discretization on non orthogonal quadrilateral grids for inhomogeneous anisotropic media two classes of discretization methods are proposed for control volume formulations on quadrilateral grids in two space dimensions curvilinear grids are considered as a special case the methods are applicable for any system of conservation laws where the flux is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy and or non orthogonality of the grid further properties of the methods will be discussed and examples of their use will be presented

a small stencil and extremum preserving scheme for anisotropic diffusion problems on arbitrary 2d and 3d meshes in this paper a nonlinear extremum preserving scheme for the heterogeneous and anisotropic diffusion problems is proposed on general 2d and 3d meshes through a certain linearity preserving approach the so called harmonic averaging points located at the interface of heterogeneity are employed to define the auxiliary unknowns this new scheme is locally conservative has only cell centered unknowns and possesses a small stencil which is five point on the structured quadrilateral meshes and seven point on the structured hexahedral meshes the stability result in h 1 norm is obtained under quite general assumptions numerical results show that our scheme is robust and extremum preserving and the optimal convergence rates are verified on general distorted meshes in case that the diffusion tensor is taken to be anisotropic at times discontinuous

a class of complex ica algorithms based on the kurtosis cost function in this paper we introduce a novel way of performing real valued optimization in the complex domain this framework enables a direct complex optimization technique when the cost function satisfies the brandwood s independent analyticity condition in particular this technique has been used to derive three algorithms namely kurtosis maximization using gradient update km g kurtosis maximization using fixed point update km f and kurtosis maximization using newton update km n to perform the complex independent component analysis ica based on the maximization of the complex kurtosis cost function the derivation and related analysis of the three algorithms are performed in the complex domain without using any complex real mapping for differentiation and optimization a general complex newton rule is also derived for developing the km n algorithm the real conjugate gradient algorithm is extended to the complex domain similar to the derivation of complex newton rule the simulation results indicate that the fixed point version km f and gradient version km g are superior to other similar algorithms when the sources include both circular and noncircular distributions and the dimension is relatively high

numerical optimization for symmetric tensor decomposition we consider the problem of decomposing a real valued symmetric tensor as the sum of outer products of real valued vectors algebraic methods exist for computing complex valued decompositions of symmetric tensors but here we focus on real valued decompositions both unconstrained and nonnegative for problems with low rank structure we discuss when solutions exist and how to formulate the mathematical program numerical results show the properties of the proposed formulations including one that ignores symmetry on a set of test problems and illustrate that these straightforward formulations can be effective even though the problem is nonconvex

first order convergence of multi point flux approximation on triangular grids and comparison with mixed finite element methods in this paper we show first order convergence of a multi point flux approximation control volume method mpfa on unstructured triangular grids in this approach the flux approximation is derived directly in the physical space in order to do this we introduce a perturbed mixed finite element method that is equivalent to the mpfa scheme and prove the first order convergence of this approach moreover we carefully compare the computational performance properties of the mpfa method with those of a lowest order raviart thomas and brezzi douglas marini mixed finite element approximation

residual based a posteriori error estimation for multipoint flux mixed finite element methods a novel residual type a posteriori error analysis technique is developed for multipoint flux mixed finite element methods for flow in porous media in two or three space dimensions the derived a posteriori error estimator for the velocity and pressure error in l 2 l2 norm consists of discretization and quadrature indicators and is shown to be reliable and efficient the main tools of analysis are a locally postprocessed approximation to the pressure solution of an auxiliary problem and a quadrature error estimate numerical experiments are presented to illustrate the competitive behavior of the estimator

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

a family of mpfa finite volume schemes with full pressure support for the general tensor pressure equation on cell centered triangular grids a new family of cell centered finite volume schemes is presented for solving the general full tensor pressure equation of subsurface flow in porous media on arbitary unstructured triangulations the new schemes are flux continuous and have full pressure support fps over each subcell with continuous pressure imposed across each control volume sub interface in contrast to earlier formulations the earlier methods are point wise continuous in pressure and flux with triangle pressure support tps which leads to a more limited quadrature range an m matrix analysis identifies bounding limits for the schemes to posses a local discrete maximum principle conditions for the schemes to be positive definite are also derived a range of computational examples are presented for unstructured triangular grids including highly irregular grids and the new fps schemes are compared against the earlier pointwise continuous tps formulations the earlier pointwise tps methods can induce strong spurious oscillations for problems involving strong full tensor anisotropy where the m matrix conditions are violated and can lead to decoupled solutions in such cases unstructured cell centered decoupling is investigated in contrast to tps the new fps formulation leads to well resolved solutions that are essentially free of spurious oscillations a substantial degree of improved convergence behavior for both pressure and velocity is also observed in all convergence tests this is particularly important for problems involving high anisotropy ratios also the new formulation proves to be highly beneficial for an upscaling example where enhancement of convergence is highly significant for certain quadrature points clearly demonstrating further advantages of the new formulation

tensor decompositions for learning latent variable models this work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models including gaussian mixture models hidden markov models and latent dirichlet allocation which exploits a certain tensor structure in their low order observable moments typically of second and third order specifically parameter estimation is reduced to the problem of extracting a certain orthogonal decomposition of a symmetric tensor derived from the moments this decomposition can be viewed as a natural generalization of the singular value decomposition for matrices although tensor decompositions are generally intractable to compute the decomposition of these specially structured tensors can be efficiently obtained by a variety of approaches including power iterations and maximization approaches similar to the case of matrices a detailed analysis of a robust tensor power method is provided establishing an analogue of wedin s perturbation theorem for the singular vectors of matrices this implies a robust and computationally tractable estimation approach for several popular latent variable models

robust independent component analysis by iterative maximization of the kurtosis contrast with algebraic optimal step size independent component analysis ica aims at decomposing an observed random vector into statistically independent variables deflation based implementations such as the popular one unit fastica algorithm and its variants extract the independent components one after another a novel method for deflationary ica referred to as robustica is put forward in this paper this simple technique consists of performing exact line search optimization of the kurtosis contrast function the step size leading to the global maximum of the contrast along the search direction is found among the roots of a fourth degree polynomial this polynomial rooting can be performed algebraically and thus at low cost at each iteration among other practical benefits robustica can avoid prewhitening and deals with real and complex valued mixtures of possibly noncircular sources alike the absence of prewhitening improves asymptotic performance the algorithm is robust to local extrema and shows a very high convergence speed in terms of the computational cost required to reach a given source extraction quality particularly for short data records these features are demonstrated by a comparative numerical analysis on synthetic data robustica s capabilities in processing real world data involving noncircular complex strongly super gaussian sources are illustrated by the biomedical problem of atrial activity aa extraction in atrial fibrillation af electrocardiograms ecgs where it outperforms an alternative ica based technique

sensitivity analysis of limit cycle oscillations many unsteady problems equilibrate to periodic behavior for these problems the sensitivity of periodic outputs to system parameters are often desired and must be estimated from a finite time span or frequency domain calculation sensitivities computed in the time domain over a finite time span can take excessive time to converge or fail altogether to converge to the periodic value additionally finite span outputs can exhibit local extrema in parameter space which the periodic outputs they approximate do not hindering their use in optimization we derive a theoretical basis for this error and demonstrate it using two examples a van der pol oscillator and vortex shedding from a low reynolds number airfoil we show that output windowing enables the accurate computation of periodic output sensitivities and may allow for decreased simulation time to compute both time averaged outputs and sensitivities we classify two distinct window types long time over a large not necessarily integer number of periods and short time over a small integer number of periods finally from these two classes we investigate several examples of window shape and demonstrate their convergence with window size and error in the period approximation respectively

the discrete duality finite volume method for convection diffusion problems in this paper we extend the discrete duality finite volume ddfv formulation to the steady convection diffusion equation the discrete gradients defined in ddfv are used to define a cell based gradient for the control volumes of both the primal and dual meshes in order to achieve a higher order accurate numerical flux for the convection term a priori analysis is carried out to show convergence of the approximation and a global first order convergence rate is derived the theoretical results are confirmed by some numerical experiments

the finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes we construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle our scheme is locally conservative and has only cell centered unknowns numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes

local convergence analysis of fastica and related algorithms the fastica algorithm is one of the most prominent methods to solve the problem of linear independent component analysis ica although there have been several attempts to prove local convergence properties of fastica rigorous analysis is still missing in the community the major difficulty of analysis is because of the well known sign flipping phenomenon of fastica which causes the discontinuity of the corresponding fastica map on the unit sphere in this paper by using the concept of principal fiber bundles fastica is proven to be locally quadratically convergent to a correct separation higher order local convergence properties of fastica are also investigated in the framework of a scalar shift strategy moreover as a parallelized version of fastica the so called qr fastica algorithm which employs the qr decomposition gram schmidt orthonormalization process instead of the polar decomposition is shown to share similar local convergence properties with the original fastica

bad behavior of godunov mixed methods for strongly anisotropic advection dispersion equations we study the performance of godunov mixed methods which combine a mixed hybrid finite element solver and a godunov like shock capturing solver for the numerical treatment of the advection dispersion equation with strong anisotropic tensor coefficients it turns out that a mesh locking phenomenon may cause ill conditioning and reduce the accuracy of the numerical approximation especially on coarse meshes this problem may be partially alleviated by substituting the mixed hybrid finite element solver used in the discretization of the dispersive diffusive term with a linear galerkin finite element solver which does not display such a strong ill conditioning to illustrate the different mechanisms that come into play we investigate the spectral properties of such numerical discretizations when applied to a strongly anisotropic diffusive term on a small regular mesh a thorough comparison of the stiffness matrix eigenvalues reveals that the accuracy loss of the godunov mixed method is a structural feature of the mixed hybrid method in fact the varied response of the two methods is due to the different way the smallest and largest eigenvalues of the dispersion diffusion tensor influence the diagonal and off diagonal terms of the final stiffness matrix one and two dimensional test cases support our findings

relationships between the constant modulus and wiener receivers the godard 1980 or the constant modulus algorithm cma is an effective technique for blind receiver design in communications however due to the complexity of the constant modulus cm cost function the performance of the cm receivers has primarily been evaluated using simulations theoretical analysis is typically based on either the noiseless case or approximations of the cost function the following question while resolvable numerically for a specific example remains unanswered in a generic manner in the presence of channel noise where are the cm local minima and what are their mean squared errors mse in this paper a geometrical approach is presented that relates the cm to wiener or minimum mse receivers given the mse and the intersymbol user interference of a wiener receiver a sufficient condition is given for the existence of a cm local minimum in the neighborhood of the wiener receiver the mse bounds on cm receiver performance are derived and shown to be tight in simulations the analysis shows that while in some cases the cm receiver performs almost as well as the nonblind wiener receiver it is also possible that due to its blind nature the cm receiver may perform considerably worse than a nonblind wiener receiver

a multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra in this paper we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell centered finite difference scheme a special non symmetric quadrature rule is employed that yields a positive definite cell centered system for the pressure by eliminating local velocities the method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids including hexahedra with non planar faces theoretical and numerical results indicate first order convergence for the pressure and face fluxes

monotone finite volume schemes for diffusion equations on polygonal meshes we construct a nonlinear finite volume fv scheme for diffusion equation on star shaped polygonal meshes and prove that the scheme is monotone i e it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients our scheme has only cell centered unknowns and it treats material discontinuities rigorously and offers an explicit expression for the normal flux numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non smooth highly anisotropic solutions and numerical results show that our scheme appears to be approximate second order accuracy for the solution and first order accuracy for the flux

bounds for the mse performance of constant modulus estimators the constant modulus cm criterion has become popular in the design of blind linear estimators of sub gaussian independent and identically distributed i i d processes transmitted through unknown linear channels in the presence of unknown additive interference in this paper we present an upper bound for the conditionally unbiased mean squared error umse of cm minimizing estimators that depends only on the source kurtoses and the umse of wiener estimators further analysis reveals that the extra umse of cm estimators can be upper bounded by approximately the square of the wiener i e minimum umse since our results hold for vector valued finite impulse response infinite impulse response fir iir linear channels vector valued fir iir estimators with a possibly constrained number of adjustable parameters and multiple interferers with arbitrary distribution they confirm the longstanding conjecture regarding the general mean square error mse robustness of cm estimators

super exponential methods for blind deconvolution a class of iterative methods for solving the blind deconvolution problem i e for recovering the input of an unknown possibly nonminimum phase linear system by observation of its output is presented these methods are universal do not require prior knowledge of the input distribution are computationally efficient and statistically stable and converge to the desired solution regardless of initialization at a very fast rate the effects of finite length of the data finite length of the equalizer and additive noise in the system on the attainable performance intersymbol interference are analyzed it is shown that in many cases of practical interest the performance of the proposed methods is far superior to linear prediction methods even for minimum phase systems recursive and sequential algorithms are also developed which allow real time implementation and adaptive equalization of time varying systems

finite volume method for 2d linear and nonlinear elliptic problems with discontinuities in this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the discrete duality finite volume ddfv framework this family of schemes allows very general meshes and inherits the main properties of the continuous problem in order to take into account the discontinuities and to prevent consistency defect in the scheme we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs we first illustrate our approach by the study of the 1d situation then we show how to design our new scheme called m ddfv and we propose its analysis we also describe an iterative solver whose convergence is proved which can be used to solve the nonlinear discrete equations defining the finite volume scheme finally we provide numerical results which confirm that the m ddfv scheme significantly improves the convergence rate of the usual ddfv method for both linear and nonlinear problems

a multipoint flux mixed finite element method on hexahedra we develop a mixed finite element method for elliptic problems on hexahedral grids that reduces to cell centered finite differences the paper is an extension of our earlier paper for quadrilateral and simplicial grids m f wheeler and i yotov siam j numer anal 44 2006 pp 2082 2106 the construction is motivated by the multipoint flux approximation method and it is based on an enhancement of the lowest order brezzi douglas duran fortin bddf mixed finite element spaces on hexahedra in particular there are four fluxes per face one associated with each vertex a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate first order convergence for pressures and subface fluxes on sufficiently regular grids as well as second order convergence for pressures at the cell centers second order convergence for face fluxes is also observed computationally

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

on the convergence of ica algorithms with symmetric orthogonalization independent component analysis ica problem is often posed as the maximization minimization of an objective cost function under a unitary constraint which presumes the prewhitening of the observed mixtures the parallel adaptive algorithms corresponding to this optimization setting where all the separators are jointly trained are typically implemented by a gradient based update of the separation matrix followed by the so called symmetrical orthogonalization procedure to impose the unitary constraint this article addresses the convergence analysis of such algorithms which has been considered as a difficult task due to the complication caused by the minimum frobenius or induced 2 norm distance mapping step we first provide a general characterization of the stationary points corresponding to these algorithms furthermore we show that fixed point algorithms employing symmetrical orthogonalization are monotonically convergent for convex objective functions we later generalize this convergence result for nonconvex objective functions at the last part of the article we concentrate on the kurtosis objective function as a special case we provide a new set of critical points based on householder reflection and we also provide the analysis for the minima maxima saddle point classification of these critical points

a convergence and asymptotic analysis of the generalized symmetric fastica algorithm this contribution deals with the fastica algorithm in the domain of independent component analysis ica the focus is on the asymptotic behavior of the generalized symmetric variant of the algorithm the latter has already been shown to possess the potential to achieve the cramer rao bound crb by allowing the usage of different nonlinearity functions in its implementation although the fastica algorithm along with its variants are among the most extensively studied methods in the domain of ica a rigorous study of the asymptotic distribution of the generalized symmetric fastica algorithm is still missing in fact all the existing results exhibit certain limitations some ignores the impact of data standardization on the asymptotic statistics others are only based on heuristic arguments in this work we aim at deriving general and rigorous results on the limiting distribution and the asymptotic statistics of the fastica algorithm we begin by showing that the generalized symmetric fastica optimizes a function that is a sum of the contrast functions of traditional one unit fastica with a correction of the sign based on this characterization we established the asymptotic normality and derived a closed form analytic expression of the asymptotic covariance matrix of the generalized symmetric fastica estimator using the method of estimating equation and m estimator computer simulations are also provided which support the theoretical results

performance analysis of the fastica algorithm and crame spl acute r rao bounds for linear independent component analysis the fastica or fixed point algorithm is one of the most successful algorithms for linear independent component analysis ica in terms of accuracy and computational complexity two versions of the algorithm are available in literature and software a one unit deflation algorithm and a symmetric algorithm the main result of this paper are analytic closed form expressions that characterize the separating ability of both versions of the algorithm in a local sense assuming a good initialization of the algorithms and long data records based on the analysis it is possible to combine the advantages of the symmetric and one unit version algorithms and predict their performance to validate the analysis a simple check of saddle points of the cost function is proposed that allows to find a global minimum of the cost function in almost 100 simulation runs second the crame spl acute r rao lower bound for linear ica is derived as an algorithm independent limit of the achievable separation quality the fastica algorithm is shown to approach this limit in certain scenarios extensive computer simulations supporting the theoretical findings are included

a combined finite volume nonconforming mixed hybrid finite element scheme for degenerate parabolic problems we propose and analyze a numerical scheme for nonlinear degenerate parabolic convection diffusion reaction equations in two or three space dimensions we discretize the diffusion term which generally involves an inhomogeneous and anisotropic diffusion tensor over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming crouzeix raviart finite element method or using the stiffness matrix of the hybridization of the lowest order raviart thomas mixed finite element method the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh checking the local peclet number we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection dominated case this technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor we prove the convergence of the scheme only supposing the shape regularity condition for the original mesh we use a priori estimates and the kolmogorov relative compactness theorem for this purpose the proposed scheme is robust only 5 point 7 point in space dimension three locally conservative efficient and stable which is confirmed by numerical experiments

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

monotonicity of control volume methods robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications here we discuss monotonicity for a simplified problem of single phase flow but where the simulation grids and media are allowed to be general posing challenges to control volume methods we discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine point control volume discretizations for conforming quadrilateral grids in 2d these criteria are less restrictive than the m matrix property it is shown that it is impossible to construct nine point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields numerical examples are presented which show the validity of the criteria for monotonicity further the impact of nonmonotonicity is studied different behavior for different discretization methods is illuminated and simple ideas are presented for improvement in terms of monotonicity

interpolation based second order monotone finite volume schemes for anisotropic diffusion equations on general grids we propose two interpolation based monotone schemes for the anisotropic diffusion problems on unstructured polygonal meshes through the linearity preserving approach the new schemes are characterized by their nonlinear two point flux approximation which is different from the existing ones and has no constraint on the associated interpolation algorithm for auxiliary unknowns thanks to the new nonlinear two point flux formulation it is no longer required that the interpolation algorithm should be a positivity preserving one the first scheme employs vertex unknowns as the auxiliary ones and a second order but not positivity preserving interpolation algorithm is utilized the second scheme uses the so called harmonic averaging points located on cell edges to define the auxiliary unknowns and a second order positivity preserving interpolation method is employed both schemes have nearly the same convergence rates as compared with their related second order linear schemes numerical results demonstrate that the new schemes are monotone and have the second order accuracy for the solution and first order for its gradient on severely distorted meshes

properties of some blind equalization criteria in noisy multiuser environments blind equalization in noisy multiuser channels has met with increasing attention with the advent of multiaccess digital communication systems we examine blind equalizer performance in cases where perfect equalization proves unattainable due to noise and interference from concurrent users in particular we obtain a characterization of stationary points and extrema for a family of blind criteria in undermodeled cases which assimilates the influence of differing source statistics and background noise correlation properties relations to mean square equalization measures are then obtained as a byproduct by re examining a gradient search procedure we obtain domains of attraction of each extremum in a special sufficient order setting we also derive a global step size bound for undermodeled cases which ensures convergence of a gradient search procedure to an extremum of a blind cost function we likewise confirm that the super exponential algorithm results from an optimal choice of this step size parameter

finite volume scheme for anisotropic diffusion in this paper we apply a special finite volume scheme limited to smooth temperature distributions and cartesian grids to test the importance of connectivity of the finite volumes the area of application is nuclear fusion plasma with field line aligned temperature gradients and extreme anisotropy we apply the scheme to the anisotropic heat conduction equation and compare its results with those of existing finite volume schemes for anisotropic diffusion also we introduce a general model adaptation of the steady diffusion equation for extremely anisotropic diffusion problems with closed field lines

monotonic solution of heterogeneous anisotropic diffusion problems anisotropic problems arise in various areas of science and engineering for example groundwater transport and petroleum reservoir simulations the pure diffusive anisotropic time dependent transport problem is solved on a finite number of nodes that are selected inside and on the boundary of the given domain along with possible internal boundaries connecting some of the nodes an unstructured triangular mesh that attains the generalized anisotropic delaunay condition for all the triangle sides is automatically generated by properly connecting all the nodes starting from an arbitrary initial one the control volume of each node is the closed polygon given by the union of the midpoint of each side with the anisotropic circumcentre of each final triangle a structure of the flux across the control volume sides similar to the standard galerkin finite element scheme is derived a special treatment of the flux computation mainly based on edge swaps of the initial mesh triangles is proposed in order to obtain a stiffness m matrix system that guarantees the monotonicity of the solution the proposed scheme is tested using several literature tests and the results are compared with analytical solutions as well as with the results of other algorithms in terms of convergence order computational costs are also investigated

robust streamline tracing using inter cell fluxes in locally refined and unstructured grids 1 we present a comprehensive study of velocity interpolation methods in polygons these methods are often used as postprocessing procedures for numerical schemes that do not directly calculate the velocity field but only provide cell boundary flux conditions such as the finite volume schemes these methods extend the widely used velocity interpolation algorithms such as the pollock s algorithm to more complex geometries such as perpendicular bisection pebi grids unstructured triangular grids and grids with local refinement once the velocity field is interpolated streamline trajectories and time of flight along the streamlines can be calculated for reservoir simulation model calibration and waterflood management for instance these velocity interpolation methods assume known lower order or higher order cell boundary fluxes which satisfy global mass conservation and normal flux continuity however they differ in the interpolation of velocities within the interior of the cells the interpolating velocity may be locally conservative or nonconservative continuous or discontinuous lower order or higher order results show that the interpolated velocity field has to be locally conservative in order to guarantee the correct volumetric transformation for the calculated streamlines and the time of flight velocity continuity is not as important as local conservation for the purpose of streamline applications compared to higher order interpolation for the streamline trajectories lower order interpolation has the advantage of an analytic solution and an efficient implementation based on our analysis we recommend a lower order locally conservative method for the most robust and numerically efficient calculation of streamline trajectories on unstructured grids

fast and robust fixed point algorithms for independent component analysis independent component analysis ica is a statistical method for transforming an observed multidimensional random vector into components that are statistically as independent from each other as possible we use a combination of two different approaches for linear ica comon s information theoretic approach and the projection pursuit approach using maximum entropy approximations of differential entropy we introduce a family of new contrast functions for ica these contrast functions enable both the estimation of the whole decomposition by minimizing mutual information and estimation of individual independent components as projection pursuit directions the statistical properties of the estimators based on such contrast functions are analyzed under the assumption of the linear mixture model and it is shown how to choose contrast functions that are robust and or of minimum variance finally we introduce simple fixed point algorithms for practical optimization of the contrast functions

superconvergence for control volume mixed finite element methods on rectangular grids we consider control volume mixed finite element methods for the approximation of second order elliptic problems on rectangular grids these methods associate control volumes covolumes with the vector variable as well as the scalar obtaining local algebraic representation of the vector equation e g darcy s law as well as the scalar equation e g conservation of mass we establish o h 2 superconvergence for both the scalar variable in a discrete l 2 norm and the vector variable in a discrete h rm div norm the analysis exploits a relationship between control volume mixed finite element methods and the lowest order raviart thomas mixed finite element methods

monotonically convergent algorithms for symmetric tensor approximation abstract reduced rank approximations to symmetric tensors find use in data compaction and in multi user blind source separation we derive iterative algorithms which feature monotonic convergence to a minimum of a frobenius norm approximation criterion for a certain rank r tucker product version of the approximation problem the approach exploits the gradient inequality for convex functions to establish monotonic convergence while sparing the cumbersome step size analysis required from a manifold gradient approach it likewise overcomes some limitations of symmetric versions of alternating least squares the computational load per iteration amounts to computing an unfolded matrix and a qr decomposition

fast fixed point neural blind deconvolution algorithm the aim of this letter is to introduce a new blind deconvolution algorithm based on fixed point optimization of a bussgang type cost function the cost function relies on approximate bayesian estimation achieved by an adaptive neuron the main feature of the presented algorithm is fast convergence that guarantees good deconvolution performances with limited computational demand as compared with algorithms of the same class

a gradient search interpretation of the super exponential algorithm this article reviews the super exponential algorithm proposed by shalvi and weinstein 1993 for blind channel equalization the principle of this algorithm hadamard exponentiation projection over the set of attainable combined channel equalizer impulse responses followed by a normalization is shown to coincide with a gradient search of an extremum of a cost function the cost function belongs to the family of functions given as the ratio of the standard l sub 2p and l sub 2 sequence norms where p 1 this family is very relevant in blind channel equalization tracing back to donoho s 1981 work on minimum entropy deconvolution and also underlying the godard 1980 or constant modulus and the earlier shalvi weinstein algorithms using this gradient search interpretation which is more tractable for analytical study we give a simple proof of convergence for the super exponential algorithm finally we show that the gradient step size choice giving rise to the super exponential algorithm is optimal

discretization on unstructured grids for inhomogeneous anisotropic media part ii discussion and numerical results a companion paper part i siam j sci comput 19 1998 pp 1700 1716 presents discretization methods for control volume formulations in two space dimensions properties of the developed methods are further discussed including rotational invariance and stability under the influence of anisotropy strong anisotropy may have to be handled by use of stretched grid cells numerical examples that demonstrate strengths and weaknesses of the methods are presented the examples focus on inhomogeneity anisotropy and rotational invariance

monotonic convergence of fixed point algorithms for ica we re examine a fixed point algorithm proposed by hyvarinen for independent component analysis wherein local convergence is proved subject to an ideal signal model using a square invertible mixing matrix here we derive step size bounds which ensure monotonic convergence to a local extremum for any initial condition our analysis does not assume an ideal signal model but appeals rather to properties of the contrast function itself and so applies even with noisy data and or more sources than sensors the results help alleviate the guesswork that often surrounds step size selection when the observed signal does not fit an idealized model

convergence analysis of a mixed finite volume scheme for an elliptic parabolic system modeling miscible fluid flows in porous media we study a finite volume discretization of a strongly coupled elliptic parabolic pde system describing miscible displacement in a porous medium we discretize each equation by a finite volume scheme which allows a wide variety of unstructured grids in any space dimension and gives strong enough convergence for handling the nonlinear coupling of the equations we prove the convergence of the scheme as the time and space steps go to 0 finally we provide numerical results to demonstrate the efficiency of the proposed numerical scheme

undermodeled equalization a characterization of stationary points for a family of blind criteria we attack specific problems related to equalizer performance in undermodeled cases in which assumptions of perfect equalizability are dismissed in favor of a more realistic situation in which no equalizer setting may achieve perfect channel equalization we derive a characterization of candidate convergent points for a family of blind criteria which appeal tacitly or wittingly to maximizing the ratio of different sequence norms of the combined channel equalizer impulse response this may be accomplished in a practical implementation by using equalizer output cumulants of different orders the popular godard and shalvi weinstein schemes are accommodated at one extreme of the family of criteria we also show that each maximum at the other extreme of the family involving progressively higher order output cumulants yields precisely a wiener response this suggests that blind algorithms using progressively higher order statistics may converge more closely to a wiener response than those using more modest order statistics we show moreover that the superexponential family of algorithms is also included and establish a convergence proof for undermodeled cases that appeals to no approximation finally some apparently novel bounds on attainable open eye measures in undermodeled cases are also derived

the fastica algorithm revisited convergence analysis the fast independent component analysis fastica algorithm is one of the most popular methods to solve problems in ica and blind source separation it has been shown experimentally that it outperforms most of the commonly used ica algorithms in convergence speed a rigorous local convergence analysis has been presented only for the so called one unit case in which just one of the rows of the separating matrix is considered however in the fastica algorithm there is also an explicit normalization step and it may be questioned whether the extra rotation caused by the normalization will affect the convergence speed the purpose of this paper is to show that this is not the case and the good convergence properties of the one unit case are also shared by the full algorithm with symmetrical normalization a local convergence analysis is given for the general case and the global behavior is illustrated numerically for two sources and two mixtures in several typical cases

discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients this paper introduces a new conservative cell centred finite volume scheme for the accurate solution of elliptic diffusion equations with strongly varying coefficients the discretisation is designed to model a wide range of heterogeneities and anisotropies and in particular the case where the diffusivity is represented by a nondiagonal matrix which may occur if the medium is anisotropic in a general direction such problems arise for example in oil reservoir simulation when renormalisation techniques are used to model the reservoir geology it is in this context that the method is described although it is applicable to a far wider class of problems in heat conduction electrostatics and potential theory the paper also describes the application of a mulitigrid scheme for efficient numerical solution of the algebraic equations derived using the new discretisation

efficient variant of algorithm fastica for independent component analysis attaining the cramer rao lower bound fastica is one of the most popular algorithms for independent component analysis ica demixing a set of statistically independent sources that have been mixed linearly a key question is how accurate the method is for finite data samples we propose an improved version of the fastica algorithm which is asymptotically efficient i e its accuracy given by the residual error variance attains the cramer rao lower bound crb the error is thus as small as possible this result is rigorously proven under the assumption that the probability distribution of the independent signal components belongs to the class of generalized gaussian gg distributions with parameter alpha denoted gg alpha for alpha 2 we name the algorithm efficient fastica efica computational complexity of a matlab implementation of the algorithm is shown to be only slightly about three times higher than that of the standard symmetric fastica simulations corroborate these claims and show superior performance of the algorithm compared with algorithm jade of cardoso and souloumiac and nonparametric ica of boscolo on separating sources with distribution gg alpha with arbitrary alpha as well as on sources with bimodal distribution and a good performance in separating linearly mixed speech signals

maximum principle in linear finite element approximations of anisotropic diffusion convection reaction problems a mesh condition is developed for linear finite element approximations of anisotropic diffusion convection reaction problems to satisfy a discrete maximum principle loosely speaking the condition requires that the mesh be simplicial and mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix where h h denotes the mesh size and varvec b b and c c are the coefficients of the convection and reaction terms in two dimensions the condition can be replaced by a weaker mesh condition an mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 perturbation of a generalized delaunay condition these results include many existing mesh conditions as special cases numerical results are presented to verify the theoretical findings

unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients multipoint flux approximation mpfa techniques are commonly applied for discretizing the porous media flow equations within the context of finite volume numerical procedures although these methods can be applied to heterogeneous anisotropic systems on generally unstructured grids the inverse of the resulting linear operator can suffer from a loss of monotonicity at high permeability anisotropy ratios resulting in spurious oscillations of the pressure solution the purpose of this paper is to develop a method for optimizing unstructured grids in two and three dimensions such that the monotonicity behavior of the mpfa technique is significantly improved the method employs anisotropic triangulation and can be readily combined with permeability upscaling procedures results are presented for a variety of examples and the technique is shown to perform well on problems involving complex grid point distributions and heterogeneous permeability fields with strong anisotropy ratios of o 100 oscillation free pressure solutions are achieved in all cases considered even for examples in which the original grid shows large oscillations

convergence analysis of a family of ellam schemes for a fully coupled model of miscible displacement in porous media we analyse the convergence of numerical schemes in the gdm ellam gradient discretisation method eulerian lagrangian localised adjoint method framework for a strongly coupled elliptic parabolic pde which models miscible displacement in porous media these schemes include but are not limited to mixed finite element ellam and hybrid mimetic mixed ellam schemes a complete convergence analysis is presented on the coupled model using only weak regularity assumptions on the solution which are satisfied in practical applications and not relying on l infty bounds which are impossible to ensure at the discrete level given the anisotropic diffusion tensors and the general grids used in applications

computation of periodic solution bifurcations in odes using bordered systems we consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous odes namely the fold the period doubling or flip bifurcation and the torus or neimark sacker bifurcation in the fold and flip cases we append one scalar equation to the standard periodic bvp that defines the periodic solution in the torus case four scalar equations are appended evaluation of these scalar equations and their derivatives requires the solution of linear bvps whose sparsity structure after discretization is identical to that of the linearization of the periodic bvp therefore the calculations can be done using existing numerical linear algebra techniques such as those implemented in the software auto and colsys

unified primal formulation based a priori and a posteriori error analysis of mixed finite element methods we derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second order elliptic problems it is based on the classical primal weak formulation the postprocessing of the potential proposed in t arbogast and z chen on the implementation of mixed methods as nonconforming methods for second order elliptic problems math comp 64 1995 943 972 and the discrete friedrichs inequality our analysis in particular avoids any explicit use of the uniform discrete inf sup condition and in a straightforward manner and under minimal necessary assumptions known convergence and superconvergence results are recovered the same framework then turns out to lead to optimal a posteriori energy error bounds in particular estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived they give a guaranteed and fully computable upper bound on the energy error represent error local lower bounds and are robust under some conditions on the diffusion dispersion tensor they are thus suitable for both overall error control and adaptive mesh refinement moreover the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method we finally prove that in parallel and simultaneously in converse to galerkin finite element methods under some circumstances the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the h 1 0 space and also establish several links between mixed finite element approximations and some generalized weak solutions

sign preserving of principal eigenfunctions in p1 finite element approximation of eigenvalue problems of second order elliptic operators this paper is concerned with the p1 finite element approximation of the eigenvalue problem of second order elliptic operators subject to the dirichlet boundary condition the focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem it is shown that when the stiffness matrix is an irreducible m matrix the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain mesh conditions leading to such a stiffness matrix are also studied a sufficient condition is that the mesh is simplicial interiorly connected and acute when measured in the metric specified by the inverse of the diffusion matrix the acute requirement can be replaced by the delaunay condition in two dimensions numerical results show that when the stiffness matrix is not an m matrix a finite element approximation can be structurally different from the continuous eigenvalue problem the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue in modulus can even be complex for the case with nonsymmetric operators

study of discrete duality finite volume schemes for the peaceman model in this paper we are interested in the finite volume approximation of a system describing miscible displacement in porous media this system is made of two coupled equations an anisotropic diffusion equation on the pressure and a convection diffusion dispersion equation on the concentration of invading fluid the anisotropic diffusion operators in both equations require a special care while discretizing by a finite volume method we focus here on the numerical approximation by some discrete duality finite volume methods after the presentation of the scheme we establish some a priori estimates satisfied by the numerical solution and prove existence and uniqueness of the solution to the scheme we show the efficiency of the schemes through numerical experiments

a triangle based mixed finite element finite volume technique for modeling two phase flow through porous media triangle based discretization techniques offer great advantages relative to standard finite difference methods for the modeling of flow through geometrically complex geological features the purpose of this paper is to develop and apply a triangle based method for the modeling of two phase flow through porous formations the formulation includes the effects of gravity compressibility and capillary pressure the technique entails a triangle based mixed finite element method for solution of the variable coefficient parabolic pressure equation and a second order tvd type total variation diminishing finite volume scheme for solution of the essentially hyperbolic saturation equation the method is applied to a variety of example problems and is shown to perform very well on problems involving geometric complexity coupled with heterogeneous generally anisotropic permeability descriptions

convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics abstract a nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium conservation of mass for the mixture leads to an elliptic equation for pressure and conservation for the displacing fluid yields a convectiondominated parabolic equation for the concentration of that fluid a sequential implicit time stepping procedure is defined in which the pressure and darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the concentration is approximated by a combination of a galerkin finite element method and the method of characteristics optimal order convergence in l2 is proved time truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the concentration equation temporal and spatial errors are lessened by direct computation of the velocity in the mixed method as opposed to differentiation of the pressure several extensions of these results are outlined

an approximation to miscible fluid flows in porous media with point sources and sinks by an eulerian lagrangian localized adjoint method and mixed finite element methods we develop an eulerian lagrangian localized adjoint method ellam mixed finite element method mfem solution technique for accurate numerical simulation of coupled systems of partial differential equations pdes which describe complex fluid flow processes in porous media an ellam which was shown previously to outperform many widely used methods in the context of linear convection diffusion pdes is presented to solve the transport equation for concentration since accurate fluid velocities are crucial in numerical simulations an mfem is used to solve the pressure equation for the pressure and darcy velocity this minimizes the numerical difficulties occurring in standard methods for approximating velocities caused by differentiation of the pressure and then multiplication by rough coefficients the ellam mfem solution technique significantly reduces temporal errors symmetrizes the governing transport equation eliminates nonphysical oscillation and or excessive numerical dispersion in many simulators conserves mass and treats boundary conditions accurately numerical experiments show that the ellam mfem solution technique simulates miscible displacements of incompressible fluid flows in porous media accurately with fairly coarse spatial grids and very large time steps which are one or two orders of magnitude larger than the time steps used in many methods moreover the ellam mfem solution technique can treat large mobility ratios discontinuous permeabilities and porosities anisotropic dispersion in tensor form and point sources and sinks