alpha trimmed means and their relationship to median filters suppose that x is a finite set of n numbers the trimmed mean of x is obtained by sorting x into ascending order removing trimming a fixed fraction alpha 0 leq alpha leq 0 5 from the high and low ends of the sorted set and computing the average of the remaining values when applied to a sliding window of length lw the trimming process is called trim filtering for 0 5 the trimmed mean of a set is the median of the set and the filtering operation is called median filtering repeated application of a median filter to the output of a previous median filter of the same length lw eventually produces a signal which is invariant to median filtering this final signal is called the root signal this paper explains the relationship between trimmed means and median filters derives a simple straightforward and fast algorithm for applying a median filter and provides a new explanation of the convergence of repeated median filtering to the root signal the latter result incorporates an approach which permits generalization of the associated concepts to a larger class of index map filters

sorting continuous time signals analog median and median type filters there are two natural orderings in signals temporal order and rank order there is no compelling reason to explore only one of these orderings either in the discrete time or in the continuous time case nevertheless the concept of rank order for continuous time signals remains virtually neglected which is in striking contrast to the discrete time case ranked order discrete time filters of which the running median is the most common example have been intensively studied for three decades the dependence of these nonlinear systems on the order statistics of the input samples stands in contrast with the tapped delay line filter which depends on temporal order only however continuous time signals can also be meaningfully sorted a fact that is explored in this paper to define and study the analog median filter and other ranked order filters the paper introduces the basic tools needed to analyze and understand these continuous time nonlinear filters the distribution function and the sorting and presents some of their properties in a tutorial way the analog median filter is defined in terms of the unique nonincreasing left continuous sorting more general filters can also be defined including filters similar to spl alpha trimmed mean filters and l filters these include filters that depend on one parameter and contain the running average and running median as special cases the rate of convergence of the digital median filter to the analog median filter is discussed and related to the signal sampling period the duration of the filter window and the smoothness of the input signal the paper introduces the concept of noise width and studies the effect of additive and multiplicative noise at the output of the analog median filter in terms of the noise width and the smoothness of the input signal

brief paper robust set membership state estimation application to underwater robotics this paper proposes a new observer for estimating the state vector of a nonlinear system this observer which is robust with respect to outliers assumes that the measurement errors as well as the number of outliers that could occur within a given time window are bounded the principle of the approach is to use interval analysis to deal properly with the nonlinearities involved in the system without any linearization or approximation and to propagate through the time in a forward and backward manner the assumptions made about outliers a test case related to the localization and control of an underwater robot is also proposed to illustrate the efficiency of the approach

using interval arithmetic to prove that a set is path connected in this paper we give a numerical algorithm able to prove whether a set s described by nonlinear inequalities is path connected or not to our knowledge no other algorithm numerical or symbolic is able to deal with this type of problem the proposed approach uses interval arithmetic to build a graph which has exactly the same number of connected components as s examples illustrate the principle of the approach

towards adaptive smoothed aggregation alpha sa for nonsymmetric problems applying smoothed aggregation sa multigrid to solve a nonsymmetric linear system a mathbf x mathbf b is often impeded by the lack of a minimization principle that can be used as a basis for the coarse grid correction process this paper proposes a petrov galerkin pg approach based on applying sa to either of two symmetric positive definite spd matrices sqrt a t a or sqrt aa t these matrices however are typically full and difficult to compute so it is not computationally efficient to use them directly to form a coarse grid correction the proposed approach approximates these coarse grid corrections by using sa to accurately approximate the right and left singular vectors of a that correspond to the lowest singular value these left and right singular vectors are used to construct the restriction and interpolation operators respectively a preliminary two level convergence theory is presented suggesting that more relaxation should be applied than for an spd problem additionally a nonsymmetric version of adaptive sa alpha sa is given that automatically constructs sa multigrid hierarchies using a stationary relaxation process on all levels numerical results are reported for convection diffusion problems in two dimensions with varying amounts of convection for constant variable and recirculating convection fields the results suggest that the proposed approach is algorithmically scalable for problems coming from these nonsymmetric scalar pdes with the exception of recirculating flow this paper serves as a first step for nonsymmetric alpha sa the long term goal of this effort is to develop nonsymmetric alpha sa for systems of pdes where the sa framework has proven to be well suited for adaptivity in spd problems

convergence analysis of multigrid algorithms for nonselfadjoint and indefinite elliptic problems a convergence estimate is established for nonsymmetric multigrid algorithms for finite element methods for nonselfadjoint and indefinite elliptic problems it is shown without further regularity assumption beyond h 1 necessary to define the weak form that the v cycle algorithm converges with rate bounded by 1 cj 1 for sufficiently small coarse mesh size h 1 where j is the number of levels in the multigrid algorithm

robust convergence of multilevel algorithms for convection diffusion equations a new approach is developed to analyze convergence of multilevel algorithms for convection diffusion equations this approach uses a multilevel recursion formula which can be applied to a variety of nonsymmetric problems here the recursion formula is applied to a robust multilevel algorithm for convection diffusion equations with convection in the x or y direction the multilevel algorithm uses semicoarsening line relaxation and prewavelets the convergence rate is proved to be less than 0 18 independent of the size of the convection term and the number of unknowns the assumptions allow the convection term to have a turning point so that an interior layer can appear in the solution of the convection diffusion equation the computational cost of the multilevel cycle is about o n log n independent of the size of the convection term where n is the number of unknowns it is proved that o log n multilevel cycles starting from the initial guess 0 lead to an o n 2 algebraic error with respect to the l infty norm independent of the size of the convection term

convergence analysis of a multigrid method for convection diffusion equations this paper is concerned with the convergence analysis of robust multigrid methods for convection diffusion problems we consider a finite difference discretization of a 2d model convection diffusion problem with constant coefficients and dirichlet boundary conditions for the approximate solution of this discrete problem a multigrid method based on semicoarsening matrix dependent prolongation and restriction and line smoothers is applied for a multigrid w cycle we prove an upper bound for the contraction number in the euclidean norm which is smaller than one and independent of the mesh size and the diffusion convection ratio for the contraction number of a multigrid v cycle a bound is proved which is uniform for a class of convection dominated problems the analysis is based on linear algebra arguments only

the analysis of smoothers for multigrid algorithms the purpose of this paper is to provide a general technique for defining and analyzing smoothing operators for use in multigrid algorithms the smoothing operators considered are based on subspace decomposition and include point line and block versions of jacobi and gauss seidel iteration as well as generalizations we shall show that these smoothers will be effective in multigrid algorithms provided that the subspace decomposition satisfies two simple conditions in many applications these conditions are trivial to verify

some new error estimates for ritz galerkin methods with minimal regularity assumptions new uniform error estimates are established for finite element approximations u h of solutions u of second order elliptic equations lu f using only the regularity assumption u 1 c f 1 using an aubin nitsche type duality argument we show for example that for arbitrary fixed e sufficiently small there exists an h 0 such that for 0 h h 0 u u h 0 e u u h 1 here s denotes the norm on the sobolev space h s other related results are established

analysis and comparison of geometric and algebraic multigrid for convection diffusion equations the discrete convection diffusion equations obtained from streamline diffusion finite element discretization are solved on both uniform meshes and adaptive meshes estimates of error reduction rates for both geometric multigrid gmg and algebraic multigrid amg are established on uniform rectangular meshes for a model problem our analysis shows that gmg with line gauss seidel smoothing and bilinear interpolation converges if h gg epsilon 2 3 and amg with the same smoother converges more rapidly than gmg if the interpolation constant beta in the approximation assumption of amg satisfies beta ll frac h sqrt epsilon alpha hs 1mm mbox where alpha big begin smallmatrix 1 h sqrt varepsilon 2 h ge sqrt varepsilon end smallmatrix on unstructured triangular meshes the performance of gmg and amg both as solvers and as preconditioners for gmres are evaluated numerical results show that gmres with amg preconditioning is a robust and reliable solver on both type of meshes

subspace correction methods for nonsymmetric parabolic problems abstract a convergence proof for general subspace correction techniques applied to an abstract nonsymmetric parabolic equation is given one of the main concerns is to give a unified convergence analysis for domain decomposition and multigrid methods for parabolic problems the analysis is also valid for nonsymmetric problems for second order parabolic problems the convection can dominate the diffusion the algorithms can be applied to domain decomposition methods with or without the coarse mesh for applications to multigrid methods the coarsest mesh does not need to be very coarse a relation between the coarse mesh size and the time step is needed to get a convergence rate independent of the mesh the number of iterations at each time step for the algorithms is also estimated some numerical experiments are presented for domain decomposition methods with minimum overlap which support the theoretical predictions the algorithms are able to capture the sharp traveling shocks for convection dominated problems

theory of order statistic filters and their relationship to linear fir filters necessary and or sufficient conditions on both the filter coefficients and the signal process are derived in order that nonrecursive order statistic os and linear filtering are equivalent operations the results indicate that an understanding of os filters hinges on a better understanding of the properties of signals containing logically monotonic components the results extend a number of previous theories characterizing the well known median and ranked order filters to a broader class of filters and input signals

uniform convergence of multigrid v cycle iterations for indefinite and nonsymmetric problems in this paper an analysis of a multigrid method for nonsymmetric and or indefinite elliptic problems is presented in this multigrid method various types of smoothers may be used one type of smoother considered is defined in terms of an associated symmetric problem and includes point and line jacobi and gauss seidel iterations smoothers based entirely on the original operator are also considered one smoother is based on the normal form that is the product of the operator and its transpose other smoothers studied include point and line jacobi and gauss seidel it is shown that the uniform estimates of j h bramble and j e pasciak math comp 60 1993 pp 447 471 for symmetric positive definite problems carry over to these algorithms more precisely the multigrid iteration for the nonsymmetric and or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine but not dependent on the number of multigrid levels

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

loop detection of mobile robots using interval analysis this paper proposes an original set membership approach for loop detection of mobile robots in the situation where proprioceptive sensors only are available to detect loops the new concepts of thet plane which is a two dimensional space with time coordinates are introduced intervals of functions or tubes are then used to represent uncertain trajectories and tests are provided in order to eliminate parts of the t plane that do not correspond to any loop an experiment with an actual underwater robot is proposed in order to illustrate the principle and the efficiency of the approach

spatial planning a configuration space approach this paper presents algorithms for computing constraints on the position of an object due to the presence of ther objects this problem arises in applications that require choosing how to arrange or how to move objects without collisions the approach presented here is based on characterizing the position and orientation of an object as a single point in a configuration space in which each coordinate represents a degree of freedom in the position or orientation of the object the configurations forbidden to this object due to the presence of other objects can then be characterized as regions in the configuration space called configuration space obstacles the paper presents algorithms for computing these configuration space obstacles when the objects are polygons or polyhedra

a generalized bpx multigrid framework covering nonnested v cycle methods more than a decade ago bramble pasciak and xu developed a framework in analyzing the multigrid methods with nonnested spaces or non inherited quadratic forms it was subsequently known as the bpx multigrid framework which was widely used in the analysis of multigrid and domain decomposition methods however the framework has an apparent limit in the analysis of nonnested v cycle methods and it produces a variable v cycle or nonuniform convergence rate v cycle methods or other nonoptimal results in analysis thus far this paper completes a long time effort in extending the bpx multigrid framework so that it truly covers the nonnested v cycle we will apply the extended bpx framework to the analysis of many v cycle nonnested multigrid methods some of them were proven previously only for two level and w cycle iterations some numerical results are presented to support the theoretical analysis of this paper

convergence analysis of a multigrid method for a convection dominated model problem the paper presents a convergence analysis of a multigrid solver for a system of linear algebraic equations resulting from the discretization of a convection diffusion problem using a finite element method we consider piecewise linear finite elements in combination with a streamline diffusion stabilization we analyze a multigrid method that is based on canonical intergrid transfer operators a direct discretization approach for the coarse grid operators and a smoother of line jacobi type a robust diffusion and h independent bound for the contraction number of the two grid method and the multigrid w cycle are proved for a special class of convection diffusion problems namely with neumann conditions on the outflow boundary dirichlet conditions on the rest of the boundary and a flow direction that is constant and aligned with gridlines our convergence analysis is based on modified smoothing and approximation properties the arithmetic complexity of one multigrid iteration is optimal up to a logarithmic term

the analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems we prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems we provide results for the so called symmetric multigrid schemes we show that for the variable v script cycle and the w script cycle schemes multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns provided that the initial grid is sufficiently fine we show that the v script cycle algorithm also converges under appropriate assumptions on the coarsest grid but at a rate which may deteriorate as the number of levels increases this deterioration for the v script cycle may occur even in the case of full elliptic regularity finally the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory

convergence analysis of the schwarz algorithm and multilevel decomposition iterative methods ii nonselfadjoint and indefinite elliptic problems convergence estimate for the schwarz alternating algorithm applied to nonselfadjoint and indefinite elliptic problems is established for a certain overlapping domain decomposition multilevel iterative algorithms are proposed and analyzed for the finite element method for the same problem with locally refined grids the author shows that the algorithm based on the hierarchical basis decomposition converges with rate bounded by 1 cj 2 where j is the number of levels the author also analyzes the algorithm based on the full multigrid decomposition and shows that the algorithm converges with rate 1 cj 1 for problems of any dimension all the results are established without regularity assumption for the differential equation

estimation of the sinusoidal signal frequency based on the marginal median dft the marginal median discrete fourier transform dft is used for estimation of complex sinusoidal signals embedded in an impulse noise environment expression for the marginal median dft of the sinusoidal signal in the neighborhood of the exact frequency is derived two specific displacement techniques are proposed in order to achieve an accurate estimation of frequency displaced from the frequency grid they are based on specific ratio of the marginal median dft magnitudes for samples in the neighborhood of the marginal median dft maximum calculated over the frequency grid efficiency and accuracy of the proposed techniques is proved for mixed gaussian and impulse noise environment

the analysis of pmhss multigrid methods for elliptic problems with smooth complex coefficients in this paper we investigate a class of multigrid methods with hss smoothers applied to second order elliptic problems whose dominant coefficient is complex valued these smoothers are based on hermitian skew hermitian splitting hss modified hss mhss and preconditioned mhss pmhss iterations it is shown that the abstract theory for multigrid method can be extended and applied to elliptic problems with complex coefficient under some assumption conditions we prove that multigrid method with pmhss smoother is convergent and optimal numerical results are reported to confirm the theoretical analysis