a convergence rate theorem for finite difference approximations to delta functions we prove a rate of convergence theorem for approximations to certain integrals over codimension one manifolds in r n the type of manifold involved here is defined by the zero level set of a smooth mapping u r email protected r our approximations are based on the two finite difference methods for discretizing delta functions presented in 16 we included a convergence proof in that paper but only proved rates of convergence in some greatly simplified situations numerical experiments indicated that our two methods were at least first and second order accurate respectively in this note we prove those empirical convergence rates for the two algorithms under fairly general hypotheses

performance advantage of complex lms for controlling narrow band adaptive arrays in narrow band adaptive array applications the mean square convergence of the discrete time real least mean square lms algorithm is slowed by image frequency noises generated in the lms loops the complex lms algorithm proposed by widrow et al is shown to eliminate these noises yielding convergence of the mean squared error mse at slightly over twice the rate this paper includes a comprehensive analysis of the mse of adaptation for lms the analysis is based upon the method developed in the 1968 dissertation by k d senne and it represents the most complete treatment of the subject published to date

on the optimum gain parameter in lms adaptation in this correspondence the optimum selection of the adaptation parameter in lms adaptive estimation is discussed for a white data covarianee matrix it is shown via analysis and numerical evaluation that the optimum selection of is a complicated function of the number of filter taps the initial weight setting the wiener weights and the number of learning samples is not chosen in general to yield the most rapid transient response for most cases of interest a smaller value of will be selected for slower adaptation and smaller misadjustment error at the end of the learning phase

two methods for discretizing a delta function supported on a level set this paper presents two new methods for discretizing a dirac delta function which is concentrated on the zero level set of a smooth function u r n r the function u is only known at the discrete set of points belonging to a regular mesh covering r n these two methods are used to approximate integrals over the manifold defined by the level set both methods are conceptually simple and easy to implement we present the results of numerical experiments indicating that as the mesh size h goes to zero the rate of convergence is at least o h for the first method and o h 2 for the second method we perform a limited analysis of the proposed algorithms including a proof of convergence for both methods

high order numerical methods to a type of delta function integrals we study second to fourth order numerical methods to a type of delta function integrals in one to three dimensions these delta function integrals arise from recent efficient level set methods for computing the multivalued solutions of nonlinear pdes we show that the natural quadrature approach with usual discrete delta functions and support size formulas to the two dimensional delta function integrals suffer from nonconvergence we then design high order numerical methods to this type of delta function integrals based on interpolation approach numerical examples are presented to verify the efficiency and accuracy of our methods

finite difference methods for approximating heaviside functions we present a finite difference method for discretizing a heaviside function h u x where u is a level set function u r email protected r that is positive on a bounded region email protected r n there are two variants of our algorithm both of which are adapted from finite difference methods that we proposed for discretizing delta functions in j d towers two methods for discretizing a delta function supported on a level set j comput phys 220 2007 915 931 j d towers discretizing delta functions via finite differences and gradient normalization preprint at http www miracosta edu home jtowers j d towers a convergence rate theorem for finite difference approximations to delta functions j comput phys 227 2008 6591 6597 we consider our approximate heaviside functions as they are used to approximate integrals over w we prove that our first approximate heaviside function leads to second order accurate quadrature algorithms numerical experiments verify this second order accuracy for our second algorithm numerical experiments indicate at least third order accuracy if the integrand f and email protected are sufficiently smooth numerical experiments also indicate that our approximations are effective when used to discretize certain singular source terms in partial differential equations we mostly focus on smooth f and u by this we mean that f is smooth in a neighborhood of w u is smooth in a neighborhood of email protected and the level set u x 0 is a manifold of codimension one however our algorithms still give reasonable results if either f or u has jumps in its derivatives numerical experiments indicate approximately second order accuracy for both algorithms if the regularity of the data is reduced in this way assuming that the level set u x 0 is a manifold numerical experiments indicate that dependence on the placement of w with respect to the grid is quite small for our algorithms specifically a grid shift results in an o h p change in the computed solution where p is the observed rate of convergence

the complex lms adaptive algorithm transient weight mean and covariance with applications to the ale the transient and steady state mean and covariance of the complex valued lms adaptive element weights are investigated when the inputs are samples from circularly normal processes it is shown that the data covariance diagonalizing matrix also diagonalizes the weight covariance matrix this result permits describing the complete transient behavior of the adaptive line enhancer ale weight covariance matrix in closed form for the case of multiple equal power narrow band statistically independent orthogonal rayleigh fading sinusoids in broad band noise based on these transient covariance results it is shown that for any stage of adaptation there exists an optimum ale gain constant that minimizes the excess mean squared prediction error this result is of particular significance when processing time limited data

a new extrapolation algorithm for band limited signals using the regularization method the ill posedness of the extrapolation problem in the presence of noise is considered a stable algorithm is constructed by solving a fredholm equation based on a regularization method the algorithm appears relatively robust since the noise eta sub delta t is taken as a function in l sup 2 t t t 0 such that the error energy integral mod eta sub delta t mod sup 2 dt

short note a proof that a discrete delta function is second order accurate it is proved that a discrete delta function introduced by smereka p smereka the numerical approximation of a delta function with application to level set methods j comput phys 211 2006 77 90 gives a second order accurate quadrature rule for surface integrals using values on a regular background grid the delta function is found using a technique of mayo a mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam j numer anal 21 1984 285 299 it can be expressed naturally using a level set function

discretization of dirac delta functions in level set methods discretization of singular functions is an important component in many problems to which level set methods have been applied we present two methods for constructing consistent approximations to dirac delta measures concentrated on piecewise smooth curves or surfaces both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size the first algorithm is based on a tensor product of regularized one dimensional delta functions it is independent of the irregularity relative to the grid in the second method the regularization is constructed from a one dimensional regularization that is extended to multi dimensions with a variable support depending on the orientation of the singularity relative to the computational grid convergence analysis and numerical results are given

delta function approximations in level set methods by distance function extension in a k tornberg b engquist numerical approximations of singular source terms in differential equations j comput phys 200 2004 462 488 it was shown for simple examples that the then m

microstatistic lms filtering adaptive microstatistic filters are developed for applications in which the second order statistics of the thresholded signals are not known or may be nonstationary a multilevel threshold decomposition such that real valued stochastic processes can be filtered is used and the computational complexity of the algorithm can be arbitrarily specified by the designer the adaptation uses the least mean squares error approach of the least mean square lms algorithm the convergence of the adaptive algorithm is proved due to the nonhomogeneous statistical characteristic of the threshold signals a different step size adaptation parameter can be assigned to each threshold level simple design guidelines are developed for finding the set of nonhomogeneous step sizes which in practice yield better convergence characteristics

robust second order accurate discretizations of the multi dimensional heaviside and dirac delta functions we present a robust second order accurate method for discretizing the multi dimensional heaviside and the dirac delta functions on irregular domains the method is robust in the following ways 1 integrations of source terms on a co dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain s boundary 2 the method depends only on the function value of a level function not on its derivatives we present the discretizations in tabulated form to make their implementations straightforward we present numerical results in two and three spatial dimensions to demonstrate the second order accuracy in the l 1 norm in the case of the solution of pdes with singular source terms in the case of evaluating the contribution of singular source terms on interfaces the method is also second order accurate in the l norm

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

high order numerical methods to two dimensional delta function integrals in level set methods in this paper we design and analyze a class of high order numerical methods to delta function integrals appearing in level set methods in two dimensional case the methods comprise approximating the mesh cell restrictions of the delta function integral in each mesh cell the two dimensional delta function integral can be rewritten as a one dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral and thus is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals proposed in x wen high order numerical methods to a type of delta function integrals j comput phys 226 2007 1952 1967 we establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub algorithms and has better accuracy under an assumption on the zero level set of the level set function which holds generally numerical examples are presented showing that the second to fourth order methods implemented in this paper achieve or exceed the expected accuracy and demonstrating the advantage of using our high order numerical methods

high order numerical quadratures to one dimensional delta function integrals we study high order numerical quadratures to one dimensional delta function integrals in this paper this is motivated by the fact that traditional numerical quadratures give only first order accuracy in general we provide criteria on discrete delta functions and support size formulas which ensure any desired accuracy of the numerical quadratures discrete delta functions and support size formulas satisfying these criteria are designed numerical examples are presented to verify the performed analysis and the high order accuracy of the proposed numerical quadratures

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

discretizing delta functions via finite differences and gradient normalization in j d towers two methods for discretizing a delta function supported on a level set j comput phys 220 2007 915 931 the author presented two closely related finite difference methods referred to here as fdm1 and fdm2 for discretizing a delta function supported on a manifold of codimension one defined by the zero level set of a smooth mapping u r n r these methods were shown to be consistent meaning that they converge to the true solution as the mesh size h 0 in the codimension one setting in this paper we concentrate on n 3 but generalize our methods to codimensions other than one now the level set function is generally a vector valued mapping u r n r m 1 m n 3 seemingly reasonable algorithms based on simple products of approximate delta functions are not generally consistent when applied to these problems motivated by this we instead use the wedge product formalism to generalize our fdm algorithms and this approach results in accurate often consistent approximations with the goal of ensuring consistency in general we propose a new gradient normalization process that is applied before our fdm algorithms these combined algorithms seem to be consistent in all reasonable situations with numerical experiments indicating o h 2 convergence for our new gradient normalized fdm2 algorithm in the full codimension setting m n our gradient normalization processing also improves accuracy when using more standard approximate delta functions this combination also yields approximations that appear to be consistent

geometric integration over irregular domains with application to level set methods we present a geometric approach for calculating integrals over irregular domains described by a level set function this procedure can be used to evaluate integrals over a lower dimensional interface and may be used to evaluate the contribution of singular source terms this approach produces results that are second order accurate and robust to the perturbation of the interface location on the grid moreover since we use a cell wise approach this procedure can be easily extended to quadtree and octree grids we demonstrate the second order accuracy and the robustness of the method in two and three spatial dimensions

level set jet schemes for stiff advection equations the semijet method abstract many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature here a semi implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems the new method offers an improvement over the semi implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided

convergence analysis of lms filters with uncorrelated gaussian data statistical analysis of the least mean squares lms adaptive algorithm with uncorrelated gaussian data is presented exact analytical expressions for the steady state mean square error mse and the performance degradation due to weight vector misadjustment are derived necessary and sufficient conditions for the convergence of the algorithm to the optimal wiener solution within a finite variance are derived it is found that the adaptive coefficient which controls the rate of convergence of the algorithm must be restricted to an interval significantly smaller than the domain commonly stated in the literature the outcome of this paper therefore places fundamental limitations on the mse performance and rate of convergence of the lms adaptive scheme

the numerical approximation of a delta function with application to level set methods it is shown that a discrete delta function can be constructed using a technique developed by anita mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam j sci comput 21 1984 285 299 for the numerical solution of elliptic equations with discontinuous source terms this delta function is concentrated on the zero level set of a continuous function in two space dimensions this corresponds to a line and a surface in three space dimensions delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function the numerical implementation of these delta functions achieves the expected order of accuracy

approximating solutions of variational inequalities for asymptotically nonexpansive mappings by using viscosity approximation methods for asymptotically nonexpansive mappings in banach spaces some sufficient and necessary conditions for a new type of iterative sequences to converging to a fixed point which is also the unique solution of some variational inequalities are obtained the results presented in the paper extend and improve some recent results in c e chidume jinlu li a udomene convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings proc amer math soc 138 2 2005 473 480 n shahzad a udomene fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in banach spaces nonlinear anal 64 2006 558 567 t c lim h k xu fixed point theories for asymptotically nonexpansive mappings nonlinear anal tma 22 1994 1345 1355 h k xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 2004 279 291

electrohydrodynamics of three dimensional vesicles a numerical approach a three dimensional numerical model of vesicle electrohydrodynamics in the presence of dc electric fields is presented the vesicle membrane is modeled as a thin capacitive interface through the use of a semi implicit gradient augmented level set jet scheme the enclosed volume and surface area are conserved both locally and globally by a new navier stokes projection method the electric field calculations explicitly take into account the capacitive interface by an implicit immersed interface method formulation which calculates the electric potential field and the transmembrane potential simultaneously the results match well with previously published experimental analytic and two dimensional computational works

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented