive analytically the subgradients the principles of the recently developed wirtinger s calculus in complex rkhs are exploited furthermore both linear and widely linear in rkhs estimation filters are considered to cope with the problem of increasing memory requirements which is present in almost all online schemes in rkhs the sparsification scheme based on projection onto closed balls has been adopted we demonstrate the effectiveness of the proposed framework in a non linear channel identification task a non linear channel equalization problem and a quadrature phase shift keying equalization scheme using both circular and non circular synthetic signal sources

perturbation theory for dual semigroups i the sun reflexive case soit a 0 le generateur infinitesimal d un co semigroupe d operateurs lineaires bornes to t sur un espace de banach x soit b un operateur lineaire borne sur x a 0 b engendre un co semigroupe t t sur x b agit hors de l espace x dans un espace plus grand y on donne une methode systematique pour construire un tel espace y

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

fast iterative solution of stabilised stokes systems part i using simple diagonal preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems the use of stabilisation methods for convenient but unstable mixed elements introduces stabilisation parameters we show how these can be chosen to obtain rapid iterative convergence we propose a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems describe the effects of stabilisation on the algebraic structure of the discrete stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends here we discuss the simple case of diagonal preconditioning our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable we demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete laplacian are approachable for the stokes problem

pruning error minimization in least squares support vector machines the support vector machine svm is a method for classification and for function approximation this method commonly makes use of an spl epsi insensitive cost function meaning that errors smaller than spl epsi remain unpunished as an alternative a least squares support vector machine lssvm uses a quadratic cost function when the lssvm method is used for function approximation a nonsparse solution is obtained the sparseness is imposed by pruning i e recursively solving the approximation problem and subsequently omitting data that has a small error in the previous pass however omitting data with a small approximation error in the previous pass does not reliably predict what the error will be after the sample has been omitted in this paper a procedure is introduced that selects from a data set the training sample that will introduce the smallest approximation error when it will be omitted it is shown that this pruning scheme outperforms the standard one

on local hermitian and skew hermitian splitting iteration methods for generalized saddle point problems in this paper we first present a local hermitian and skew hermitian splitting lhss iteration method for solving a class of generalized saddle point problems the new method converges to the solution under suitable restrictions on the preconditioning matrix then we give a modified lhss mlhss iteration method and further extend it to the generalized saddle point problems obtaining the so called generalized mlhss gmlhss iteration method numerical experiments for a model navier stokes problem are given and the results show that the new methods outperform the classical uzawa method and the inexact parameterized uzawa method

efficient dictionary refining kernel adaptive filter with fundamental insights first we study a basic kernel adaptive filtering algorithm based on stochastic restricted gradient which is a natural extension of the naive online regularized risk minimization algorithm norma our error surface analysis shows that the algorithm has a decorrelation property due to its explicit use of kernel matrix it turns out that its normalized version projects the current coefficient vector onto the same hyperplane but with a different metric as the kernel normalized least mean square knlms algorithm this gives us a fundamental insight bridging two classes of kernel adaptive filtering algorithms second we use this insight to derive an efficient forward backward splitting algorithm the same metric as used for the normalized algorithm is employed in the forward step whereas the ordinary euclidean metric is employed in the backward step leading to efficient adaptive refinements of the filter dictionary we show a monotone approximation property of the proposed algorithm with respect to some modified cost function under certain conditions numerical examples show the efficacy of the proposed algorithm

a relaxed deteriorated pss preconditioner for nonsymmetric saddle point problems from the steady navier stokes equation for nonsymmetric saddle point problems arising from the steady navier stokes equations pan ng and bai presented a deteriorated positive definite and skew hermitian splitting dpss preconditioner pan et al 2006 to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the unconditional convergence property of the dpss iteration method is proved and a relaxed dpss rdpss preconditioner is proposed the rdpss preconditioner is much closer to the coefficient matrix than the dpss preconditioner in certain norm which straightforwardly results in an rdpss iteration method the convergence conditions of the rdpss iteration are analyzed and the optimal parameter which minimizes the spectral radius of the rdpss iteration matrix is derived using the rdpss preconditioner to accelerate some krylov subspace methods like gmres is also studied the eigenproperty of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the rdpss preconditioner

adaptive multiregression in reproducing kernel hilbert spaces the multiaccess mimo channel case this paper introduces a wide framework for online i e time adaptive supervised multiregression tasks the problem is formulated in a general infinite dimensional reproducing kernel hilbert space rkhs in this context a fairly large number of nonlinear multiregression models fall as special cases including the linear case any convex continuous and not necessarily differentiable function can be used as a loss function in order to quantify the disagreement between the output of the system and the desired response the only requirement is the subgradient of the adopted loss function to be available in an analytic form to this end we demonstrate a way to calculate the subgradients of robust loss functions suitable for the multiregression task as it is by now well documented when dealing with online schemes in rkhs the memory keeps increasing with each iteration step to attack this problem a simple sparsification strategy is utilized which leads to an algorithmic scheme of linear complexity with respect to the number of unknown parameters a convergence analysis of the technique based on arguments of convex analysis is also provided to demonstrate the capacity of the proposed method the multiregressor is applied to the multiaccess multiple input multiple output channel equalization task for a setting with poor resources and nonavailable channel information numerical results verify the potential of the method when its performance is compared with those of the state of the art linear techniques which in contrast use space time coding more antenna elements as well as full channel information

extension of wirtinger s calculus to reproducing kernel hilbert spaces and the complex kernel lms over the last decade kernel methods for nonlinear processing have successfully been used in the machine learning community the primary mathematical tool employed in these methods is the notion of the reproducing kernel hilbert space rkhs however so far the emphasis has been on batch techniques it is only recently that online techniques have been considered in the context of adaptive signal processing tasks moreover these efforts have only been focussed on real valued data sequences to the best of our knowledge no adaptive kernel based strategy has been developed so far for complex valued signals furthermore although the real reproducing kernels are used in an increasing number of machine learning problems complex kernels have not yet been used in spite of their potential interest in applications that deal with complex signals with communications being a typical example in this paper we present a general framework to attack the problem of adaptive filtering of complex signals using either real reproducing kernels taking advantage of a technique called complexification of real rkhss or complex reproducing kernels highlighting the use of the complex gaussian kernel in order to derive gradients of operators that need to be defined on the associated complex rkhss we employ the powerful tool of wirtinger s calculus which has recently attracted attention in the signal processing community wirtinger s calculus simplifies computations and offers an elegant tool for treating complex signals to this end in this paper the notion of wirtinger s calculus is extended for the first time to include complex rkhss and use it to derive several realizations of the complex kernel least mean square cklms algorithm experiments verify that the cklms offers significant performance improvements over several linear and nonlinear algorithms when dealing with nonlinearities

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

the asymptotic shapes of periodic solutions of a singular delay differential system let f r r be given so that f 0 0 and f x 1 x ax2 bx3 o x3 as x 0 we characterize those small values of e 0 and r for which there are periodic solutions of periods approximately 2k with k n of the following system arising from a network of neurons formula the periodic solutions are synchronized if k is even and phase locked if k is odd we show that as e 0 these periodic solutions approach square waves if a 0 and b 0 or if a 0

adaptive nonlinear estimation based on parallel projection along affine subspaces in reproducing kernel hilbert space we propose a novel algorithm using a reproducing kernel for adaptive nonlinear estimation the proposed algorithm is based on three ideas projection along subspace selective update and parallel projection the projection along subspace yields excellent performances with small dictionary sizes the selective update effectively reduces the complexity without any serious degradation of performance the parallel projection leads to fast convergence tracking accompanied by noise robustness a convergence analysis in the non selective update case is presented by using the adaptive projected subgradient method simulation results exemplify the benefits from the three ideas as well as showing the advantages over the state of the art algorithms the proposed algorithm bridges the quantized kernel least mean square algorithm of chen and the sparse sequential algorithm of dodd

on inexact preconditioners for nonsymmetric matrices inexact versions of the block triangular preconditioners for nonsymmetric matrices of block two by two structures in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 and i c f ipsen siam j sci comput 23 2001 pp 1050 1051 are presented and the two preconditioners for symmetric block two by two matrices in c durazzi and v ruggiero numer linear algebra appl 10 2003 pp 673 688 are extended to general nonsymmetric matrices moreover we precisely describe the spectral properties of the preconditioned matrices and the finite step termination properties of the preconditioned krylov subspace iteration methods with an optimal or galerkin property with respect to these preconditioners several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners

natural preconditioning and iterative methods for saddle point systems the solution of quadratic or locally quadratic extremum problems subject to linear ized constraints gives rise to linear systems in saddle point form this is true whether in the continuous or the discrete setting so saddle point systems arising from the discretization of partial differential equation problems such as those describing electromagnetic problems or incompressible flow lead to equations with this structure as do for example interior point methods and the sequential quadratic programming approach to nonlinear optimization this survey concerns iterative solution methods for these problems and in particular shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods these preconditioners are related to the original extremum problem and their effectiveness in terms of rapidity of convergence is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matri

a relaxed dimensional factorization preconditioner for the incompressible navier stokes equations in this paper we introduce a relaxed dimensional factorization rdf preconditioner for saddle point problems properties of the preconditioned matrix are analyzed and compared with those of the closely related dimensional splitting ds preconditioner recently introduced by benzi and guo 7 numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the rdf preconditioner with respect to both mesh size and viscosity

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

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

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

iterative methods for the saddle point problem arising from the hc ei formulation of the eddy current problem this paper is concerned with the solution of the linear system arising from a finite element approximation of the time harmonic eddy current problem we consider the bf h c bf e i formulation introduced and analyzed in a m alonso rodriguez r hiptmair and a valli numer methods partial differential equations 21 2005 pp 742 763 where an optimal error estimate for the finite element approximation using edge elements of the first order is proved now we propose and analyze iterative procedures for the solution of the resulting linear system based on the physical decomposition of the computational domain in an insulating region and a conducting one if the insulator does not contain any nonbounding cycle we prove that the dirichlet neumann iteration converges with a rate that is independent of the mesh size in the case of a connected conductor with general topology we propose to use either a modified version of the dirichlet neumann iteration or an uzawa like method we compare the per

online sparse system identification and signal reconstruction using projections onto weighted ell_ 1 balls this paper presents a novel projection based adaptive algorithm for sparse signal and system identification the sequentially observed data are used to generate an equivalent sequence of closed convex sets namely hyperslabs each hyperslab is the geometric equivalent of a cost criterion that quantifies data mismatch sparsity is imposed by the introduction of appropriately designed weighted l1 balls and the related projection operator is also derived the algorithm develops around projections onto the sequence of the generated hyperslabs as well as the weighted l1 balls the resulting scheme exhibits linear dependence with respect to the unknown system s order on the number of multiplications additions and an o llog2l dependence on sorting operations where l is the length of the system signal to be estimated numerical results are also given to validate the performance of the proposed method against the least absolute shrinkage and selection operator lasso algorithm and two very recently developed adaptive sparse schemes that fuse arguments from the lms rls adaptation mechanisms with those imposed by the lasso rational

ikeda hopf bifurcation revisited abstract a large variety of passive optical systems subject to a delayed feedback have appeared in the literature and are described mathematically by the same class of scalar delay differential equations ddes these equations include ikeda dde and their solutions are determined in terms of a control parameter distinct from the delay we concentrate on the first hopf bifurcation generated by a fixed delay and determine a general expression for its direction of bifurcation we then examine our result in the two limits of small and large delays for small delays we show that a hopf bifurcation to nearly sinusoidal oscillations is possible provided that the feedback rate is sufficiently high bifurcation from infinity for large delays we complement the early work by chow et al proc roy soc edinburgh a 120 1992 223 229 and hale and huang j diff equ 114 1994 1 23 by comparing analytical and numerical bifurcation diagrams as the oscillations progressively change from sine to square wave

low complexity constrained affine projection algorithms this paper proposes low complexity constrained affine projection cap algorithms the algorithms are suitable for linearly constrained filtering problems often encountered in communications systems the cap algorithms derived in this paper trade convergence speed and computational complexity in the same way as the conventional affine projection ap algorithm in addition data selective versions of the cap algorithm are derived based on the concept of set membership filtering the set membership constrained affine projection sm cap algorithms include several constraint sets in order to construct a space of feasible solutions for the coefficient updates the sm cap algorithms include a data dependent step size that provides fast convergence and low mean squared error the paper also discusses important aspects of convergence and stability of constrained normalized adaptation algorithms and shows that normalization may introduce bias in the final solution

eigenvalue estimates for saddle point matrices of hermitian and indefinite leading blocks we study the eigenvalue bounds for the nonsingular saddle point matrices of hermitian and indefinite 1 1 and 2 2 blocks without imposing the restrictions that the 1 1 blocks are positive definite on the kernels of the 2 1 blocks

a note on preconditioning nonsymmetric matrices the preconditioners for indefinite matrices of kkt form in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 are extended to general nonsymmetric matrices

online kernel based classification using adaptive projection algorithms the goal of this paper is to derive a novel online algorithm for classification in reproducing kernel hilbert spaces rkhs by exploiting projection based adaptive filtering tools the paper brings powerful convex analytic and set theoretic estimation arguments in machine learning by revisiting the standard kernel based classification as the problem of finding a point which belongs to a closed halfspace a special closed convex set in an rkhs in this way classification in an online setting where data arrive sequentially is viewed as the problem of finding a point classifier in the nonempty intersection of an infinite sequence of closed halfspaces in the rkhs convex analysis is also used to introduce sparsification arguments in the design by imposing an additional simple convex constraint on the norm of the classifier an algorithmic solution to the resulting optimization problem where new convex constraints are added every time instant is given by the recently introduced adaptive projected subgradient method apsm which generalizes a number of well known projection based adaptive filtering algorithms such as the classical normalized least mean squares nlms and the affine projection algorithm apa under mild conditions the generated sequence of estimates enjoys monotone approximation strong convergence asymptotic optimality and a characterization of the limit point further we show that the additional convex constraint on the norm of the classifier naturally leads to an online sparsification of the resulting kernel series expansion we validate the proposed design by considering the adaptive equalization problem of a nonlinear channel and by comparing it with classical as well as with recently developed stochastic gradient descent techniques

quantized kernel least mean square algorithm in this paper we propose a quantization approach as an alternative of sparsification to curb the growth of the radial basis function structure in kernel adaptive filtering the basic idea behind this method is to quantize and hence compress the input or feature space different from sparsification the new approach uses the redundant data to update the coefficient of the closest center in particular a quantized kernel least mean square qklms algorithm is developed which is based on a simple online vector quantization method the analytical study of the mean square convergence has been carried out the energy conservation relation for qklms is established and on this basis we arrive at a sufficient condition for mean square convergence and a lower and upper bound on the theoretical value of the steady state excess mean square error static function estimation and short term chaotic time series prediction examples are presented to demonstrate the excellent performance

delay induced instabilities in nonlinear feedback systems extending general methods developed in synergetics to construct order parameter equations we work out a systematic elimination procedure for the stable modes in nonlinear delay systems it will be shown that in the vicinity of an instability the dynamical behavior of the infinite dimensional delay system is approximately governed by a low dimensional set of order parameter equations which turn out to be of the form of ordinary differential equations i e they no longer contain memory terms the general formalism will then be compared with experimental data obtained from the nonlinear operation of a phase locked loop where the finite propagation time of the signal in the feedback loop is taken into account

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

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

subcritical hopf bifurcation in dynamical systems described by a scalar nonlinear delay differential equation a subcritical hopf bifurcation in a dynamical system modeled by a scalar nonlinear delay differential equation is studied theoretically and experimentally the subcritical hopf bifurcation leads to a significant domain of bistability where stable steady and time periodic states coexist

a geometric approach to support vector machine svm classification the geometric framework for the support vector machine svm classification problem provides an intuitive ground for the understanding and the application of geometric optimization algorithms leading to practical solutions of real world classification problems in this work the notion of reduced convex hull is employed and supported by a set of new theoretical results these results allow existing geometric algorithms to be directly and practically applied to solve not only separable but also nonseparable classification problems both accurately and efficiently as a practical application of the new theoretical results a known geometric algorithm has been employed and transformed accordingly to solve nonseparable problems successfully

the augmented complex kernel lms recently a unified framework for adaptive kernel based signal processing of complex data was presented by the authors which besides offering techniques to map the input data to complex reproducing kernel hilbert spaces developed a suitable wirtinger like calculus for general hilbert spaces in this short paper the extended wirtinger s calculus is adopted to derive complex kernel based widely linear estimation filters suitable for applications involving noncircular data furthermore we illuminate several important characteristics of the widely linear filters we show that although in many cases the gains from adopting widely linear estimation filters as alternatives to ordinary linear ones are rudimentary for the case of kernel based widely linear filters significant performance improvements can be obtained

substructuring preconditioners for saddle point problems arising from maxwell s equations in three dimensions this paper is concerned with the saddle point problems arising from edge element discretizations of maxwell s equations in a general three dimensional nonconvex polyhedral domain a new augmented technique is first introduced to transform the problems into equivalent augmented saddle point systems so that they can be solved by some existing preconditioned iterative methods then some substructuring preconditioners are proposed with very simple coarse solvers for the augmented saddle point systems with the preconditioners the condition numbers of the preconditioned systems are nearly optimal namely they grow only as the logarithm of the ratio between the subdomain diameter and the finite element mesh size

a modified dimensional split preconditioner for generalized saddle point problems abstract for large and sparse saddle point linear systems arising from 2d linearized navier stokes equations benzi and guo recently studied a dimensional split ds preconditioner appl numer math 61 2011 66 76 by further applying it to generalized saddle point problems in this paper we present a modified dimensional split mds preconditioner this new preconditioner is based on a splitting of the generalized saddle point matrix resulting in an unconditional convergent fixed point iteration the basic iteration is accelerated by a krylov subspace method like restarted gmres the implementation of the mds preconditioner is discussed and a similar case is also analyzed finally numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the mds preconditioner

adaptive robust distributed learning in diffusion sensor networks in this paper the problem of adaptive distributed learning in diffusion networks is considered the algorithms are developed within the convex set theoretic framework more specifically they are based on computationally simple geometric projections onto closed convex sets the paper suggests a novel combine project adapt protocol for cooperation among the nodes of the network such a protocol fits naturally with the philosophy that underlies the projection based rationale moreover the possibility that some of the nodes may fail is also considered and it is addressed by employing robust statistics loss functions such loss functions can easily be accommodated in the adopted algorithmic framework all that is required from a loss function is convexity under some mild assumptions the proposed algorithms enjoy monotonicity asymptotic optimality asymptotic consensus strong convergence and linear complexity with respect to the number of unknown parameters finally experiments in the context of the system identification task verify the validity of the proposed algorithmic schemes which are compared to other recent algorithms that have been developed for adaptive distributed learning

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

complex support vector machines for regression and quaternary classification the paper presents a new framework for complex support vector regression svr as well as support vector machines svm for quaternary classification the method exploits the notion of widely linear estimation to model the input out relation for complex valued data and considers two cases 1 the complex data are split into their real and imaginary parts and a typical real kernel is employed to map the complex data to a complexified feature space and 2 a pure complex kernel is used to directly map the data to the induced complex feature space the recently developed wirtinger s calculus on complex reproducing kernel hilbert spaces is employed to compute the lagrangian and derive the dual optimization problem as one of our major results we prove that any complex svm svr task is equivalent with solving two real svm svr tasks exploiting a specific real kernel which is generated by the chosen complex kernel in particular the case of pure complex kernels leads to the generation of new kernels which have not been considered before in the classification case the proposed framework inherently splits the complex space into four parts this leads naturally to solving the four class task quaternary classification instead of the typical two classes of the real svm in turn this rationale can be used in a multiclass problem as a split class scenario based on four classes as opposed to the one versus all method this can lead to significant computational savings experiments demonstrate the effectiveness of the proposed framework for regression and classification tasks that involve complex data

adaptive constrained learning in reproducing kernel hilbert spaces the robust beamforming case this paper establishes a new paradigm for convexly constrained adaptive learning in reproducing kernel hilbert spaces rkhs although the technique is of a general nature we present it in the context of the beamforming problem a priori knowledge like beampattern specifications and constraints concerning robustness against steering vector errors takes the form of multiple closed convex sets in a high possibly infinite dimensional rkhs every robustness constraint is shown to be equivalent to a min max optimization task formed by means of the robust statistics epsiv insensitive loss function such a multiplicity of specifications turns out to obtain a simple expression by using the rich frame of fixed point sets of certain mappings defined in a hilbert space moreover the cost function that the final solution has to optimize is expressed as an infinite sequence of convex nondifferentiable loss functions springing from the sequence of the incoming training data a novel adaptive beamforming design of linear complexity with respect to the number of unknown parameters to such a constrained nonlinear learning problem is derived by employing a very recently developed version of the adaptive projected subgradient method apsm the method produces a sequence that under mild conditions exhibits properties like the strong convergence to a beamformer that satisfies all of the imposed constraints and in the meantime asymptotically minimizes the sequence of the loss functions imposed by the training data the numerical examples demonstrate that the proposed method displays increased resolution in cases where the classical linear beamforming solutions collapse moreover it leads to solutions which are in agreement with the imposed a priori knowledge as opposed to unconstrained online kernel regression techniques

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

adaptive learning in cartesian product of reproducing kernel hilbert spaces we propose a novel adaptive learning algorithm based on iterative orthogonal projections in the cartesian product of multiple reproducing kernel hilbert spaces rkhss the objective is to estimate or track nonlinear functions that are supposed to contain multiple components such as i linear and nonlinear components and ii high and low frequency components in this case the use of multiple rkhss permits a compact representation of multicomponent functions the proposed algorithm is where two different methods of the author meet multikernel adaptive filtering and the algorithm of hyperplane projection along affine subspace hypass in a particular case the sum space of the rkhss is isomorphic under a straightforward correspondence to the product space and hence the proposed algorithm can also be regarded as an iterative projection method in the sum space the efficacy of the proposed algorithm is shown by numerical examples

generalized shift splitting preconditioners for nonsingular and singular generalized saddle point problems based on the shift splitting technique a class of generalized shift splitting preconditioners are proposed for both nonsingular and singular generalized saddle point problems the generalized shift splitting preconditioner is induced by a generalized shift splitting of the generalized saddle point matrix resulting in a generalized shift splitting fixed point iteration theoretical analyses show that the generalized shift splitting iteration method is convergent and semi convergent unconditionally for solving the nonsingular and the singular generalized saddle point problems respectively numerical experiments of a model navier stokes problem are implemented to demonstrate the feasibility and effectiveness of the proposed preconditioners

constrained adaptation algorithms employing householder transformation this paper presents a tutorial like detailed explanation of linearly constrained minimum variance filtering in order to introduce an efficient implementation that utilizes householder transformation ht through a graphical description of the algorithms further insight on linearly constrained adaptive filters was made possible and the main differences among several algorithms were highlighted the method proposed herein based on the ht allows direct application of any unconstrained adaptation algorithm as in a generalized sidelobe canceller gsc but unlike the gsc the ht based approach always renders efficient implementations a complete and detailed comparison with the gsc model and a thorough discussion of the advantages of the ht based approach are also given simulations were run in a beamforming application where a linear array of 12 sensors was used it was verified that not only the ht approach yields efficient implementation of constrained adaptive filters but in addition the beampatterns achieved with this method were much closer to the optimal solution than the beampatterns obtained with gsc models with similar computational complexity

finite element simulation of eddy current problems using magnetic scalar potentials we propose a new implementation of the finite element approximation of eddy current problems using as the principal unknown the magnetic field in the non conducting region a scalar magnetic potential is introduced the method can deal automatically with any topological configuration of the conducting region and being based on the search of a scalar magnetic potential in the non conducting region has the advantage of making use of a reduced number of unknowns several numerical tests are presented for illustrating the performance of the proposed method in particular the numerical simulation of a new type of transformer of complicated topological shape is shown

analysis on block diagonal and triangular preconditioners for a pml system of an electromagnetic scattering problem abstract we shall propose several block triangular preconditioners for a pml system of an electromagnetic wave scattering problem and analyze the spectral behavior of the preconditioned systems when the pml system is discretized by edge element methods it results in a discrete system with its stiffness matrix being complex symmetric but indefinite which can be formulated into a real symmetric but indefinite saddle point system in order to preserve the symmetry of the coefficient matrix we present block triangular preconditioners with two sided preconditioning for the discrete pml system we will estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively on the other hand one may also like to apply some iteration methods for nonsymmetric linear systems in applications although the discrete systems are symmetric to this end we propose a block triangular preconditioner to precondition the systems only from one side and analyze the spectrum of the preconditioned systems in addition we have also established a spectral estimate of the preconditioned system by an effective preconditioner that was recently developed in literature numerical experiments are presented to demonstrate the effectiveness and robustness of these new preconditioners and our theoretical predictions on the spectral bounds of the preconditioned systems

multikernel adaptive filtering this paper exemplifies that the use of multiple kernels leads to efficient adaptive filtering for nonlinear systems two types of multikernel adaptive filtering algorithms are proposed one is a simple generalization of the kernel normalized least mean square knlms algorithm 2 adopting a coherence criterion for dictionary designing the other is derived by applying the adaptive proximal forward backward splitting method to a certain squared distance function plus a weighted block l1 norm penalty encouraging the sparsity of an adaptive filter at the block level for efficiency the proposed multikernel approach enjoys a higher degree of freedom than those approaches which design a kernel as a convex combination of multiple kernels numerical examples show that the proposed approach achieves significant gains particularly for nonstationary data as well as insensitivity to the choice of some design parameters

pairwise optimal weight realization acceleration technique for set theoretic adaptive parallel subgradient projection algorithm the adaptive parallel subgradient projection psp algorithm was proposed in 2002 as a set theoretic adaptive filtering algorithm providing fast and stable convergence robustness against noise and low computational complexity by using weighted parallel projections onto multiple time varying closed half spaces in this paper we present a novel weighting technique named pairwise optimal weight realization power for further acceleration of the adaptive psp algorithm a simple closed form formula is derived to compute the projection onto the intersection of two closed half spaces defined by a triplet of vectors using the formula inductively the proposed weighting technique realizes a good direction of update the resulting weights turn out to be pairwise optimal in a certain sense the proposed algorithm has the inherently parallel structure composed of q primitive functions hence its total computational complexity o qrn is reduced to o rn with q concurrent processors r a constant positive integer numerical examples demonstrate that the proposed technique for r 1 yields significantly faster convergence than not only adaptive psp with uniform weights affine projection algorithm and fast newton transversal filters but also the regularized recursive least squares algorithm

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

preconditioners for regularized saddle point problems with an application for heterogeneous darcy flow problems saddle point problems arise in the modeling of many important practical situations preconditioners for the corresponding matrices of block triangular form based on coupled inner outer iteration methods are analyzed and applied to a darcy flow problem possibly with strong heterogeneity and non symmetric saddle point systems using proper regularized forms of the given matrix and its preconditioner it is shown that the eigenvalues cluster about one or two points on the real axis for large values of the regularization parameters and that eigenvalue bounds do not depend on this variation therefore just two outer iterations can suffice on the other hand several iterations may be needed to solve the inner iteration systems

shift splitting preconditioners for saddle point problems in this paper we first present a shift splitting preconditioner for saddle point problems the preconditioner is based on a shift splitting of the saddle point matrix resulting in an unconditional convergent fixed point iteration based on the idea of the splitting we further propose a local shift splitting preconditioner some properties of the local shift splitting preconditioned matrix are studied these preconditioners extend those studied by bai yin and su for solving non hermitian positive definite linear systems bai et al 2006 finally numerical experiments of a model stokes problem are presented to show the effectiveness of the proposed preconditioners

efficient online subspace learning with an indefinite kernel for visual tracking and recognition we propose an exact framework for online learning with a family of indefinite not positive kernels as we study the case of nonpositive kernels we first show how to extend kernel principal component analysis kpca from a reproducing kernel hilbert space to krein space we then formulate an incremental kpca in krein space that does not require the calculation of preimages and therefore is both efficient and exact our approach has been motivated by the application of visual tracking for which we wish to employ a robust gradient based kernel we use the proposed nonlinear appearance model learned online via kpca in krein space for visual tracking in many popular and difficult tracking scenarios we also show applications of our kernel framework for the problem of face recognition

robust reduced rank adaptive algorithm based on parallel subgradient projection and krylov subspace in this paper we propose a novel reduced rank adaptive filtering algorithm exploiting the krylov subspace associated with estimates of certain statistics of input and output signals we point out that when the estimated statistics are erroneous e g due to sudden changes of environments the existing krylov subspace based reduced rank methods compute the point that minimizes a ldquowrongrdquo mean square error mse in the subspace the proposed algorithm exploits the set theoretic adaptive filtering framework for tracking efficiently the optimal point in the sense of minimizing the ldquotruerdquo mse in the subspace therefore compared with the existing methods the proposed algorithm is more suited to adaptive filtering applications a convergence analysis of the algorithm is performed by extending the adaptive projected subgradient method apsm numerical examples demonstrate that the proposed algorithm enjoys better tracking performance than the existing methods for system identification problems

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

on parameterized block triangular preconditioners for generalized saddle point problems in this paper we discuss two classes of parameterized block triangular preconditioners for the generalized saddle point problems these preconditioners generalize the common block diagonal and triangular preconditioners we will give distributions of the eigenvalues of the preconditioned matrix and provide estimates for the interval containing the real eigenvalues numerical experiments of a model stokes problem are presented

an adaptive projected subgradient approach to learning in diffusion networks we present an algorithm that minimizes asymptotically a sequence of nonnegative convex functions over diffusion networks in the proposed algorithm at each iteration the nodes in the network have only partial information of the cost function but they are able to achieve consensus on a possible minimizer asymptotically to account for possible node failures position changes and or reachability problems because of moving obstacles jammers etc the algorithm can cope with changing network topologies and cost functions a desirable feature in online algorithms where information arrives sequentially many projection based algorithms can be straightforwardly extended to probabilistic diffusion networks with the proposed scheme the system identification problem in distributed networks is given as one example of a possible application

from sine waves to square waves in delay equations suppose f f 0 0 and the fixed point zero undergoes a generic supercritical period doubling bifurcation at 0 we characterise those small values of e 0 for which there are periodic solutions of period approximately two of the equationas e 0 these solutions approach square waves

parameter estimates for the relaxed dimensional factorization preconditioner and application to hemodynamics we present new results on the relaxed dimensional factorization rdf preconditioner for solving saddle point problems from incompressible flow simulations first introduced in benzi et al 2011 this method contains a parameter 0 0 to be chosen by the user previous works provided an estimate of in the 2d case using local fourier analysis a novel algebraic estimation technique for finding a suitable value of the rdf parameter in both the 2d and the 3d case with arbitrary geometries is proposed this technique is tested on a variety of discrete saddle point problems arising from the approximation of the navier stokes equations using a marker and cell scheme and a finite element one we also show results for a large scale problem relevant for hemodynamics simulation that we solve in parallel using up to 8196 cores

rmcp relaxed mixed constraint preconditioners for saddle point linear systems arising in geomechanics abstract a major computational issue in the finite element fe integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system because of ill conditioning the iterative solution which is recommended in large size 3d settings requires the computation of a suitable preconditioner to guarantee convergence in this paper the coupled system is solved by a krylov subspace method preconditioned by a relaxed mixed constraint preconditioner rmcp which is a generalization based on a parameter of the mixed constraint preconditioner mcp developed in 7 choice of optimal is driven by the spectral distribution of suitable symmetric positive definite spd matrices numerical tests performed on realistic 3d problems reveal that rmcp accelerates krylov subspace solvers by a factor up to three with respect to mcp

online learning with kernels kernel based algorithms such as support vector machines have achieved considerable success in various problems in batch setting where all of the training data is available in advance support vector machines combine the so called kernel trick with the large margin idea there has been little use of these methods in an online setting suitable for real time applications in this paper we consider online learning in a reproducing kernel hilbert space by considering classical stochastic gradient descent within a feature space and the use of some straightforward tricks we develop simple and computationally efficient algorithms for a wide range of problems such as classification regression and novelty detection in addition to allowing the exploitation of the kernel trick in an online setting we examine the value of large margins for classification in the online setting with a drifting target we derive worst case loss bounds and moreover we show the convergence of the hypothesis to the minimizer of the regularized risk functional we present some experimental results that support the theory as well as illustrating the power of the new algorithms for online novelty detection

multiaccess interference suppression in orthogonal space time block coded mimo systems by adaptive projected subgradient method this paper introduces adaptive filters that are effective to suppress multiple access interference mai in orthogonal space time block coded multiple input multiple output ostbc mimo systems we define an optimal linear filter that minimizes the mean square error between the filter output and a scaled version of the desired output under a constraint defined by the available channel state information csi the adaptive filters refine a given estimate of the optimal filter by suppressing a sequence of closed convex functions with the adaptive projected subgradient method apsm at each iteration to provide robustness against imperfect csi the adaptive filters use not only the available csi but also estimates of previously transmitted symbols which usually belong to a small finite set in digital communication systems the resulting algorithms employ computationally efficient projections onto hyperplanes or hyperslabs and do not require any matrix inversion an efficient recursive scheme based on such an algorithm is also presented convergence analysis and simulation results show the excellent performance of the proposed schemes

the dimensional splitting iteration methods for solving saddle point problems arising from time harmonic eddy current models abstract a dimensional splitting iteration method is proposed for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the time harmonic eddy current models which is by making use of the special positive semidefinite splittings of the saddle point matrix it is proved that the proposed iteration method is unconditionally convergent for both cases of simple topology and general topology numerical results show that the corresponding preconditioner is superior to the existing preconditioners when those preconditioners are used to accelerate the convergence rate of krylov subspace methods

new choices of preconditioning matrices for generalized inexact parameterized iterative methods for large sparse saddle point problems chen and jiang recently studied a class of generalized inexact parameterized iterative methods see f chen y l jiang a generalization of the inexact parameterized uzawa methods for saddle point problems appl math comput 206 2008 765 771 in this paper the methods are modified and some choices of preconditioning matrices are given these preconditioning matrices have advantages in solving large sparse linear system numerical experiments of a model stokes problem are presented

decomposition through formalization in a product space when an optimization problem is posed in a product space it is classical to decompose this problem the goal of this paper is to show how such an approach can be used when the problem to be solved is not naturally posed in a product space by associating systematically to this problem an equivalent one posed in ann fold cartesian product space we obtain by decomposition of the latter both a splitting of operators and a desintegration of constraints for the former applications to three rather classical mathematical programming problems are given

periodic solutions of a singularly perturbed delay differential equation abstract a singularly perturbed differential delay equation of the form 1 x t x t f x t 1 exhibits slowly oscillating periodic solutions sops near the first period doubling bifurcation point of the underlying map obtained by setting 0 for extremely small values of these periodic solutions resemble square waves which consist of sharp o transition layers connecting intervals of approximately unit length in this article we obtain analytic expressions for these square wave periodic solutions by solving the corresponding transition layer equations and show that they are in excellent agreement with numerical solutions for a range of values of and we also derive analytic expressions for other periodic solutions which are odd harmonics of the sops and numerically exhibit their instability near the first period doubling bifurcation point of the map the numerical computations were performed using a high accuracy chebyshev spectral scheme we give a brief description together with a study of its accuracy and efficiency

uniform numerical method for singularly perturbed delay differential equations this paper deals with the singularly perturbed initial value problem for a linear first order delay differential equation a numerical method is constructed for this problem which involves an appropriate piecewise uniform mesh on each time subinterval the difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter numerical results are presented

an augmented lagrangian approach to essential boundary conditions in meshless methods the problem of boundary conditions enforcement in meshless methods has been solved in the literature by several approaches in the present paper the moving least squares mls approximation is introduced in the total potential energy functional for the elastic solid problem and an augmented lagrangian term is added to satisfy essential boundary conditions the method can be easily extended to any kind of constraint for the approximation variables the solution is found by iterating alternatively on approximation variables and on lagrange multipliers the advantages of the proposed formulation are a the ability to deal with the same approach with any constraint type b the number of the variables is not increased by the lagrange multipliers c the hessian of the functional w r t the approximation variables is banded well conditioned and strictly positive definite and d the cost of the augmented lagrangian iteration is a very small fraction of the global computing time therefore the augmented lagrangian element free alef approach represents an attractive and very efficient numerical tool not only for the boundary conditions enforcement but also for the solution of interface and non linear problems copyright 2001 john wiley sons ltd