the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

domain decomposition for least squares finite element methods for the stokes equations in this paper we consider the solutions of the discrete linear system resulting from least squares finite element discretization applied to the stokes equations additive and multiplicative domain decomposition methods are used to solve the linear system and the uniform convergence results are established for both methods

additive schwarz algorithms for parabolic problems in this paper we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems we introduce additive schwarz domain decomposition methods for parabolic differential equations and also consider the dependence of convergence rates of these algorithms on parameters of time step and space mesh the resulting preconditioned linear system of equations is solved by the conjugate gradient method finally we give a numerical example

a fuzzy operator for the enhancement of blurred and noisy images rule based fuzzy operators are a novel class of operators specifically designed in order to apply the principles of approximate reasoning to digital image processing this paper shows how a fuzzy operator that is able to perform detail sharpening but is insensitive to noise can be designed the results obtainable by the proposed technique in the enhancement of a real image are presented

noise reduction by fuzzy image filtering a new fuzzy filter is presented for the noise reduction of images corrupted with additive noise the filter consists of two stages the first stage computes a fuzzy derivative for eight different directions the second stage uses these fuzzy derivatives to perform fuzzy smoothing by weighting the contributions of neighboring pixel values both stages are based on fuzzy rules which make use of membership functions the filter can be applied iteratively to effectively reduce heavy noise in particular the shape of the membership functions is adapted according to the remaining noise level after each iteration making use of the distribution of the homogeneity in the image a statistical model for the noise distribution can be incorporated to relate the homogeneity to the adaptation scheme of the membership functions experimental results are obtained to show the feasibility of the proposed approach these results are also compared to other filters by numerical measures and visual inspection

a parallel domain decomposition algorithm of mixed element equation for second order elliptic dirichlet boundary value problem a parallel domain decomposition algorithm of schwarz type is introduced to solve the mixed finite element equation for the dirichlet boundary value problem of a second order partial differential equation of elliptic type and convergence of this algorithm is analyzed

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

nonlinear cochlear mechanics an earlier paper characterizing the linear mechanical response of the organ of corti j acoust soc am 138 1102 1121 2015 is extended to the nonlinear domain assuming the existence of nonlinear oscillators nonlocally coupled through the pressure they help create the oscillator equations are derived and examined when the stimuli are modulated tones and clicks the nonlinearities are constrained by the requirements of oscillator stability and the invariance of zero crossings in the click response to changes in click amplitude the nonlinear oscillator equations for tones are solved in terms of the fluid pressure that drives them and its time derivative presumably a proxy for forces created by outer hair cells the pressure equation is reduced to quadrature the integrand depending on the oscillators responses the resulting nonlocally coupled nonlinear equations for the pressure and oscillator amplitudes and phases are solved numerically in terms of the fluid pressure at the stapes methods fo

adaptive trimmed mean filters for image restoration an adaptive smoothing filter is proposed for reducing noise in digital signals of any dimensionality the adaptive procedure is based on the selection of an appropriate inner or outer trimmed mean filter according to local measurements of the tail behavior impulsivity of the noise process the set of trimmed means used provides robustness against a wide range of noise possibilities ranging from very shallow tailed to very heavy tailed a monte carlo analysis using a family of generalized exponential distributions supports the choice of the trimmed mean selected for measured values of an impulsivity statistic the assumption underlying the definition of the filter is that the signal to be filtered is locally smoothly varying and that the noise process is uncorrelated and derives from an unknown unimodal symmetric distribution for image processing applications a second statistic is used to mark the location of abrupt intensity changes or edges in the vicinity of an edge the trend preserving median filter is used since the impulsivity and edge statistics used in defining the adaptive filter are both functions of order statistics the extra computation required for their calculation is minimal examples are provided of the filter as applied to images corrupted by a variety of noises

an application of the abstract multilevel theory to nonconforming finite element methods a multilevel product algorithm and its additive version or a multilevel preconditioner are proposed for the crouzeix raviart nonconforming finite element methods for second order elliptic equations the authors show that the preconditioned matrix has a condition number bounded uniformly from above if the domain can be partitioned into overlapping convex subdomains a uniform convergence for the multilevel product algorithm is also established the results are regularity free for the continuous problem and can be applied to problems with rough solutions and local grid refinement numerical results that illustrate the theory are presented

rate of convergence for some constraint decomposition methods for nonlinear variational inequalities some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset one of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods for domain decomposition and multigrid methods the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems in order to analyse the convergence rate we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints a special nonlinear interpolation operator is introduced for decomposing the functions

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

multiplicative schwarz methods for parabolic problems based on domain decomposition we give a multiplicative schwarz domain decomposition method for semi linear parabolic problems we consider the relation between the convergence rate and discretization parameters including the diameter of the sub domain we give the error estimate which tells us that the convergence of the approximate solution is independent of the iteration number at each time level finally we give a numerical example

convergence rate of a schwarz multilevel method for the constrained minimization of nonquadratic functionals in l badea convergence rate of a multiplicative schwarz method for strongly nonlinear inequalities v barbu i lasiecka d tiba and c varsan eds kluwer academic publishers boston 2003 the convergence of a subspace correction method applied to the constrained minimization of a functional in a general reflexive banach space has been proved provided that the convex set verifies a certain assumption this assumption is weaker than that in which the convex set is decomposed according to the space decomposition as a sum of subsets in the sobolev spaces the proposed method becomes a multiplicative schwarz method for the solution of the variational inequalities coming from the minimization of nonquadratic functionals we prove in this paper that this assumption holds for the one two and multilevel multiplicative schwarz methods in the finite element space and we explicitly write the constants in the error estimations depending on the overlapping and mesh parameters our error estimates are similar with those obtained for the minimization of quadratic functionals in l badea x c tai and j wang siam j numer anal 2003 pp 1052 1073 or with those obtained for the one obstacle problem in x c tai numer math 93 2003 pp 755 786

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

an anisotropic h adaptive finite element method for compressible navier stokes equations abstract application of the finite element method with significantly stretched elements to solve compressible navier stokes equations is presented stretched elements are introduced in an adaptive process as a result of selective directional subsecting of elements an adaptive strategy of such anisotropic h type refinements was developed it is based on minimization of the interpolation errors reduction of the residual error indicators and an adaptive recovery of the viscous fluxes with the possibility to control their accuracy an iterative solver with the convergence characteristics independent of the aspect ratio of elements is presented the solver is the gmres algorithm with the domain decomposition schwarz type preconditioner the proposed techniques are used to solve a benchmark problem a hypersonic high reynolds number flow with strong shock boundary layer interaction

uniform convergence of the schwarz alternating method for solving singularly perturbed advection diffusion equations our purpose in this paper is to study the convergence rate of some variants of the schwarz alternating method for solving two types of singularly perturbed advection diffusion equations one in which the advection term first order term dominates and the other in which the lowest order term zeroth order term dominates for such problems boundary or other layer regions may be present for singular perturbation problems in which the advection dominates we choose two overlapping subregions one corresponding to an outer or elliptic region and the other corresponding to an inner or hyperbolic region our main result for this problem shows that if the subdomains can be chosen to follow the flow i e if the boundary interface of one of the subdomains corresponds to an outflow boundary for the streamlines of the flow then the schwarz iterates converge in the maximum norm with an error reduction factor per iteration that exponentially decays with increasing overlap or decreasing diffusion for the singular perturbation problem in which the lowest order term dominates we consider many overlapping subregions with no special requirement on outflow or inflow boundaries our main result here which is valid for two variants of the classical schwarz algorithm shows that the error reduction factor per iteration decays exponentially in the maximum norm with increasing overlap and increasing dominance of the zeroth order term for both problems our proofs rely on the maximum principle and the construction of barrier functions

fuzzy random impulse noise reduction method a new two step fuzzy filter that adopts a fuzzy logic approach for the enhancement of images corrupted with impulse noise is presented in this paper the filtering method entitled as fuzzy random impulse noise reduction method frinr consists of a fuzzy detection mechanism and a fuzzy filtering method to remove random valued impulse noise from corrupted images based on the criteria of peak signal to noise ratio psnr and subjective evaluations we have found experimentally that the proposed method provides a significant improvement on other state of the art methods

intensity invariance of fine time structure in basilar membrane click responses implications for cochlear mechanics basilar membrane and auditory nerve responses to impulsive acoustic stimuli whether measured directly in response to clicks or obtained indirectly using cross or reverse correlation and or fourier analysis manifest a striking symmetry near invariance with stimulus intensity of the fine time structure of the response over almost the entire dynamic range of hearing this paper explores the origin and implications of this symmetry for cochlear mechanics intensity invariance is investigated by applying the eq nl theorem de boer aud neurosci 3 377 388 1997 to define a family of linear cochlear models in which the strength of the active force generators is controlled by a real valued intensity dependent parameter with 0 1 the invariance of fine time structure is conjectured to imply that as is varied the poles of the admittance of the cochlear partition remain within relatively narrow bands of the complex plane oriented perpendicular to the real frequency axis physically the conjecture implies that the local resonant frequencies of the cochlear partition are nearly independent of intensity cochlear model responses computed by extending the model obtained by solution of the inverse problem in squirrel monkey at low sound levels zweig j acoust soc am 89 1229 1254 1991 with three different forms of the intensity dependence of the partition admittance support the conjecture intensity invariance of cochlear resonant frequencies is shown to be consistent with the well known half octave shift describing the shift with intensity in the peak or best frequency of the basilar membrane frequency response shifts in best frequency do not arise locally via changes in the underlying resonant frequencies of the partition but globally through the intensity dependence of the driving pressure near invariance of fine time structure places strong constraints on the mechanical effects of force generation by outer hair cells in particular the symmetry requires that the feedback forces generated by outer hair cells ohcs not significantly affect the natural resonant frequencies of the cochlear partition these results contradict many if not most cochlear models in which ohc forces produce significant changes in the reactance and resonant frequencies of the partition

global and uniform convergence of subspace correction methods for some convex optimization problems this paper gives some global and uniform convergence estimates for a class of subspace correction based on space decomposition iterative methods applied to some unconstrained convex optimization problems some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

an optimal fuzzy system for color image enhancement a gaussian membership function is proposed to fuzzify the image information in spatial domain we introduce a global contrast intensification operator gint which contains three parameters viz intensification parameter t fuzzifier fh and the crossover point muc for enhancement of color images we define fuzzy contrast based quality factor qf and entropy based quality factor qe and the corresponding visual factors for the desired appearance of images by minimizing the fuzzy entropy of the image information with respect to these quality factors the parameters t fh and muc are calculated globally by using the proposed technique a visible improvement in the image quality is observed for under exposed images as the entropy of the output image is decreased the terminating criterion is decided by both the visual and quality factors for over exposed and under plus over exposed images the proposed fuzzification function needs to be modified by taking maximum intensity as the fourth parameter the type of the images is indicated by the visual factor which is less than 1 for under exposed images and more than 1 for over exposed images

parallel galerkin domain decomposition procedures for wave equation parallel galerkin domain decomposition procedures for wave equation are given these procedures use implicit method in the sub domains and simple explicit flux calculation on the inter boundaries of sub domains by integral mean method or extrapolation method thus the parallelism can be achieved by these procedures the explicit nature of the flux prediction induces a time step constraint that is necessary to preserve the stability l 2 norm error estimates are derived for these procedures experimental results are presented to confirm the theoretical results

additive schwarz algorithms for parabolic convection diffusion equations in this paper we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems we introduce three additive schwarz type domain decomposition methods for general not necessarily selfadjoint linear second order parabolic partial differential equations and also study the convergence rates of these algorithms the resulting preconditioned linear system of equations is solved by the generalized minimal residual method numerical results are also reported

multigrid convergence for second order elliptic problems with smooth complex coefficients the finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither hermitian nor definite when the coefficient function is complex valued for such problems under a uniqueness assumption we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number providing an efficient solver for computing discrete approximations numerical experiments while confirming the theory also reveal pronounced sensitivity of gauss seidel iterations on the ordering of the unknowns for certain problems

convergence of algebraic multigrid based on smoothed aggregation we prove aconvergence estimate for the algebraic multigrid method with prolongations defined by aggregation using zero energy modes followed by a smoothing the method input is the problem matrix and a matrix of the zero energy modes the estimate depends only polylogarithmically on the mesh size and requires only a weak approximation property for the aggregates which can be a priori verified computationally construction of the prolongator in the case of a general second order system is described and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements

schwarz methods of neumann neumann type for three dimensional elliptic finite element problems several domain decomposition methods of neumann neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements we will only consider problems in three dimensions several new variants of the basic algorithm are introduced in a schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi level methods the neumann neumann algorithms have several advantages over other domain decomposition methods the subregions which define the subproblems only share the boundary degrees of freedom with their neighbors the subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries in its original form however the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases in the new variants of the algorithms considered in this paper the preconditioners include global components of low rank to overcome this difficulty bounds are established for the condition number of the iteration operator which are independent of the number of subregions and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions 1995 john wiley sons inc

analysis of a parallel schwarz algorithm for elliptic problems a parallel algorithm which is derived directly from the schwarz alternating algorithm with an overlapping domain decomposition for solving boundary value problems of elliptic partial differential equations is analyzed a new general analysis framework different from lions ones is established to describe the recursive relations of error functions between approximate solutions and the exact solution at the different iterative steps some abstract theories on convergence and convergent rate of the synchronously parallel algorithm are established under different conditions

additive schwarz methods for parabolic problems based on domain decomposition we give an additive schwarz domain decomposition method for semi linear parabolic problems we consider the dependence of convergence rate of this algorithm on parameters of time step and space mesh we give the error estimate which tell us that the convergence of the approximate solution is independent of the iteration number at each time level

local multilevel methods for adaptive finite element methods for nonsymmetric and indefinite elliptic boundary value problems in this paper we propose some local multilevel algorithms for solving linear systems arising from adaptive finite element approximations of nonsymmetric and indefinite elliptic boundary value problems two types of local smoothers are constructed one is based on the original nonsymmetric problems and the other is defined in terms of the associated symmetric problems it is shown that the local multilevel methods for the nonsymmetric and indefinite elliptic boundary value problems are optimal which means that the convergence rates of the local multilevel methods are independent of mesh sizes and mesh levels provided that the coarsest grid is sufficiently fine numerical experiments are reported to confirm our theory

multiplicative schwarz methods for parabolic problems the class of multiplicative schwarz methods originated from the classical schwarz alternating method it has been shown to be one of the most powerful methods for solving finite element or finite d

numerical analysis of vanka type solvers for steady stokes and navier stokes flows we consider vanka type smoothers for solving stokes and navier stokes problems in each iteration step this smoother requires the solution of several small local subproblems over finite element blocks it is shown that for particular choices for the blocks the algorithm always converges to the solution of the stokes problem and under suitable conditions to the solution of the navier stokes problem the convergence properties are analyzed and numerical examples are presented

domain decomposition for a least square finite element method for second order elliptic problem in this paper we consider the solutions of the discrete linear system resulting from least square mixed finite element discretization applied to a second order elliptic problems additive and multiplicative domain decomposition methods are used to solve the linear system and the uniform convergence results are established for both methods

finding the impedance of the organ of corti measurements of the nonlinear response of the basilar membrane to a pure tone are shown to have a simple form for moderate membrane velocities v x f vu vu x f vu x f f fc x where the response v is the velocity of the membrane at measurement position x vu is the umbo velocity f is the frequency of the stimulus and fc x is the local characteristic frequency the frequency dependence of the functions x f and x f is determined from the data and x f and ln x f are shown to be analytic functions in the lower half of the complex frequency plane with re x f a monotonically increasing function of f at fixed x the linear limit of basilar membrane motion is characterized by a transfer function t x f v1 1 estimated by extrapolating v x f vu vu to a small membrane velocity v1 t x f and ln t x f are shown to be analytic functions in the lower half of the complex frequency plane the inverse of the amplitude of the transfer function which has both a deep dip at f fc x