generalizing the hough transform to detect arbitrary shapes abstract the hough transform is a method for detecting curves by exploiting the duality between points on a curve and parameters of that curve the initial work showed how to detect both analytic curves 1 2 and non analytic curves 3 but these methods were restricted to binary edge images this work was generalized to the detection of some analytic curves in grey level images specifically lines 4 circles 5 and parabolas 6 the line detection case is the best known of these and has been ingeniously exploited in several applications 7 8 9 we show how the boundaries of an arbitrary non analytic shape can be used to construct a mapping between image space and hough transform space such a mapping can be exploited to detect instances of that particular shape in an image furthermore variations in the shape such as rotations scale changes or figure ground reversals correspond to straightforward transformations of this mapping however the most remarkable property is that such mappings can be composed to build mappings for complex shapes from the mappings of simpler component shapes this makes the generalized hough transform a kind of universal transform which can be used to find arbitrarily complex shapes

an extension of a bound for functions in sobolev spaces with applications to m s spline interpolation and smoothing given a function f on a bounded open subset of mathbb r n with a lipschitz continuous boundary we obtain a sobolev bound involving the values of f at finitely many points of overline omega this result improves previous ones due to narcowich et al math comp 74 743 763 2005 and wendland and rieger numer math 101 643 662 2005 we then apply the sobolev bound to derive error estimates for interpolating and smoothing m s splines in the case of smoothing noisy data as well as exact data are considered

l p bernstein estimates and approximation by spherical basis functions the purpose of this paper is to establish l p error estimates a bernstein inequality and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n sphere in particular the bernstein inequality estimates l p bessel potential sobolev norms of functions in this space in terms of the minimal separation and the l p norm of the function itself an important step in its proof involves measuring the l p stability of functions in the approximating space in terms of the l p norm of the coefficients involved as an application of the bernstein inequality we derive inverse theorems for sbf approximation in the l p norm finally we give a new characterization of besov spaces on the n sphere in terms of spaces of sbfs

multiscale rbf collocation for solving pdes on spheres in this paper we discuss multiscale radial basis function collocation methods for solving certain elliptic partial differential equations on the unit sphere the approximate solution is constructed in a multi level fashion each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh two variants of the collocation method are considered sometimes called symmetric and unsymmetric although here both are symmetric a convergence theory is given which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions

h 2 convergence of least squares kernel collocation methods the strong form asymmetric kernel based collocation method commonly referred to as the kansa method is easy to implement and hence is widely used for solving engineering problems and partial differential equations despite the lack of theoretical support the simple least squares ls formulation on the other hand makes the study of its solvability and convergence rather nontrivial in this paper we focus on general second order linear elliptic differential equations in omega subset mathbb r d under dirichlet boundary conditions with kernels that reproduce h m omega and some smoothness assumptions on the solution we provide conditions for a constrained ls method and a class of weighted ls algorithms to be convergent theoretically for max 2 lceil d 1 2 rceil leq nu leq m we identify some h nu omega convergent ls formulations that have an optimal error behavior like h m nu for d leq3 the proposed methods are optimal in h 2 omega we demonstrate the effect

convergence of unsymmetric kernel based meshless collocation methods this paper proves convergence of variations of the unsymmetric kernel based collocation method introduced by kansa in 1986 since then this method has been very successfully used in many applications though it may theoretically fail in special situations and though it had no error bound or convergence proof up to now thus it is necessary to add assumptions or to make modifications our modifications prevent numerical failure by dropping strict collocation and allow a rigorous mathematical analysis proving error bounds and convergence rates these rates improve with the smoothness of the solution the domain and the kernel providing the trial spaces but they are currently not yet optimal and deserve refinement they are based on rates of approximation to the residuals by nonstationary meshless kernel based trial spaces and they are independent of the type of differential operator the results are applicable to large classes of linear problems in strong form provided that there is a smooth solution and the test and trial discretizations are chosen with some care our analysis does not require assumptions like ellipticity and it can be extended to ill posed problems

on unsymmetric collocation by radial basis functions solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by kansa in 1990 it rewrites the problem as a generalized interpolation problem and the solution is obtained by solving a possibly large linear system the method has been used successfully in a variety of applications but a proof of nonsingularity of the linear system was still missing this paper shows that a general proof of this fact is impossible however numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort

a high order meshless galerkin method for semilinear parabolic equations on spheres we describe a novel meshless galerkin method for numerically solving semilinear parabolic equations on spheres the new approximation method is based upon a discretization in space using spherical basis functions in a galerkin approximation as our spatial approximation spaces are built with spherical basis functions they can be of arbitrary order and do not require the construction of an underlying mesh we will establish convergence of the meshless method by adapting to the sphere a convergence result due to thomee and wahlbin to do this requires proving new approximation results including a novel inverse or nikolskii inequality for spherical basis functions we also discuss how the integrals in the galerkin method can accurately and more efficiently be computed using a recently developed quadrature rule these new quadrature formulas also apply to galerkin approximations of elliptic partial differential equations on the sphere finally we provide several numerical examples

robust object detection with interleaved categorization and segmentation this paper presents a novel method for detecting and localizing objects of a visual category in cluttered real world scenes our approach considers object categorization and figure ground segmentation as two interleaved processes that closely collaborate towards a common goal as shown in our work the tight coupling between those two processes allows them to benefit from each other and improve the combined performance the core part of our approach is a highly flexible learned representation for object shape that can combine the information observed on different training examples in a probabilistic extension of the generalized hough transform the resulting approach can detect categorical objects in novel images and automatically infer a probabilistic segmentation from the recognition result this segmentation is then in turn used to again improve recognition by allowing the system to focus its efforts on object pixels and to discard misleading influences from the background moreover the information from where in the image a hypothesis draws its support is employed in an mdl based hypothesis verification stage to resolve ambiguities between overlapping hypotheses and factor out the effects of partial occlusion an extensive evaluation on several large data sets shows that the proposed system is applicable to a range of different object categories including both rigid and articulated objects in addition its flexible representation allows it to achieve competitive object detection performance already from training sets that are between one and two orders of magnitude smaller than those used in comparable systems

approximate interpolation with applications to selecting smoothing parameters in this paper we study the global behavior of a function that is known to be small at a given discrete data set such a function might be interpreted as the error function between an unknown function and a given approximant we will show that a small error on the discrete data set leads under mild assumptions automatically to a small error on a larger region we will apply these results to spline smoothing and show that a specific a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant this has also a surprising application in stabilizing the interpolation process by splines and positive definite kernels

preconditioning landweber iteration in hilbert scales in this paper we investigate convergence of landweber iteration in hilbert scales for linear and nonlinear inverse problems as opposed to the usual application of hilbert scales in the framework of regularization methods we focus here on the case s 0 which for tikhonov regularization corresponds to regularization in a weaker norm in this case the hilbert scale operator l 2 s appearing in the iteration acts as a preconditioner which significantly reduces the number of iterations needed to match an appropriate stopping criterion additionally we carry out our analysis under significantly relaxed conditions i e we only require inlinemediaobject not available see fulltext instead of inlinemediaobject not available see fulltext which is the usual condition for regularization in hilbert scales the assumptions needed for our analysis are verified for several examples and numerical results are presented illustrating the theoretical ones

unsymmetric meshless methods for operator equations a general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well posed linear operator equations is presented the results provide optimal convergence rates if the test and trial spaces satisfy a stability condition operators need not be elliptic and the problems can be posed in weak or strong form without changing the theory non stationary kernel based trial and test spaces are shown to fit into the framework disregarding the operator equation as a special case unsymmetric meshless kernel based methods solving weakly posed problems with distributional data are treated in some detail this provides a foundation of certain variations of the meshless local petrov galerkin technique of s n atluri and collaborators

error analysis of method of lines mol via generalized interpolating moving least squares gimls approximation this paper gives an error analysis for the method of lines mol using generalized interpolating moving least squares gimls approximation in this study error bound for the time dependent linear and nonlinear second order differential equations in d dimension will be obtained when the gimls method is used for approximating the spatial variables also the well known courantfriedrichslewy cfl condition will be derived in both cases linear and nonlinear equations finally numerical examples will be reported to confirm the ability of the proposed technique

an autopolyploidy based genetic algorithm for enhanced evolution of linear polyfractal arrays there has been considerable recent interest in techniques for the optimization of large n antenna arrays unfortunately the successful development of such techniques has been hindered by the large number of independent parameters that must be optimized and the complexity of the calculations needed for the electromagnetic evaluation of large n arrays one promising new design methodology for large n arrays which has recently been introduced is based on properties of a subset of fractal random arrays called polyfractal arrays polyfractal arrays have many embedded self similar structures thereby allowing very large and seemingly complex array layouts to be described with only a small set of independent parameters in addition by effectively utilizing the self similarity of polyfractal arrays a considerable reduction can be achieved in the amount of time required to evaluate the radiation patterns of large n arrays this paper introduces a type of nature based design process that applies a specially formulated genetic algorithm ga technique to evolve optimal polyfractal array layouts the most unique aspect of this optimization technique is a new autopolyploidy based chromosome expansion that maximizes the efficiency of the gas simple polyfractal geometries are used in the initial stage or first epoch of the optimization because the number of independent parameters is small and the computation times are relatively fast after the optimization converges for the first epoch more complicated descriptions of these polyfractal arrays are introduced to provide additional independent parameters for the optimizer as it progresses through later epochs of evolution this process has been shown to be very effective in creating optimized large n arrays the largest example considered here being a 1616 element linear array with a 24 30 db sidelobe level and a 0 056deg half power beamwidth

stable and convergent unsymmetric meshless collocation methods in the theoretical part of this paper we introduce a simplified proof technique for error bounds and convergence of a variation of kansa s well known unsymmetric meshless collocation method for a numerical implementation of the convergent variation a previously proposed greedy technique is coupled with linear optimization this algorithm allows a fully adaptive on the fly data dependent meshless selection of test and trial spaces the new method satisfies the assumptions of the background theory and numerical experiments demonstrate its stability

discrete least squares radial basis functions approximations abstract we consider discrete least squares methods using radial basis functions a general l2 tikhonov regularization with w 2 m penalty is considered we provide error estimates that are comparable to kernel based interpolation in cases which the function it is approximating is within and is outside of the native space of the kernel our proven theories concern the denseness condition of collocation points and selection of regularization parameters in particular the unregularized least squares method is shown to have w 2 convergence for d 2 on smooth domain r d for any properly regularized least squares method the same convergence estimates hold for a large range of 0 these results are extended to the case of noisy data numerical demonstrations are provided to verify the theoretical results in terms of applications we also apply the proposed method to solve a heat equation whose initial condition has huge oscillation in the domain

error and stability estimates for surface divergence free rbf interpolants on the sphere recently a new class of surface divergence free radial basis function interpolants has been developed for surfaces in 3 in this paper several approximation results for this class of interpolants will be derived in the case of the sphere s 2 in particular sobolev type error estimates are obtained as well as optimal stability estimates for the associated interpolation matrices in addition a bernstein estimate and an inverse theorem are also derived numerical validation of the theoretical results is also given

a kernel based embedding method and convergence analysis for surfaces pdes we analyze a least squares strong form kernel collocation formulation for solving second order elliptic pdes on smooth connected and compact surfaces with bounded geometry the methods do not require any partial derivatives of surface normal vectors or metric based on some standard smoothness assumptions for high order convergence we provide the sufficient denseness conditions on the collocation points to ensure the methods are convergent in addition to some convergence verifications we also simulate some reaction diffusion equations to exhibit the pattern formations

the computation of potentials near and on the boundary by an extraction technique for boundary element methods here we present the two dimensional version of a bootstrapping algorithm for the computation of arbitrary derivatives of the cauchy data and for the extraction of potentials near and on the boundary for the laplacian if corresponding boundary integral equations are used the method employs the derivatives of the green representation formula in terms of cauchy singular or weakly singular integrals and also their compositions with derivatives it allows the robust and superconvergent numerical evaluation of potentials and their derivatives up to and on the boundary curve furthermore we determine a criterion whether to use the traditional green representation formula or our extraction technique for the point evaluation of the potentials in the domain

calculus on surfaces with general closest point functions the closest point method for solving partial differential equations pdes posed on surfaces was recently introduced by ruuth and merriman j comput phys 227 2008 pp 1943 1961 and successfully applied to a variety of surface pdes in this paper we study the theoretical foundations of this method the main idea is that surface differentials of a surface function can be replaced with cartesian differentials of its closest point extension i e its composition with a closest point function we introduce a general class of these closest point functions a subset of differentiable retractions show that these are exactly the functions necessary to satisfy the above idea and give a geometric characterization of this class finally we construct some closest point functions and demonstrate their effectiveness numerically on surface pdes

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

modular broadband phased arrays based on a nonuniform distribution of elements along the peano gosper space filling curve the peano gosper space filling curve provides an excellent framework for designing broadband planar antenna arrays with highly modular architectures uniformly distributing elements along the curve leads to an element distribution with a triangular lattice that has an irregular fractal boundary contour this boundary contour allows for a modular subarray configuration and better sidelobe suppression than conventional triangular lattice arrays with a regular boundary contour while they have a greater bandwidth than square lattice distributions arrays based on a triangular lattice still possess a rather limited bandwidth for beam steering applications due to the formation of grating lobes in this communication it will be shown that the beam steering capabilities of the peano gosper array can be enhanced by introducing perturbations into the basic recursive array generation scheme with the proper implementation the perturbed arrays retain the attractive features of modularity and recursive beamforming that are associated with the standard peano gosper array examples will be presented for several stages of peano gosper arrays that were designed for 2 1 broadband performance while scanning within a 30 conical volume full wave simulations will be used to examine the effects of mutual coupling on these aperiodic array layouts

beyond the sum of parts voting with groups of dependent entities the high complexity of multi scale category level object detection in cluttered scenes is efficiently handled by hough voting methods however the main shortcoming of the approach is that mutually dependent local observations are independently casting their votes for intrinsically global object properties such as object scale object hypotheses are then assumed to be a mere sum of their part votes popular representation schemes are however based on a dense sampling of semi local image features which are consequently mutually dependent we take advantage of part dependencies and incorporate them into probabilistic hough voting by deriving an objective function that connects three intimately related problems i grouping mutually dependent parts ii solving the correspondence problem conjointly for dependent parts and iii finding concerted object hypotheses using extended groups rather than based on local observations alone early commitments are avoided by not restricting parts to only a single vote for a locally best correspondence and we learn a weighting of parts during training to reflect their differing relevance for an object experiments successfully demonstrate the benefit of incorporating part dependencies through grouping into hough voting the joint optimization of groupings correspondences and votes not only improves the detection accuracy over standard hough voting and a sliding window baseline but it also reduces the computational complexity by significantly decreasing the number of candidate hypotheses

identification of nonlinear spatiotemporal dynamical systems with nonuniform observations using reproducing kernel based integral least square regulation the identification of nonlinear spatiotemporal dynamical systems given by partial differential equations has attracted a lot of attention in the past decades several methods such as searching principle based algorithms partially linear kernel methods and coupled lattice methods have been developed to address the identification problems however most existing methods have some restrictions on sampling processes in that the sampling intervals should usually be very small and uniformly distributed in spatiotemporal domains these are actually not applicable for some practical applications in this paper to tackle this issue a novel kernel based learning algorithm named integral least square regularization regression ilsrr is proposed which can be used to effectively achieve accurate derivative estimation for nonlinear functions in the time domain with this technique a discretization method named inverse meshless collocation is then developed to realize the dimensional reduction of the system to be identified thereafter with this novel inverse meshless collocation model the ilsrr and a multiple kernel based learning algorithm a multistep identification method is systematically proposed to address the identification problem of spatiotemporal systems with pointwise nonuniform observations numerical studies for benchmark systems with necessary discussions are presented to illustrate the effectiveness and the advantages of the proposed method

a finite volume method for solving parabolic equations on logically cartesian curved surface meshes we present a second order finite volume scheme for the constant coefficient diffusion equation on curved parametric surfaces while our scheme is applicable to general quadrilateral surface meshes based on smooth or piecewise smooth coordinate transformations our primary motivation for developing the present scheme is to solve diffusion problems on a particular set of circular and spherical meshes introduced in d a calhoun c helzel and r j leveque siam rev 50 2008 pp 723 752 for the discretization of hyperbolic problems these grids are generated from mappings of a single cartesian grid and were designed to have nearly uniform cells sizes and avoid the pole singularity associated with polar or spherical grid mappings the present method for parabolic equations offers several advantages it does not require analytic metric terms shows second order accuracy on our disk and sphere grids can be easily coupled to existing finite volume solvers for logically cartesian meshes and handles general mixed boundary conditions our parabolic scheme should appeal to researchers in the fields of geophysical fluid dynamics computational biology and any other discipline that requires the solution of parabolic equations on quadrilateral surface meshes in this article we present several numerical examples demonstrating the accuracy of the scheme and then use the scheme to solve advection reaction diffusion equations modeling biological pattern formation on surfaces

numerical analysis of the allen cahn equation and approximation for mean curvature flows we propose and analyze a semi discrete in time scheme and a fully discrete scheme for the allen cahn equation u t u 2 f u 0 arising from phase transition in materials science where is a small parameter known as an interaction length the primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods in particular by focusing on the dependence of the error bounds on optimal order and quasi optimal order error bounds are shown for the semi discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u 0 in particular all our error bounds depend on only in some lower polynomial order for small the cruxes of the analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of de mottoni and schatzman 18 19 and chen 12 and to establish a discrete counterpart of it for a linearized allen cahn operator to handle the nonlinear term finally as a nontrivial byproduct the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow

the meshless kernel based method of lines for parabolic equations using the heat equation as a simple example we give a rigorous theoretical analysis of the method of lines implemented as a meshless method based on spatial trial spaces spanned by translates of positive definite kernels the technique can be generalized to other parabolic problems and some numerical illustrations are given

design of broadband planar arrays based on the optimization of aperiodic tilings antenna arrays based on aperiodic tilings have been shown to exhibit low sidelobe levels and modest bandwidths over which grating lobes are suppressed in addition compared to conventional periodic arrays these arrays are naturally thinned i e mean inter element spacing is greater than the generation of these arrays involves placing array elements at the locations of the vertices of an aperiodic tiling to obtain a realizable design the entire array is then scaled and truncated to achieve a desired minimum element spacing and aperture size this paper demonstrates that it is possible to greatly extend the bandwidth of these arrays by incorporating a simple perturbation scheme into the basic array generation process the implementation of this perturbation scheme is straightforward and it lends itself well to being combined with an optimization technique such as the genetic algorithm it is successfully used to generate arrays that have large bandwidths peak sidelobe level db with no grating lobes of up to a minimum element spacing of moreover the flexibility of this technique will be further demonstrated by introducing a slight variation of the basic scheme that is capable of generating arrays with extremely wide bandwidths an example will be presented for an array design that has a bandwidth corresponding to a minimum element spacing of up to

on the extraction technique in boundary integral equations in this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the cauchy data of regular three dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations the method rests on the derivatives of the generalized green s representation formula which are expressed in terms of singular boundary integrals as hadamard s finite parts their regularization together with asymptotic pseudohomogeneous kernel expansions yields a constructive method for obtaining generalized jump relations these expansions are obtained via composition of taylor expansions of the local surface representation the density functions differential operators and the fundamental solution of the original problem together with the use of local polar coordinates in the parameter domain for boundary integral equations obtained by the direct method this method allows the recursive numerical extraction of potentials and their derivatives near and up to the boundary surface

meshless collocation error estimates with application to dynamical systems in this paper we derive error estimates for generalized interpolation in particular collocation in sobolev spaces we employ our estimates in collocation problems using radial basis functions and extend and improve previously known results for elliptic problems finally we use meshless collocation to approximate lyapunov functions for dynamical systems

a high order approximation method for semilinear parabolic equations on spheres we describe a novel discretisation method for numerically solving systems of semilinear parabolic equations on euclidean spheres the new approximation method is based upon a discretisation in space using spherical basis functions and can be of arbitrary order this together with the fact that the solutions of semilinear parabolic problems are known to be infinitely smooth at least locally in time allows us to prove stability and convergence of the discretisation in a straight forward way

multiscale analysis in sobolev spaces on the sphere we consider a multiscale approximation scheme at scattered sites for functions in sobolev spaces on the unit sphere mathbb s n the approximation is constructed using a sequence of scaled compactly supported radial basis functions restricted to mathbb s n a convergence theorem for the scheme is proved and the condition number of the linear system is shown to stay bounded by a constant from level to level thereby establishing for the first time a mathematical theory for multiscale approximation with scaled versions of a single compactly supported radial basis function at scattered data points

space fractional advection dispersion equations by the kansa method the paper makes the first attempt at applying the kansa method a radial basis function meshless collocation method to the space fractional advection dispersion equations which have recently been observed to accurately describe solute transport in a variety of field and lab experiments characterized by occasional large jumps with fewer parameters than the classical models of integer order derivative however because of non local property of integro differential operator of space fractional derivative numerical solution of these novel models is very challenging and little has been reported in literature it is stressed that local approximation techniques such as the finite element and finite difference methods lose their sparse discretization matrix due to this non local property thus the global methods appear to have certain advantages in numerical simulation of these non local models because of their high accuracy and smaller size resultant matrix equation compared with the finite difference method popular in the solution of fractional equations the kansa method is a recent meshless global technique and is promising for high dimensional irregular domain problems in this study the resultant matrix of the kansa method is accurately calculated by the gauss jacobi quadrature rule numerical results show that the kansa method is highly accurate and computationally efficient for space fractional advection dispersion problems

generalized space filling gosper curves and their application to the design of wideband modular planar antenna arrays the peano gosper space filling curve provides an excellent framework for designing planar antenna array distributions with modular architectures and suppressed sidelobes over a relatively wide bandwidth the curve consists of a self avoiding path that intersects a triangular lattice and its construction is based on the iterative application of a generating curve there exist a number of other recently discovered curves coined generalized gosper curves that possess properties similar to those of the peano gosper curve but are based on larger and more complex generating curves in this paper these generalized curves will be examined as the framework for antenna array layouts it will be shown that arrays based on these curves have a number of excellent characteristics while offering modular array configurations and sizes that are not inherent to peano gosper arrays moreover when combined with a simple recursive perturbation technique the element distributions of these arrays can be efficiently adjusted to generate designs with bandwidths that far exceed that of standard periodic and triangular lattice arrays the efficacy of this technique will be demonstrated through design examples including one that has more than a 10 1 bandwidth full wave simulations of a wideband patch array will also be used to investigate the effects of mutual coupling on these array distributions

the orthogonal gradients method a radial basis functions method for solving partial differential equations on arbitrary surfaces much work has been done on reconstructing arbitrary surfaces using the radial basis function rbf method but one can hardly find any work done on the use of rbfs to solve partial differential equations pdes on arbitrary surfaces in this paper we investigate methods to solve pdes on arbitrary stationary surfaces embedded in r 3 using the rbf method we present three rbf based methods that easily discretize surface differential operators we take advantage of the meshfree character of rbfs which give us a high accuracy and the flexibility to represent the most complex geometries in any dimension two out of the three methods which we call the orthogonal gradients ogr methods are the result of our work and are hereby presented for the first time

extrinsic meshless collocation methods for pdes on manifolds we proposed ways to implement meshless collocation methods extrinsically for solving elliptic pdes on smooth closed connected and complete riemannian manifolds with arbitrary codimensions our m

the peano gosper fractal array this paper investigates the radiation characteristics of a new type of array that is based on the family of space filling and self avoiding fractals known as peano gosper curves the elements of the fractal array are uniformly distributed along a peano gosper curve which leads to a planar array configuration with parallelogram cells that is bounded by a closed koch curve these unique properties are exploited in order to develop a design methodology for deterministic arrays that have no grating lobes even when the minimum spacing between elements is increased to at least one wavelength this leads to a class of arrays that are relatively broad band when compared to more conventional periodic planar arrays with square or rectangular cells and regular boundary contours an efficient iterative procedure for calculating the radiation patterns of these peano gosper fractal arrays to arbitrary stage of growth p is also introduced in this paper

all well posed problems have uniformly stable and convergent discretizations this paper considers a large class of linear operator equations including linear boundary value problems for partial differential equations and treats them as linear recovery problems for functions from their data well posedness of the problem means that this recovery is continuous discretization recovers restricted trial functions from restricted test data and it is well posed or stable if this restricted recovery is continuous after defining a general framework for these notions this paper proves that all well posed linear problems have stable and refinable computational discretizations with a stability that is determined by the well posedness of the problem and independent of the computational discretization provided that sufficiently many test data are used the solutions of discretized problems converge when enlarging the trial spaces and the convergence rate is determined by how well the data of the function solving the analytic problem can be approximated by the data of the trial functions this allows new and very simple proofs of convergence rates for generalized finite elements symmetric and unsymmetric kansa type collocation and other meshfree methods like meshless local petrov galerkin techniques it is also shown that for a fixed trial space weak formulations have a slightly better convergence rate than strong formulations but at the expense of numerical integration since convergence rates are reduced to those coming from approximation theory and since trial spaces are arbitrary this also covers various spectral and pseudospectral methods all of this is illustrated by examples

eigenvalue stability of radial basis function discretizations for time dependent problems differentiation matrices obtained with infinitely smooth radial basis function rbf collocation methods have under many conditions eigenvalues with positive real part preventing the use of such methods for time dependent problems we explore this difficulty at theoretical and practical levels theoretically we prove that differentiation matrices for conditionally positive definite rbfs are stable for periodic domains we also show that for gaussian rbfs special node distributions can achieve stability in 1 d and tensor product nonperiodic domains as a more practical approach for bounded domains we consider differentiation matrices based on least squares rbf approximations and show that such schemes can lead to stable methods on less regular nodes by separating centers and nodes least squares techniques open the possibility of the separation of accuracy and stability characteristics

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

a nash hormander iteration and boundary elements for the molodensky problem we investigate the numerical approximation of the nonlinear molodensky problem which reconstructs the surface of the earth from the gravitational potential and the gravity vector the method based on a smoothed nash hormander iteration solves a sequence of exterior oblique robin problems and uses a regularization based on a higher order heat equation to overcome the loss of derivatives in the surface update in particular we obtain a quantitative a priori estimate for the error after m m steps justify the use of smoothing operators based on the heat equation and comment on the accurate evaluation of the hessian of the gravitational potential on the surface using a representation in terms of a hypersingular integral a boundary element method is used to solve the exterior problem numerical results compare the error between the approximation and the exact solution in a model problem

fractile arrays a new class of tiled arrays with fractal boundaries in this paper a new class of antenna arrays are introduced which we call fractile arrays a fractile array is defined as any array with a fractal boundary contour that tiles the plane without gaps or overlaps it will be shown that the unique geometrical features of fractiles may be exploited in order to make available a family of deterministic arrays that offer several highly desirable performance advantages over their conventional periodic planar array counterparts most notably fractile arrays have no grating lobes even when the minimum spacing between elements is increased to at least one wavelength this has led to the development of a new design methodology for modular broadband low sidelobe arrays that is based on fractal tilings several examples of fractile arrays will be considered including peano gosper terdragon 6 terdragon and fudgeflake arrays efficient iterative procedures for calculating the radiation patterns of these fractile arrays to arbitrary stage of growth p are also introduced in this paper