optimal identical binary quantizer design for distributed estimation we consider the design of identical one bit probabilistic quantizers for distributed estimation in sensor networks we assume the parameter range to be finite and known and use the maximum cramer rao lower bound crb over the parameter range as our performance metric we restrict our theoretical analysis to the class of antisymmetric quantizers and determine a set of conditions for which the probabilistic quantizer function is greatly simplified we identify a broad class of noise distributions which includes gaussian noise in the low snr regime for which the often used threshold quantizer is found to be minimax optimal aided with theoretical results we formulate an optimization problem to obtain the optimum minimax crb quantizer for a wide range of noise distributions we demonstrate the superior performance of the new quantizer particularly in the moderate to high snr regime

the approximation power of moving least squares a general method for near best approximations to functionals on r d using scattered data information is discussed the method is actually the moving least squares method presented by the backus gilbert approach it is shown that the method works very well for interpolation smoothing and derivatives approximations for the interpolation problem this approach gives mclain s method the method is near best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation the interpolation approximation in r d is shown to be a c function and an approximation order result is proven for quasi uniform sets of data points

the piecewise polynomial partition of unity functions for the generalized finite element methods a partition of unity pu function is an essential component of the generalized finite element method gfem the popular shepard pu functions which are rational functions are easy to construct but have difficulties in dealing with essential boundary conditions and require lengthy computing time for reasonable accuracy in numerical integration in this paper we introduce two simple pu functions the first is a highly regular piecewise polynomial consisting of two distinct polynomials that is effective for uniformly partitioned patches the second is a highly regular piecewise polynomial consisting of three distinct polynomials which is for arbitrary partitioned patches

on the approximability and the selection of particle shape functions particle methods also known as meshless or meshfree methods have become popular in approximating solutions of partial differential equations especially in the engineering community these methods do not employ a mesh or use it minimally in the construction of shape functions there is a wide variety of classes of shape functions that can be used in particle methods in this paper we primarily address the issue of selecting a class of shape functions among this wide variety that would yield efficient approximation of the unknown solution we have also made several comments and observations on the order of convergence of the interpolation error when these shape functions are used specifically we have shown that the interpolation error estimate for certain classes of shape functions may not indicate the actual order of convergence of the approximation error

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

compactly supported fundamental functions for spline interpolation in this paper various ways of constructing locally supported fundamental splines leading to highly accurate local interpolation schemes are proposed and analyzed

reproducing kernel element method part ii globally conforming im cn hierarchies in this part of the work a minimal degrees of freedom arbitrary smooth globally compatible i m c n interpolation hierarchy is constructed in the framework of reproducing kernel element method rkem for arbitrary multiple dimensional domains this is the first interpolation hierarchical structure that has been constructed with both minimal degrees of freedom and higher order smoothness or continuity over multi dimensional domain the proposed hierarchical structure possesses the generalized kronecker property i e o a w b i ox a x j dijdab jaj jbj 6 m

on distributed linear estimation with observation model uncertainties we consider distributed estimation of a gaussian source in a heterogenous bandwidth constrained sensor network where the source is corrupted by independent multiplicative and additive observation noises we assume the additive observation noise is zero mean gaussian with known variance however the system designer is unaware of the distribution of multiplicative observation noise and only knows its first and second order moments for multibit quantizers we derive an accurate closed form approximation for the mean square error mse of the linear minimum mse estimator at the fusion center for both error free and erroneous communication channels we propose several rate allocation methods named as longest root to leaf path greedy integer relaxation and individual rate allocation to minimize the mse given a network bandwidth constraint and minimize the required network bandwidth given a target mse we also derive the bayesian cramer rao lower bound crlb for an arbitrarily distributed multiplicative observation noise and compare the mse performance of our proposed methods against the crlb our results corroborate that for the low power multiplicative observation noise and adequate network bandwidth the gaps between the mse of greedy and integer relaxation methods and the crlb are negligible while the mse of individual rate allocation and uniform methods is not satisfactory through analysis and simulations we also explore why maximum likelihood and maximum a posteriori estimators based on one bit quantization perform poorly for the low power additive observation noise

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

reproducing kernel element method part i theoretical formulation in this paper and its sequels we introduce and analyze a new class of methods collectively called the reproducing kernel element method rkem the central idea in the development of the new method is to combine the strengths of both finite element methods fem and meshfree methods two distinguished features of rkem are the arbitrarily high order smoothness and the interpolation property of the shape functions these properties are desirable especially in solving galerkin weak forms of higher order partial differential equations and in treating dirichlet boundary conditions so unlike the fem there is no need for special treatment with the rkem in solving high order equations compared to meshfree methods dirichlet boundary conditions do not present any difficulty in using the rkem a rigorous error analysis and convergence study of the method are presented the performance of the method is illustrated and assessed through some numerical examples

the generalized product partition of unity for the meshless methods the partition of unity is an essential ingredient for meshless methods named by gfem pufem partition of unity fem xfem extended fem rkpm reproducing kernel particle method rppm reproducing polynomial particle method the method of hp clouds in the literature there are two popular choices for partition of unity a piecewise linear fem mesh and the shepard type partition of unity however the partition of unity pu by a fem mesh leads to the singular or nearly singular matrices and non smooth approximation functions the shepard type partition of unity requires lengthy computing time and its implementation is difficult in order to alleviate these difficulties oh et al introduced the smooth piecewise polynomial pu functions with flat top that lead to small matrix condition numbers and almost everywhere partition of unity that can handle essential boundary conditions nevertheless we could not have the smooth closed form pu functions with flat top for general polygonal patches 2d and general polyhedral patches 3d in this paper we introduce one of the most simple and efficient partition of unity called the generalized product partition of unity the product pu functions constructed by this method are the closed form smooth piecewise polynomials with flat top and could handle background meshes general polygonal patches as well as general polyhedral patches arising in practical applications of meshless methods

parameter estimation from quantized observations in multiplicative noise environments the problem of distributed parameter estimation from binary quantized observations is studied when the unquantized observations are corrupted by combined multiplicative and additive gaussian noise these results are applicable to sensor networks where the sensors observe a parameter in combined additive and nonadditive noise and to a case where dispersed receivers are employed with analog communication over fading channels where the receivers employ binary quantization before noise free digital communications to a fusion center we first discuss the case in which all the quantizers use an identical threshold the parameter identifiability condition is given and surprisingly it is shown that unless the common threshold is chosen properly and the parameter lies in an open interval the parameter will not generally be identifiable in contrast to the additive noise case the best achievable mean square error mse performance is characterized by deriving the corresponding cramer rao lower bound crlb a closed form expression describing the corresponding maximum likelihood ml estimator is presented the stability of the performance of the ml estimators is improved when a nonidentical threshold strategy is utilized to estimate the unknown parameter the thresholds are designed by maximizing the minimum asymptotic relative efficiency are between quantized and unquantized ml estimators although the ml estimation problem is nonconvex it is shown that one can relax the optimization to make it convex the solution to the relaxed problem is used as an initial solution in a gradient algorithm to solve the original problem next the case where both the variances of the additive noise and multiplicative noise are unknown is studied the corresponding crlb is obtained and the ml estimation problem is transformed to a convex optimization which can be solved efficiently finally numerical simulations are performed to substantiate the theoretical analysis

generalized finite element method using mesh based handbooks application to problems in domains with many voids this paper describes a new version of the generalized finite element method originally developed int j numer methods engrg 47 2000 1401 comput methods appl mech engrg 181 2000 43 the design and implementation of the generalized finite element method ph d thesis texas am comput methods appl mech engrg 190 2001 4081 which is well suited for problems set in domains with a large number of internal features e g voids inclusions cracks etc the main idea is to employ handbook functions constructed on subdomains resulting from the mesh discretization of the problem domain the proposed new version of the gfem is shown to be robust with respect to the spacing of the features and is capable of achieving high accuracy on meshes which are rather coarse relative to the distribution of the features

almost everywhere partition of unity to deal with essential boundary conditions in meshless methods abstract due to overlapping nature of supports of partition of unity functions and the lack of the kronecker delta property of meshless shape functions it is difficult to deal with essential boundary conditions in meshless methods in this paper in order to alleviate this difficulty we introduce almost everywhere partition of unity that is a partition of unity except a few points along boundary in two dimensional case actually the gradient of partition of unity functions become infinitely large at these exceptional points however we prove that the presence of these bad points does not change the convergence rates of computed solutions comparing with the computed solutions obtained by the lagrange multiplier method the penalty method and the nitche s method we demonstrate the proposed method is more effective in dealing with essential boundary conditions in meshless methods

meshfree particle methods for thin plates abstract in this paper we are concerned with meshfree particle methods for the solutions of the classical plate model the vertical displacement of a thin plate is governed by a fourth order elliptic equation and thus the approximation functions for numerical solutions are required to have continuous partial derivatives hence the conventional finite element method has difficulties to solve the fourth order problems meshfree methods have the advantage of constructing smooth approximation functions however most of the earlier works on meshfree methods for plate problems used either moving least squares method with penalty method or coupling fem with meshfree method to deal with essential boundary conditions in this paper by using generalized product partition of unity introduced by oh et al we introduce meshfree particle methods in which approximation functions have high order polynomial reproducing property and the kronecker delta property we also prove error estimates for the proposed meshfree methods moreover to demonstrate the effectiveness of our method results of the proposed method are compared with existing results for various shapes of plates with variety of boundary conditions and loads

an immersed boundary method for nonuniform grids we propose a new direct forcing immersed boundary method for simulating the flow around an arbitrarily shaped body in nonuniform grids a new formulation of the weight function for the distribution of the immersed boundary forcing with an efficient strategy that distributes the lagrangian points on the surface of the immersed body is developed we validate our method for various flows induced by a translational or angular motion of a sphere by measuring the lift drag and moment coefficients for an arbitrary shape particle we test our method for flow around an ellipsoid in the uniform flow all tests show good agreement with available references finally we investigate the interaction between a tilted ellipsoid and the wall in the uniform flow and the poiseuille flow

the closed form reproducing polynomial particle shape functions for meshfree particle methods abstract it has been known that reproducing kernel particle rkp shape functions with kronecker delta property are not available in simple forms thus in this paper we construct highly regular piecewise polynomial reproducing polynomial particle rpp shape functions that satisfy the kronecker delta property like rkp shape functions the rpp shape functions reproduce the complete polynomials of degree k for an integer k 0 moreover the particles associated with these rpp shape functions are either uniformly distributed in or non uniformly distributed in 0 or in a compact set a b

error analysis of the reproducing kernel particle method interest in meshfree or meshless methods has grown rapidly in recent years in solving boundary value problems bvps arising in mechanics especially in dealing with difficult problems involving large deformation moving discontinuities etc in this paper we provide a theoretical analysis of the reproducing kernel particle method rkpm which belongs to the family of meshfree methods one goal of the paper is to set up a framework for error estimates of rkpm we introduce the concept of a regular family of particle distributions and derive optimal order error estimates for rkp interpolants on a regular family of particle distributions the interpolation error estimates can be used to yield error estimates for rkp solutions of bvps

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

a geometric schur weyl duality for quotients of affine hecke algebras abstract after establishing a geometric schur weyl duality in a general setting we recall this duality in type a in the finite and affine case we extend the duality in the affine case to positive parts of the affine algebras the positive parts have nice ideals coming from geometry allowing duality for quotients some of the quotients of the positive affine hecke algebra are then identified to some cyclotomic hecke algebras and the geometric setting allows the construction of canonical bases

on quantizer design for distributed bayesian estimation in sensor networks we consider the problem of distributed estimation under the bayesian criterion and explore the design of optimal quantizers in such a system we show that for a conditionally unbiased and efficient estimator at the fusion center and when local observations have identical distributions it is optimal to partition the local sensors into groups with all sensors within a group using the same quantization rule when all the sensors use identical number of decision regions use of identical quantizers at the sensors is optimal when the network is constrained by the capacity of the wireless multiple access channel over which the sensors transmit their quantized observations we show that binary quantizers at the local sensors are optimal under certain conditions based on these observations we address the location parameter estimation problem and present our optimal quantizer design approach we also derive the performance limit for distributed location parameter estimation under the bayesian criterion and find the conditions when the widely used threshold quantizer achieves this limit we corroborate this result using simulations we then relax the assumption of conditionally independent observations and derive the optimality conditions of quantizers for conditionally dependent observations using counter examples we also show that the previous results do not hold in this setting of dependent observations and therefore identical quantizers are not optimal index terms

moving least square reproducing kernel methods i methodology and convergence this paper formulates the moving least square interpolation scheme in a framework of the so called moving least square reproducing kernel mlsrk representation in this study the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation which as a generalization of the early discrete approach establishes a continuous basis for a partition of unity this new formulation possesses the quality of simplicity and it is easy to implement moreover the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy in this contribution a generic m consistency relation has been found which is the essential property of the mlsrk approximation an interpolation error estimate is given to assess the convergence rate of the approximation it is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the sobolev norms as a meshless method the convergence rate is measured by a new control variable dilation parameter of the window function instead of the mesh size h as usually done in the finite element analysis to illustrate the procedure convergence has been shown for the numerical solution of the second order elliptic differential equations in a galerkin procedure invoked with this interpolant in the numerical example a two point boundary problem is solved by using the method and an optimal convergence rate is observed with respect to various norms

reproducing polynomial singularity particle methods and adaptive meshless methods for two dimensional elliptic boundary value problems abstract oh et al h s oh j g kim j w jeong the closed form reproducing polynomial particle shape functions for meshfree particle methods comput methods appl mech engrg 196 2007 3435 3461 introduced the reproducing polynomial particle rpp shape functions that are piecewise polynomial and satisfy the kronecker delta property in this paper we introduce reproducing polynomial particle methods rppm that is the galerkin approximation method associated with the use of the rpp approximation space planting particles in the computation domain in a patchwise uniform manner we also introduce patchwise rppm furthermore constructing partition of unity functions with flat top adaptively we introduce adaptive patchwise rppm and compare it with reproducing singularity particle methods rspm that is the galerkin approximation method associated with the use of reproducing singularity particle rsp shape functions