preconditioning and boundary conditions without h 2 estimates l 2 condition numbers and the distribution of the singular values this work deals with the behavior in the l sub 2 norm of the condition number and distribution of the l sub 2 singular values of the preconditioned operators b sub h sup minus 1 a sub h and a sub h b sub h sup minus 1 sub h where a sub h and b sub h are finite element discretizations of second order elliptic operators a and b in an earlier work manteuffel and parter proved that b sub h sup minus 1 a sub h a sub h b sub h sup minus 1 have a uniformly bounded l sub 2 condition number if and only if a sup and b sup a and b have the same boundary conditions this earlier work used the h sub 2 regularity of a and b as well as optimal l sub 2 error estimates and a quasi uniform grid for the finite element spaces in the present paper the authors first extend these condition number results to the case in which neither h sub 2 regularity and hence optimal l sub 2 error estimates nor the quasi uniformity assumption need to be satisfied instead it is assumed that the principle part of the preconditioning operator b is a scalar multiple 1 mu of the principle part of a

finite difference preconditioning cubic spline collocation method of elliptic equations we discuss a finite difference preconditioner for the c 1 interpolatory cubic spline collocation method for a uniformly elliptic operator a defined by au delta u a 0 u in omega the unit square with homogeneous dirichlet boundary conditions using the generalized field of values arguments we discuss the eigenvalues of the preconditioned matrix hat l n 2 1 hat a n 2 where hat a n 2 is the matrix of the collocation discretization operator a n 2 corresponding to a and hat l n 2 is the matrix of the finite difference operator l n 2 corresponding to the uniformly elliptic operator l given by l v delta v v in omega with homogeneous dirichlet boundary conditions finally we mention a bound of h 1 singular values of hat l 1 n 2 hat a n 2 for a general elliptic operator au delta u a 1 u x a 2 u y a 0 u in omega

eigenfaces vs fisherfaces recognition using class specific linear projection we develop a face recognition algorithm which is insensitive to large variation in lighting direction and facial expression taking a pattern classification approach we consider each pixel in an image as a coordinate in a high dimensional space we take advantage of the observation that the images of a particular face under varying illumination but fixed pose lie in a 3d linear subspace of the high dimensional image space if the face is a lambertian surface without shadowing however since faces are not truly lambertian surfaces and do indeed produce self shadowing images will deviate from this linear subspace rather than explicitly modeling this deviation we linearly project the image into a subspace in a manner which discounts those regions of the face with large deviation our projection method is based on fisher s linear discriminant and produces well separated classes in a low dimensional subspace even under severe variation in lighting and facial expressions the eigenface technique another method based on linearly projecting the image space to a low dimensional subspace has similar computational requirements yet extensive experimental results demonstrate that the proposed fisherface method has error rates that are lower than those of the eigenface technique for tests on the harvard and yale face databases

chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning abstract a new chebyshev pseudospectral algorithm for second order elliptic equations using finite element preconditioning is proposed and tested on various problems bilinear and biquadratic lagrange elements are considered as well as bicubic hermite elements the numerical results show that bilinear elements produce spectral accuracy with the minimum computational work l shaped regions are treated by a subdomain approach

a reduced order model based preconditioner for the efficient solution of transient diffusion equations this paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive definite linear systems arising from the numerical discretization of transient parabolic and selfadjoint partial differential equations the preconditioners are obtained by nesting appropriate projections of reduced order models into the classical iteration of the preconditioned conjugate gradient pcg the main idea is to employ the reduced order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases this approach is particularly appealing for transient systems where the full model solution has to be computed at each time step in these cases the natural reduced space is the one generated by full model solutions at previous time steps when increasing the size of the projection space the proposed methodology highly reduces the system conditioning number and the number of pcg iterations at every time step the cost of the application of the preconditioner linearly increases with the size of the projection basis and a trade off must be found to effectively reduce the pcg computational cost the quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models c 2016 the authors international journal for numerical methods in engineering published by john wiley sons ltd

preconditioners for spectral discretizations of helmholtz s equation with sommerfeld boundary conditions some preconditioners for the iterative solution of helmholtz s equation discretized with spectral legendre collocation methods are introduced and studied the preconditioners are based either on a finite element discretization of helmholtz s equation on the spectral collocation mesh or on replacing the sommerfeld like boundary condition on a subset of the boundary with either neumann or dirichlet boundary conditions the convergence rate of the resulting iterative methods is only mildly dependent on the spectral degree n and the wave number

preconditioning cubic spline collocation discretizations of elliptic equations this work considers the uniformly elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 and partial u over partial nu alpha u on gamma 1 and its discretization based on hermite cubic spline spaces and collocation at the gauss points using an interpolatory basis with support on the gauss points one obtains the matrix hat a n h 1 over n we discuss the h 1 condition numbers and the distribution of tilde beta n singular values of the preconditioned matrices tilde beta n 1 hat a n where tilde beta n is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator b given by b v delta v b 0 v in omega with boundary conditions v 0 on gamma 0 partial v over partial nu beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by gauss points or the space of continuous functions which are linear on the triangles of the triangulation of omega using the gauss points when a b we obtain results on the eigenvalues of tilde beta n 1 hat a n in the general case we obtain bounds and clustering results on the tilde beta n singular values of tilde beta n 1 hat a n these results are related to the results of manteuffel and parter mp parter and wong pw and wong w for finite element discretizations as well as the results of parter and rothman pr for discretizations based on legendre spectral collocation

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

preconditioning legendre spectral collocation methods for elliptic problems i finite difference operators in 1979 orszag and morchoisne independently proposed a finite difference preconditioning of the chebyshev collocation discretization of the poisson equation over the years there has been intensive research both experimental and theoretical on the finite difference and finite element preconditioning of both chebyshev and legendre spectral collocation methods for this problem in this work we present results on a frequently employed variant of finite element preconditioning of legendre spectral collocation methods for the helmholtz equation with dirichlet boundary conditions we show that there exist two positive constants 0 0 and w n m a n m u u ell 2 w n m m 1 n m s n m u u ell 2 leq lambda 1 where w n m rm diagonal w k tilde w j the diagonal matrix of legendre gauss lobatto lgl weights and a n m is the collocation matrix m n m is the mass matrix and s n m is the stiffness matrix of the finite element method these estimates imply that the eigenvalues gamma k of s 1 n m m n m a n m satisfy rm re gamma k geq lambda 0 0 gamma k leq lambda 1 these results depend on a thorough study of the ratio 2w k x k 1 x k 1 where x k are lgl collocation points

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

an overlapping schwarz method for spectral element solution of the incompressible navier stokes equations efficient solution of the navier stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems for unsteady incompressible flows the pressure operator is the leading contributor to stiffness as the characteristic propagation speed is infinite in the context of operator splitting formulations it is the pressure solve which is the most computationally challenging despite its elliptic origins we examine several preconditioners for the consistent l2poisson operator arising in the pn pn 2spectral element formulation of the incompressible navier stokes equations we develop a finite element based additive schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior gauss points for large two dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation

fast direct solvers for piecewise hermite bicubic orthogonal spline collocation equations fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise hermite bicubics is employed for the approximate solution of poisson s

preconditioning legendre spectral collocation approximations to elliptic problems this work deals with the h 1 condition numbers and the distribution of the tilde beta n m singular values of the preconditioned operators tilde beta n m 1 w n m hat a n m hat a n m is the matrix representation of the legendre spectral collocation discretization of the elliptic operator a defined by au delta u a 1 u x a 2 u y a 0 u in omega the unit square with boundary conditions u 0 on gamma 0 frac partial u partial nu a alpha u on gamma 1 tilde beta n m is the stiffness matrix associated with the finite element discretization of the positive definite elliptic operator b defined by bv delta v b 0 v in omega with boundary conditions v 0 on gamma 0 frac partial v partial nu b beta v on gamma 1 the finite element space is either the space of continuous functions which are bilinear on the rectangles determined by the legendre gauss lobatto lgl points or the s

pseudospectral least squares method for the second order elliptic boundary value problem the least squares legendre and chebyshev pseudospectral methods are presented for a first order system equivalent to a second order elliptic partial differential equation continuous and discrete homogeneous least squares functionals using legendre and chebyshev weights are shown to be equivalent to the h1 omega norm and chebyshev weighted div curl norm over appropriate polynomial spaces respectively the spectral error estimates are derived the block diagonal finite element preconditioner is developed for the both cases several numerical tests are demonstrated on the spectral discretization errors and on performances of the finite element preconditioner

analysis on eigenvalues for preconditioning cubic spline collocation method of elliptic equations abstract in the work of solving a uniformly elliptic differential equations au u a 1 u x a 2 u y a 0 u f in the unit square with boundary conditions by the c 1 cubic spline collocation method one may need to investigate efficient preconditioning techniques for this purpose using the generalized field of values argument we show the uniform bounds of the eigenvalues of the preconditioned matrix when a full finite element preconditioning is considered

on the eigenvalues of second order elliptic difference operators while there is now an extensive literature on the approximation of the eigenvalue problem for compact operators these theories do not include the classical finite difference approximation in the self adjoint case the problem is resolved through the variational characterizations of the eigenvalue and eigenfunctions the problem is unresolved in the non self adjoint case in this work we consider a model problem delta u au x bu y lambda nu the unit square the operator n is a first order operator the left hand side is approximated by straightforward central differences the operator n is approximated weakly

a pseudo spectral scheme for the incompressible navier stokes equations using unstructured nodal elements a pseudo spectral scheme for solving the incompressible navier stokes equations using unstructured nodal triangles is proposed efficient algorithms are developed with numerical evidence that indicates optimal rates of convergence can be achieved navier stokes simulations of kovasznay shear layer roll up and row past a cylinder are included to show comparisons between the different nodal sets considered and an alternative modal approach c 2000 academic press

preconditioning nonconforming finite element methods for treating dirichlet boundary conditions ii this work deals with theh 1 condition numbers and the distribution of theb h singular values of the preconditioned operators b h 1 a h 0 h 1 wherea h andb h are finite element discretizations of second order elliptic operators a andb respectively b is also assumed to be self adjoint and positive definite for conforming finite elements parter and wong have shown that the singular values cluster in a positive finite interval goldstein also has derived results on the spectral distribution ofb h 1 a h using a different approach as a generalization of the results of parter and wong the current work includes nonconforming finite element methods which deal with dirichlet boundary conditions it will be shown that in this more general setting the singular values also cluster in a positive finite interval in particular if the leading part ofb is the same as the leading part ofa then the singular values cluster about the point 1 two specific methods are given as applications of this theory they are the penalty method of babuska and the method of nearly zero boundary conditions of nitsche finally it will be shown that the same results can be proven by an approach generalized from the work of goldstein

exponential decay of c 1 cubic splines vanishing at two symmetric points in each knot interval in the course of working on the preconditioning of cubic collocation at gauss points one has to deal with the exponential decay of certain cubic splines such results were obtained by kim and parter in the case of uniform spacing in this paper we extend these results to arbitrary grids and state its application to the preconditioned cubic collocation method

preconditioning nonconforming finite element methods this paper is concerned with the h sup 1 condition number and singular value distribution of preconditioned nonself adjoint discrete elliptic operators in particular a wide class of nonconforming finite element methods are investigated the main results give a precise interval on which the singular values cluster results of this kind have an important bearing on the convergence behavior of the conjugate gradient method applied to the normal equations

on the combination of self organized systems to generate pseudo random numbers cellular automata ca is a self organizing structure with complex behavior which can be used to generating pseudo random numbers pure ca has a simple structure and has no ability to produce long sequences of random numbers but in order to resolve this problem a programmable controllable ca pcca using actuating factor or combination of different self organizing criticality phenomena can be used the purpose of the aforementioned methods is the use of the ca speed while preventing the automata self organizing factor for specific use in generating random numbers in this paper a pcca by using sandpile model is proposed the sandpile model is a complex system that operates at a critical state between chaos and order this state is known as self organized criticality soc and is characterized by displaying invariant scale behavior in the precise case of the sandpile model by randomly and continuously dropping sand grains on top of a two dimensional grid lattice a power law relationship between the frequency and size of sand avalanches is observed the avalanche behavior and the pure ca behavior are combined in a novel method which can be used as the pseudo random number generator prng experimental results show that this generator is able to reach a random behavior from a pseudo chaotic one by combining two self organized systems in addition to independency of the initial state it has been also indicated that the generated numbers are independent and its probability distribution is likely to be uniform accordingly some tests of prngs such as entropy chi square diehard and several basic statistical tests have been performed and all of them have been successfully passed

theory and applications of cellular automata in cryptography this paper deals with the theory and application of cellular automata ca for a class of block ciphers and stream ciphers based on ca state transitions certain fundamental transformations are defined which are block ciphering functions of the proposed enciphering scheme these fundamental transformations are found to generate the simple alternating group of even permutations which in turn is a subgroup of the permutation group these functions are implemented with a class of programmable cellular automata pca built around rules 51 153 and 195 further high quality pseudorandom pattern generators built around rule 90 and 150 programmable cellular automata with a rule selector i e combining function has been proposed as running key generators in stream ciphers both the schemes provide better security against different types of attacks with a simple regular modular and cascadable structure of ca hardware implementation of such schemes idealy suit vlsi implementation

permutation and sampling with maximum length ca or pseudorandom number generation abstract in this paper we study the effect of dynamic permutation and sampling on the randomness quality of sequences generated by cellular automata ca dynamic permutation and sampling have not been explored in previous ca work and a suitable implementation is shown using a two ca model three different schemes that incorporate these two operations are suggested weighted permutation vector sampling with controlled multiplexing weighted permutation vector sampling with irregular decimation and permutation programmed ca sampling the experiment results show that the resulting sequences have varying degrees of improvement in diehard results and linear complexity compared to the ca

a preconditioned conjugate gradient method for nonselfadjoint or indefinite orthogonal spline collocation problems we study the computation of the orthogonal spline collocation solution of a linear dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form lu sum a ij x u x ix j sum b i x u x i c x u we apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator and prove that the resulting algorithm has a convergence rate independent of the partition step size we solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm on a uniform n n partition the cost of the algorithm for computing the collocation solution within a tolerance epsilon is o n 2 ln n ln epsilon

finite element preconditioning for legendre spectral collocation approximations to elliptic equations and systems spectral collocation approximations based on legendre gauss lobatto lgl points for helmholtz equations as well as for the linear elasticity system in rectangular domains are studied the collocati

preconditioning of the p version of the finite element method the p version finite element method for linear second order elliptic equations in an arbitrary sufficiently smooth incl polygonal bounded domain is studied in the framework of the domain decomposition dd method two types of square reference elements are used with coordinate functions given by the products of the integrated legendre polynomials estimates for the condition numbers and some useful inequalities are given we consider preconditioning of the problems arising on subdomains and of the schur complement as well as the derivation and analysis of the dd preconditioner for the entire system this is done for a class of curvilinear finite elements we obtain several dd preconditioners for which the generalized condition numbers vary from o log p 3 to o 1 this paper is based on 19 21 27 we have omitted most of the proofs in order to shorten it and have described instead what could be done as well as outlined some additional ideas the full proofs omitted can in most cases be found in 19 20 27

preconditioning c1 lagrange polynomial spline collocation method of elliptic equations by finite element method this paper studies a preconditioning polynomial spline collocation method for an elliptic differential equation by a finite element stiffness matrix for which we first show that the decay rate of the c 1 lagrange polynomial spline increases with increasing degree and second show that the distribution of eigenvalues for the established preconditioning system is well bounded in terms of the number of unknowns for a fixed degree of polynomial spline

rapid and brief communications modified linear discriminant analysis in this paper a modified fisher linear discriminant analysis flda is proposed and aims to not only overcome the rank limitation of flda that is at most only finding a discriminant vector for 2 class problem based on fisher discriminant criterion but also relax singularity of the within class scatter matrix and finally improves classification performance of flda experiments on nine publicly available datasets show that the proposed method has better or comparable performance on all the datasets than flda

preconditioning chebyshev spectral collocation method for elliptic partial differential equations in this paper we analyze a preconditioning technique for the solution of chebyshev spectral collocation equations with dirichlet boundary conditions we obtain bounds on the eigenvalues for the helmholtz equation these eigenvalue bounds are obtained as a consequence of estimates on the field of values tilde a n 2 u u l 2 tilde q n 2 u u l 2 where tilde a n 2 is the weighted collocation matrix and tilde q n 2 is the preconditioner the preconditioner tilde q n 2 is robust in the sense that it provides bounds on the h 1 0 w condition number of tilde q n 2 1 tilde l n 2 when tilde l n 2 is the weighted collocation matrix associated with the general elliptic operator lu delta u a 1u x a 2u y a 0u

on the generation of high quality random numbers by two dimensional cellular automata finding good random number generators rngs is a hard problem that is of crucial import in several fields ranging from large scale statistical physics simulations to hardware self test in this paper we employ the cellular programming evolutionary algorithm to automatically generate two dimensional cellular automata ca rngs applying an extensive suite of randomness tests to the evolved cas we demonstrate that they rapidly produce high quality random number sequences moreover based on observations of the evolved cas we are able to handcraft even better rngs which not only outperform previously demonstrated high quality rngs but can be potentially tailored to satisfy given hardware constraints

preconditioning chebyshev spectral collocation by finite difference operators in 1979 orszag proposed a finite difference preconditioning of the chebyshev collocation discretization of the poisson equation in 1984 haldenwang labrosse abboudi and deville gave analytic formulae for the eigenvalues of this preconditioned operator in the one dimensional case experimental results over many years have shown the effectiveness of this procedure and appropriate bounds on the eigenvalues in two dimensions however there have been no mathematical proofs describing the behavior of the eigenvalues in two or more dimensions in this work we consider the generalized field of values u hat a nu u l nu where hat a n is the matrix of the chebyshev collocation scheme and l n is the matrix of the finite difference operator for the case of the chebyshev collocation of the helmoltz operator au delta u au a ge 0 preconditioned by the finite difference operator associated with the helmholtz operator bu delta u bu b ge 0 we prove that there are two constants 0 0 and u hat a nu u l nu le lambda 1 these results extend to higher dimensions and to bounds on l n 1 hat a n 1 w and hat a n 1 l n 1 w in the general case where au delta u a 1u x a 2u y au