symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients we present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients based on the maple software package we approximate the leading terms in the truncation error of the taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation a test problem is solved numerically to validate the derived fourth order compact difference scheme this symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations

boundary penalty finite element methods for blending surfaces ii biharmonic equations abstract in this paper the biharmonic equations are discussed and the boundary penalty finite methods bp fems using piecewise cubic hermite elements are chosen to seek their approximate solutions satisfying the normal derivative and periodical boundary conditions theoretical analysis is made to discover that when the penalty power 2 3 or 4 and 0

an efficient sixth order solution for anisotropic poisson equation with completed richardson extrapolation and multiscale multigrid method we present an efficient numerical method for anisotropic poisson equations the sixth order accuracy is achieved through applying completed richardson extrapolation on two fourth order solutions computed from different scale grids with unequal mesh size discretization theoretical analysis is conducted to demonstrate that the richardson extrapolation is able to obtain a sixth order solution by removing the leading truncation error terms of the fourth order solution from grid with unequal mesh sizes the gain in efficiency is obtained through adopting partial semi coarsening multigrid method to solve the resulting linear systems and multiscale multigrid computation to speed up the whole solution numerical experiments are conducted to verify the accuracy and efficiency of the proposed method and the results are compared with the existing fourth order methods for solving 2d and 3d anisotropic poisson equations

high order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems we derive a new fourth order compact finite difference scheme which allows different meshsize in different coordinate directions for the two dimensional convection diffusion equation a multilevel local mesh refinement strategy is used to deal with the local singularity problem a corresponding multilevel multigrid method is designed to solve the resulting sparse linear system numerical experiments are conducted to show that the local mesh refinement strategy works well with the high order compact discretization scheme to recover high order accuracy for the computed solution our solution method is also shown to be effective and robust with respect to the level of mesh refinement and the anisotropy of the problems

spiral arc spline approximation to a planar spiral a biarc is a one parameter family of g 1 curves that can satisfy g 1 hermite data at two points an arc spline approximation to a smooth planar curve can be found by reading g 1 hermite data from the curve and tting a biarc between each pair of data points the resulting collection of biarcs forms a g 1 arc spline that interpolates the entire set of g 1 hermite data if the smooth curve is a spiral it is desirable that the arc spline approximation also be a spiral several methods are described for choosing the free parameters of the biarcs so that the arc spline approximation to a smooth spiral is a spiral c 1999 elsevier science b v all rights reserved

fast and high accuracy multigrid solution of the three dimensional poisson equation we employ a fourth order compact finite difference scheme fos with the multigrid algorithm to solve the three dimensional poisson equation we test the influence of different orderings of the grid space and different grid transfer operators on the convergence and efficiency of our high accuracy algorithm fourier smoothing analysis is conducted to show that fos has a smaller smoothing factor than the traditional second order central difference scheme cds a new method of fourier smoothing analysis is proposed for the partially decoupled red black gauss seidel relaxation with fos numerical results are given to compare the computed accuracy and the computational efficiency of fos with multigrid against cds with multigrid

solution of high order compact discretized 3d elliptic partial differential equations by an accelerated multigrid method abstract a robust and efficient numerical solver for three dimensional linear elliptic partial differential equations without cross derivative terms but variable coefficients is presented the proposed algorithm is based on a general meshsize fourth order compact finite difference scheme and an accelerated geometric multigrid solver the former not only increases the accuracy of the discretization but also does not affect the bandwidth of the common second order finite difference scheme which is relevant for distributed memory parallel implementations of the multigrid preconditioner the acceleration technique on the other hand allows for a dramatic reduction of the number of cycles and smoothing steps required to achieve convergence furthermore it is found that depending on the coefficients of the elliptic partial differential equation the multigrid preconditioned solver can be the only way to obtain a reliable numerical solution or attain convergence

global superconvergence of finite element methods for biharmonic equations and blending surfaces abstract in this paper new numerical algorithms of finite element methods fem are reported for both biharmonic equations and 3d blending surfaces to achieve the global superconvergence o h3 o h4 in h2 norms this is significant compared with the optimal convergence o h2 the algorithms are simple because only an a posteriori interpolant solution is needed such a global convergence method was originated by lin and his colleagues in 1 3 for only the clamped boundary conditions recently we extended the global superconvergence to other boundary conditions such as the simple support condition the periodic boundary and the natural boundary condition moreover we apply in 4 6 this global superconvergence to the fem using the penalty techniques for biharmonic equations and blending problems also to reach o h3 and o h4 for quasiuniform and uniform ij respectively currently we develop in 7 8 and in this paper the fem using the penalty plus hybrid techniques to reduce the condition number down to o h 4 o h 5 of the associated matrix while retaining superconvergence o h3 o h4 since instability is severe for biharmonic equations any reduction of the condition number is crucial by the new algorithms in this paper not only can a great deal of cpu time be saved but also the complicated biharmonic equations and blending surfaces may be solved in double precision numerical experiments are carried out to support the theoretical conclusions

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

the use of cornu spirals in drawing planar curves of controlled curvature abstract cornu spirals or clothoids have been used in highway design for many years in the past the spirals have been found manually by draftsmen the purpose of this paper is to show that it is practical to find these spirals with a micro computer the design curve will be made up of arcs of circles and segments of cornu spirals joined in such a way that the curvature is continuous throughout and takes its largest values on the arcs of circles thus the radii of the circles used will limit and control the curvature of the whole design curve

an excmg accelerated multiscale multigrid computation for 3d poisson equation abstract multiscale multigrid msmg method is an effective computational framework for efficiently computing high accuracy solutions for elliptic partial differential equations in the current msmg method compared to the cpu cost on computing sixth order solutions by applying extrapolation and other techniques on two fourth order solutions from different scales grids much more cpu time is spent on computing fourth order solutions themselves on coarse and fine grids particularly for high dimensional problems here we propose to embed extrapolation cascadic multigrid excmg method into the msmg framework to accelerate the whole process numerical results on 3d poisson equations show that the new excmg msmg method is more efficient than the existing msmg method and the excmg method for sixth order solution computation

sixth order compact scheme combined with multigrid method and extrapolation technique for 2d poisson equation we develop a sixth order finite difference discretization strategy to solve the two dimensional poisson equation which is based on the fourth order compact discretization multigrid method richardson extrapolation technique and an operator based interpolation scheme we use multigrid v cycle procedure to build our multiscale multigrid algorithm which is similar to the full multigrid method fmg the multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid a sixth order accurate coarse grid solution is computed by using the richardson extrapolation technique then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method compared to sun zhang s sixth order richardson extrapolation compact rec discretization strategy using alternating direction implicit adi method and the standard fourth order compact difference foc scheme using a multigrid method

a transformation free hoc scheme and multigrid method for solving the 3d poisson equation on nonuniform grids a high order compact hoc difference scheme is proposed to solve the three dimensional 3d poisson equation on nonuniform orthogonal cartesian grids involving no coordinate transformation from the physical space to the computational space theoretically the proposed scheme has third to fourth order accuracy its fourth order accuracy is achieved under uniform grid settings then a multigrid method is developed to solve the linear system arising from this hoc difference scheme and the corresponding multigrid restriction and interpolation operators are constructed using the volume law numerical experiments are conducted to show the computed accuracy of the hoc scheme and the computational efficiency of the multigrid method

statistical modeling of curves using shapes and related features motivated by the problems of analyzing protein backbones diffusion tensor magnetic resonance imaging dt mri fiber tracts in the human brain and other problems involving curves in this study we present some statistical models of parameterized curves in in terms of combinations of features such as shape location scale and orientation for each combination of interest we identify a representation manifold endow it with a riemannian metric and outline tools for computing sample statistics on these manifolds an important characteristic of the chosen representations is that the ensuing comparison and modeling of curves is invariant to how the curves are parameterized the nuisance variables including parameterization are removed by forming quotient spaces under appropriate group actions in the case of shape analysis the resulting spaces are quotient spaces of hilbert spheres and we derive certain wrapped truncated normal densities for capturing variability in observed curves we demonstrate

approximating smooth planar curves by arc splines when a smooth curve is used to describe the path of a computer controlled cutting machine the path is usually approximated by many straight line segments it is preferable to describe the cutting path as an arc spline a tangent continuous piecewise curve made of circular arcs and straight line segments this paper presents an algorithm for finding an arbitrarily close arc spline approximation of a smooth curve

multigrid method and fourth order compact difference discretization scheme with unequal meshsizes for 3d poisson equation a fourth order compact difference discretization scheme with unequal meshsizes in different coordinate directions is employed to solve a three dimensional 3d poisson equation on a cubic domain two multgrid methods are developed to solve the resulting sparse linear systems one is to use the full coarsening multigrid method with plane gauss seidel relaxation which uses line gauss seidel relaxation to compute each planewise solution the other is to construct a partial semi coarsening multigrid method with the traditional point or plane gauss seidel relaxations numerical experiments are conducted to test the computed accuracy of the fourth order compact difference scheme and the computational efficiency of the multigrid methods with the fourth order compact difference scheme

boundary penalty techniques we examine techniques for enforcing dirichlet boundary data in finite element solution of boundary value problems the common programming strategy of adding a large number to certain diagonal entries and scaling the right side is compared with penalty methods by selectively underintegrating the continuous penalty formulation and selecting the penalty parameter e 1 optimally as a function of mesh size h we deduce the large number strategy and hence show that the optimal rate is preserved numerical experiments to study the rates for continuous penalty underintegrated penalty and large number strategies corroborate the argument

on the convergence rate of the boundary penalty method the convergence rate of the boundary penalty finite element method is discussed for the poisson equation with inhomogeneous dirichlet boundary conditions on a polygonal domain it is proved that under certain circumstances an optimal convergence rate may be achieved which agrees with the rate obtained recently in the numerical experiments by utku and carey

shape analysis of elastic curves in euclidean spaces this paper introduces a square root velocity srv representation for analyzing shapes of curves in euclidean spaces under an elastic metric in this srv representation the elastic metric simplifies to the il2 metric the reparameterization group acts by isometries and the space of unit length curves becomes the unit sphere the shape space of closed curves is the quotient space of a submanifold of the unit sphere modulo rotation and reparameterization groups and we find geodesics in that space using a path straightening approach these geodesics and geodesic distances provide a framework for optimally matching deforming and comparing shapes these ideas are demonstrated using 1 shape analysis of cylindrical helices for studying protein structure 2 shape analysis of facial curves for recognizing faces 3 a wrapped probability distribution for capturing shapes of planar closed curves and 4 parallel transport of deformations for predicting shapes from novel poses

epipolar parameterization for reconstructing 3d rigid curve abstract this paper describes a new method for reconstructing a 3d rigid curve from a sequence of uncalibrated images using 31 epipolar parameterization the approach can be divided into the following two steps first a nonlinear discrete method is presented for point by point reconstruction of the curve instead of whole curve reconstruction projective and euclidean geometric tools are used second the parametric representation of the curve is defined by 31 b spline curves a linear method is proposed to interpolate the reconstructed points to obtain a complete curve thus it is proved that the 31 curve interpolation is equivalent to determining a set of control points of 3d regularized b spline curves it is shown that the 3d epipolar parameterization is an efficient method for reconstructing a 31 curve from image sequences experimental results are presented for real data

the family of biarcs that matches planar two point g1 hermite data the biarc is a curve made by joining two circular arcs in a g 1 fashion there is a one parameter family of biarcs that can match given planar two point g 1 hermite data this note considers the range of g 1 hermite data that can be matched identifies the region in which the members of the family of biarcs lie and shows that there is exactly one member biarc passing through each point in that region

topics in surface modeling contains recent ideas and results in three areas of growing importance in curve and surface design algebraic methods variational surface design and some special applications leading researchers from throughout the world have contributed their latest work and provided several promising solutions to open issues in surface modeling

penalty resolution of the babuska circle paradox abstract if the solution for a simply supported circular plate is approximated by a sequence of finite element solutions on successively refined polygonal domains the finite element approximations converge to the solution of a different problem in the present study we present a finite element formulation with boundary penalty that produces valid approximations to the simply supported plate the penalty term is shown to require the use of reduced integration the dependence of the penalty parameter on mesh size h is also examined numerical experiments confirm the validity of the method and determine rates of convergence a second approach involving a modified corner condition is also considered and error estimates determined this scheme is implemented also using a discrete penalty technique the results and rates are compared with the boundary penalty method and their relative merits discussed

high accuracy solutions of incompressible navier stokes equations abstract in recent years we have developed high accuracy finite difference approximations for partial differential equations of elliptic type with particular emphasis on the convection diffusion equation these approximations are of compact type have a local truncation error of fourth order and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations in this paper we extend these high accuracy approximations to the solution of navier stokes equations solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors it is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows

comparison of second and fourth order discretizations for multigrid poisson solvers we combine a compact high order difference approximation with multigrid v cycle algorithm to solve the two dimensional poisson equation with dirichlet boundary conditions this scheme along with several different orderings of grid space and projection operators is compared with the five point formula to show the dramatic improvement in computed accuracy on serial and vector machines

weighted subspace methods and spatial smoothing analysis and comparison the effect of using a spatially smoothed forward backward covariance matrix on the performance of weighted eigen based state space methods esprit and weighted music for direction of arrival doa estimation is analyzed expressions for the mean squared error in the estimates of the signal zeros and the doa estimates along with some general properties of the estimates and optimal weighting matrices are derived a key result is that optimally weighted music and weighted state space methods esprit have identical asymptotic performance moreover by properly choosing the number of subarrays the performance of unweighted state space methods can be significantly improved it is also shown that the mean squared error in the doa estimates is independent of the exact distribution of the source amplitudes this results in a unified framework for dealing with doa estimation using a uniformly spaced linear sensor array and the time series frequency estimation problems

euler spiral for shape completion in this paper we address the curve completion problem e g the geometric continuation of boundaries of objects which are temporarily interrupted by occlusion also known as the gap completion or shape completion problem this problem is a significant element of perceptual grouping of edge elements and has been approached by using cubic splines or biarcs which minimize total curvature squared elastica as motivated by a physical analogy our approach is motivated by railroad design methods of the early 1900 s which connect two rail segments by transition curves and by the work of knuth on mathematical typography we propose that in using an energy minimizing solution completion curves should not penalize curvature as in elastica but curvature variation the minimization of total curvature variation leads to an euler spiral solution a curve whose curvature varies linearly with arclength the construction of this curve from a pair of points and tangents at these points is reduced to a nonlinear system of equations involving fresnel integrals whose solution relies on optimization from a suitable initial condition constrained to satisfy given boundary conditions since the choice of an appropriate initial curve is critical in this optimization we analytically derive an optimal solution in the class of biarc curves which is then used as the initial curve the resulting interpolations yield intuitive interpolation across gaps and occlusions and are extensible in contrast to scale invariant elastica in addition euler spiral segments can be used in other applications of curve completions e g modeling boundary segments between curvature extrema or skeletal branch geometry

variational methods in elasticity and plasticit k washizu pergamon oxford 3rd ed 1982 650 pp 100 00 isbn 008 026723 8

nested and self adaptive bezier parameterizations for shape optimization this article is a sequel of j a desideri hierarchical optimum shape algorithms using embedded bezier parameterizations in y kuznetsov et al ed numerical methods for scientific computing variational problems and applications cimne barcelona 2003 in which we defined formally a hierarchical shape optimization method based on a multi level shape representation by nested bezier parameterizations famosa and j a desideri a janka multi level shape parameterization for aerodynamic optimization application to drag and noise reduction of transonic supersonic business jet in e heikkola et al ed european congress on computational methods in applied sciences and engineering eccomas 2004 jyvaskyla 24 28 july 2004 where we conducted some preliminary numerical experiments of shape optimization in aerodynamics here we are testing the full multi level optimum shape algorithm analogous in logical structure to the classical full multigrid method second we propose a technique for parameterization self adaptivity both methodological enhancements are assessed by novel numerical experiments on an inverse shape model problem confirming both are very effective

estimation of frequencies and damping factors by two dimensional esprit type methods in this paper we consider two dimensional 2 d signals modeled by the sum of damped cisoids we propose two high resolution approaches to estimate their frequencies and damping factors both high resolution methods are based on the shift invariance structure of the signal subspace related to each dimension the first one estimates the frequency components in both dimensions as in the matrix enhancement and matrix pencil memp method before pairing them with a new algorithm the second one consists of the direct estimation of the signal frequency pairs without an additional step to pair the frequencies related to each dimension we show how these methods can estimate the scattering points of radar images

numerical simulation of 2d square driven cavity using fourth order compact finite difference schemes abstract fourth order compact finite difference schemes are employed with multigrid techniques to simulate the two dimensional square driven cavity flow with small to large reynolds numbers the governing navier stokes equation is linearized in streamfunction and vorticity formulation the fourth order compact approximation schemes are coupled with fourth order approximations for velocities and vorticity boundaries numerical solutions are obtained for square driven cavity flow at high reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods

gauss green cubature and moment computation over arbitrary geometries we have implemented in matlab a gauss like cubature formula over arbitrary bivariate domains with a piecewise regular boundary which is tracked by splines of maximum degree p spline curvilinear polygons the formula is exact for polynomials of degree at most 2n 1 using n cmn 2 nodes 1 c p m being the total number of points given on the boundary it does not need any decomposition of the domain but relies directly on univariate gauss legendre quadrature via green s integral formula several numerical tests are presented including computation of standard as well as orthogonal moments over a nonstandard planar region

bayesian multiscale modeling of closed curves in point clouds modeling object boundaries based on image or point cloud data is frequently necessary in medical and scientific applications ranging from detecting tumor contours for targeted radiation therapy to the classification of organisms based on their structural information in low contrast images or sparse and noisy point clouds there is often insufficient data to recover local segments of the boundary in isolation thus it becomes critical to model the entire boundary in the form of a closed curve to achieve this we develop a bayesian hierarchical model that expresses highly diverse 2d objects in the form of closed curves the model is based on a novel multiscale deformation process by relating multiple objects through a hierarchical formulation we can successfully recover missing boundaries by borrowing structural information from similar objects at the appropriate scale furthermore the model s latent parameters help interpret the population indicating dimensions of significant structural variability

a compact streamfunction velocity scheme on nonuniform grids for the 2d steady incompressible navier stokes equations in this paper an effective compact finite difference approximation which carries streamfunction and its first derivatives velocities as the unknown variables for the streamfunction velocity formulation of the steady two dimensional incompressible navier stokes equation is developed on non uniform orthogonal cartesian grids to solve the resulting system of equations a multigrid iterative strategy on nonuniform grids is introduced by using the interpolation techniques numerical experiments involving two test problems with analytical solutions and the lid driven square cavity flow problem are carried out to display the superiority of the currently developed method on nonuniform grid numerical results show that the present method on nonuniform grids gets as similarly efficient convergence rate as on uniform grids viz second order accuracy and the resolution of the computed solutions for the problems with the sharp changes can be significantly improved when the nonuniform grid strategy is utilized the backward facing step flow is also calculated by the present method to exhibit the capability to simulate the distant field using fewer grid points the solution for the natural convection problem reveals further the wide applications of the present method not only in the flow problems but also in the heat transfer problems all of these numerical results demonstrate the accuracy and efficiency of the currently proposed schemes

approximating cornu spirals by arc splines while modeling planar parametric curves it is desirable to approximate them efficiently by arc splines with fewest segments within the prescribed tolerance planar arc splines are ideal for describing the path of a numerically controlled cutting machine because they are easy to offset in this paper we derive a vector solution to interpolate a g1 continuous biarc curve between a given pair of points with specified start joint and end tangents using this solution as the underlying idea we present an efficient algorithm for approximating a cornu spiral with a sequence of arcs within the prescribed maximum deviation the method is simple and generic and can be applied to other types of planar parametric curves as well

a rational high order compact difference method for the steady state stream functionvorticity formulation of the navierstokes equations a rational high order compact rhoc finite difference fd method on the nine point stencil is proposed for solving the steady state two dimensional navierstokes equations in the stream functionvorticity form the resulting system of algebra equations can be solved by using the point successive over or under relaxation sor iteration numerical experiments involving two linear and two nonlinear problems with their analytical solutions and two flow problems including the lid driven cavity and backward facing step flows are carried out to validate the performance of the newly proposed method numerical solutions of the driven cavity problem with different grid mesh sizes maximum being 513513 for reynolds numbers ranging from 0 to 17500 are obtained and compared with some of the accurate results available in the literature

on nonconforming combinations of various finite element methods for solving elliptic boundary value problems nonconforming combinations of various finite element methods f e m for solving elliptic boundary value problems are presented here different order piecewise polynomials are constrained to be continuous only at the element nodes on the common boundary gamma 0 of s 1 and s 2 on which the m geqq 1 and k geqq m 1 order lagrange f e m are used respectively three important results are reported in this paper 1 error estimates of numerical solutions are derived and a unique combination is found that has an optimal convergence rate of average errors varepsilon h of numerical solutions and their generalized derivatives such a combination is that of the linear and quadratic f e m by using a coupling strategy h o h 1 2 where h and h are the maximal boundary lengths of linear and quadratic regular elements in s 1 and s 2 respectively 2 for the nonconforming combination of the linear and quadratic lagrange f e m the asymptotic convergence rates are

high order compact solvers for the three dimensional poisson equation new compact approximation schemes for the laplace operator of fourth and sixth order are proposed the schemes are based on a pade approximation of the taylor expansion for the discretized laplace operator the new schemes are compared with other finite difference approximations in several benchmark problems it is found that the new schemes exhibit a very good performance and are highly accurate especially on large grids they outperform noncompact schemes

a two point g1 hermite interpolating family of spirals a one parameter family of spirals that can match planar two point g 1 hermite data is presented these spirals can be used as an alternative to the biarc which is also a one parameter family of curves that can match two point g 1 hermite data some suggestions on choosing the free parameter of the family of spirals is given it is shown that there is a unique g 1 hermite interpolating spiral that passes through a given point in an allowable region three examples of the use of these spirals are given curve completion with spirals design with spirals and approximation of the clothoid by spirals

the solution of a time dependent problem by the b spline method abstract in this paper we apply the b spline method for solving the boundary value problems involving non homogeneous heat convection diffusion wave and telegraph equations the accuracy of the method for the equation is o t h 2 several examples are given to verify the reliability and efficiency of the method

multigrid method and fourth order compact scheme for 2d poisson equation with unequal mesh size discretization abstract a fourth order compact difference scheme with unequal mesh sizes in different coordinate directions is employed to discretize a two dimensional poisson equation in a rectangular domain multigrid methods using a partial semicoarsening strategy and line gauss seidel relaxation are designed to solve the resulting sparse linear systems numerical experiments are conducted to test the accuracy of the fourth order compact difference scheme and to compare it with the standard second order difference scheme convergence behavior of the partial semicoarsening and line gauss seidel relaxation multigrid methods is examined experimentally

parameterization invariant shape comparisons of anatomical surfaces we consider 3 d brain structures as continuous parameterized surfaces and present a metric for their comparisons that is invariant to the way they are parameterized past comparisons of such surfaces involve either volume deformations or non rigid matching under fixed parameterizations of surfaces we propose a new mathematical representation of surfaces called q maps such that l2 distances between such maps are invariant to re parameterizations this property allows for removing the parameterization variability by optimizing over the re parameterization group resulting in a proper parameterization invariant distance between shapes of surfaces we demonstrate this method in shape analysis of multiple brain structures for 34 subjects in the detroit fetal alcohol and drug exposure cohort study which results in a 91 classification rate for attention deficit hyperactivity disorder cases and controls this method outperforms some existing techniques such as spherical harmonic point distribution model spharm pdm or iterative closest point icp

high snr asymptotics for signal subspace methods in sinusoidal frequency estimation high snr limit second order properties of multiple signal classification music minimum norm mn and subspace rotation sur signal subspace methods for sinusoidal frequency estimation are discussed an alternative to large sample analysis of the methods is presented the two most important variants of these methods are considered in connection with the choice of the sample covariance matrix the simpler technique follows the principle of a linear prediction and the more complex one is based on the idea of a forward backward prediction explicit expressions for the high snr covariance elements of the estimation errors associated with all the methods are derived the expressions for the covariances are used to analyze and compare the statistical performances of music mn and sur estimation methods in both of the variants to discuss the problem of the optimal dimension of the data covariance matrix and to study the limit statistical efficiency of the methods performances of the large sample and high snr asymptotics derived using monte carlo simulations are presented

high accuracy multigrid solution of the 3d convection diffusion equation we present an explicit fourth order compact finite difference scheme for approximating the three dimensional 3d convection diffusion equation with variable coefficients this 19 point formula is defined on a uniform cubic grid fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1 we design a parallelization oriented multigrid method for fast solution of the resulting linear system using a four color gauss seidel relaxation technique for robustness and efficiency and a scaled residual injection operator to reduce the cost of multigrid inter grid transfer operator numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method several test problems are solved and highly accurate solutions of the 3d convection diffusion equations are obtained for small to medium values of the grid reynolds number effects of using different residual projection operators are compared on both vector and serial computers

matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise a study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented comparison of this method to a polynomial method svd prony method shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes it is found through perturbation analysis and simulation that for signals with unknown damping factors the pencil method is less sensitive to noise than the polynomial method an expression of the cramer rao bound for the exponential signals is presented

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

finite element approximation of the dirichlet problem using the boundary penalty method summary this paper considers a finite element approximation of the dirichlet problem for a second order self adjoint elliptic equation a u f in a region f2cn n 2 or 3 by the boundary penalty method if the finite element space defined over d h a union of elements has approximation power h in the l 2 norm then i for f2 d h convex polyhedral we show that choosing the penalty parameter e h x with 2 k yields optimal h 1 and l 2 error bounds if u hk i q

estimating two dimensional frequencies by matrix enhancement and matrix pencil a new method called the matrix enhancement and matrix pencil memp method is presented for estimating two dimensional 2 d frequencies in the memp method an enhanced matrix is constructed from the data samples and then the matrix pencil approach is used to extract out the 2 d sinusoids from the principal eigenvectors of the enhanced matrix the memp method yields the estimates of the 2 d frequencies efficiently without solving the roots of a 2 d polynomial or searching in a 2 d space it is shown that the memp method can be faster than a 2 d fft method if the number of the 2 d sinusoids is much smaller than the data set simulation results are provided to show that the accuracy of the memp method can be very close to the cramer rao lower bound

high convergence rates of digital image transformation by numerical integration using spline functions abstract this paper introduces an intriguing topic in image processing where accuracy of images even in details is important and adopts an intriguing methodology dealing with discrete topics by continuous mathematics and numerical approximation the key idea is that a pixel of images at different levels can be quantified by a greyness value which can then be regarded as the mean of an integral of continuous functions over a certain region and evaluated by numerical integration approximately however contrasted to the traditional integration the integrand has different smooth nature in different subregions due to piecewise interpolation and approximation new treatments of approximate integration and new discrete algorithms of images have been developed the cycle conversion t 1 t of image transformations is said if an image is distorted by a transformation t and then restored back to itself by the inverse transformation t 1 combined algorithms of discrete techniques proposed in 1 3 only have the convergence rates o 1 n and o 1 n 2 of sequential greyness errors where n is the division number for a pixel split into n 2 subpixels this paper reports new combination algorithms using spline functions to achieve the high convergence rates o 1 n 3 and o 1 n 4 of digital image transformations under the cycle conversion both error analysis and numerical experiments have been provided to verify the high convergence rates high convergence rates of discrete algorithms are important in saving cpu time particularly to multi greyness images moreover the computational figures for real images of 256 256 with 256 greyness levels in which n 2 is good enough for practical requirements display validity and effectiveness of the new algorithms in this paper

high snr perturbation in single 2 d frequency estimate using matrix pencil an exact and most revealing expression of the high signal to noise snr first order perturbation variance of the single 2d frequency estimate using the matrix enhancement and matrix pencil memp method is provided this expression is a general form consistent with several previously reported expressions for the 1d frequency case based on this expression the statistical efficiency of the memp method is discussed the optimum efficiency of the memp method is shown to be achieved by its 1d specialized version

a general meshsize fourth order compact difference discretization scheme for 3d poisson equation a fourth order compact difference scheme with unrestricted general meshsizes in different coordinate directions is derived to discretize three dimensional poisson equation on a regular cubic domain the difference scheme derivation procedure makes use of the symbolic representation of the finite difference schemes and is easier to understand in such complex three dimensional manipulations we use a preconditioned conjugate gradient method to solve the resulting sparse linear systems and verify the formal order of convergence of the derived fourth order finite difference scheme

an efficient two dimensional frequency estimator the problem of two dimensional frequency estimation of a sinusoid embedded in complex white gaussian noise is addressed a frequency estimator and its statistical performance are derived the estimator is demonstrated to be an efficient estimator it achieves the cramer rao bound for high signal to noise ratios the results of a computer simulator which compares the variance with that of the maximum likelihood estimator and the theoretical two dimensional cramer rao bound are presented

a fourth order kernel free boundary integral method for implicitly defined surfaces in three space dimensions abstract the kernel free boundary integral kfbi method is a finite difference version of the traditional boundary integral method for elliptic and parabolic partial differential equations on complex domains it evaluates boundary or volume integrals involved in the solution of boundary integral equations bies by solving equivalent but simple interface problems on regular grids so that the integral kernel or green function is never needed or computed this is the essential difference of the kfbi method from the traditional ones it takes advantage of the well conditioning property of discrete bies so that the number of krylov subspace iterations is essentially independent of discretization parameter or system dimension this paper presents a fourth order kernel free boundary integral method for second order elliptic partial differential equations on complex domains in three space dimensions whose boundaries are given by implicitly defined surfaces it represents the domain boundary and discretizes data on it by intersection points of the surface with cartesian grid lines the approach has a variety of advantages the current work solves simple interface problems with corrected 27 point compact finite difference schemes and calculates the discrete equations with a fast fourier transform based elliptic solver numerical examples show that the proposed method is efficient as well as accurate

multigrid method based on the transformation free hoc scheme on nonuniform grids for 2d convection diffusion problems in this paper a multigrid method based on the high order compact hoc difference scheme on nonuniform grids which has been proposed by kalita et al j c kalita a k dass d c dalal a transformation free hoc scheme for steady convection diffusion on non uniform grids int j numer methods fluids 44 2004 33 53 is proposed to solve the two dimensional 2d convection diffusion equation the hoc scheme is not involved in any grid transformation to map the nonuniform grids to uniform grids consequently the multigrid method is brand new for solving the discrete system arising from the difference equation on nonuniform grids the corresponding multigrid projection and interpolation operators are constructed by the area ratio some boundary layer and local singularity problems are used to demonstrate the superiority of the present method numerical results show that the multigrid method with the hoc scheme on nonuniform grids almost gets as equally efficient convergence rate as on uniform grids and the computed solution on nonuniform grids retains fourth order accuracy while on uniform grids just gets very poor solution for very steep boundary layer or high local singularity problems the present method is also applied to solve the 2d incompressible navier stokes equations using the stream function vorticity formulation and the numerical solutions of the lid driven cavity flow problem are obtained and compared with solutions available in the literature

boundary penalty finite element methods for blending surfaces iii superconvergence and stability and examples abstract this paper is part iii of the study on blending surfaces by partial differential equations pdes the blending surfaces in three dimensions 3d are taken into account by three parametric functions x r t y r t and z r t the boundary penalty techniques are well suited to the complicated tangent i e normal derivative boundary conditions in engineering blending by following the previous papers parts i and ii in li j comput math 16 1998 457 480 j comput appl math 110 1999 155 176 the corresponding theoretical analysis is made to discover that when the penalty power 2 3 or 3 5 and 0 1 5 in the boundary penalty finite element methods bp fems optimal convergence rates superconvergence and optimal numerical stability can be achieved respectively several interesting samples of 3d blending surfaces are provided to display the remarkable advantages of the proposed approaches in this paper unique solutions of blending surfaces optimal blending surfaces in minimum energy ease in handling the complicated boundary constraint conditions and less cpu time and computer storage needed this paper and li j comput math 16 1998 457 480 j comput appl math provide a foundation of blending surfaces by pde solutions a new trend of computer geometric design

looking for shapes in two dimensional cluttered point clouds we study the problem of identifying shape classes in point clouds these clouds contain sampled points along contours and are corrupted by clutter and observation noise taking an analysis by synthesis approach we simulate high probability configurations of sampled contours using models learned from training data to evaluate the given test data to facilitate simulations we develop statistical models for sources of nuisance variability 1 shape variations within classes 2 variability in sampling continuous curves 3 pose and scale variability 4 observation noise and 5 points introduced by clutter the variability in sampling closed curves into finite points is represented by positive diffeomorphisms of a unit circle we derive probability models on these functions using their square root forms and the fisher rao metric using a monte carlo approach we simulate configurations from a joint prior on the shape sample space and compare them to the data using a likelihood function average likelihoods of simulated configurations lead to estimates of posterior probabilities of different classes and hence bayesian classification

brief paper control curve design for nonlinear or fuzzy proportional actions using spline based functions this work explores a novel approach fora systematic design of a nonlinear mapping system typicallyfor nonlinear or fuzzy pid control applications the paper investigates a nonlinear design of proportionalactions using spline based functions specifically the controller uses bezier curves to formthe nonlinear mapping in order to emulate fuzzy pid systems while other researchers have addressed the fuzzy systemsfor approximations of given functions we believethat in general control problems these approximationsshould be considered in dealing with the properties of unknowncontrol actions proportional action is selected asa basic function for nonlinear control curve designs the reasons for this selection are discussed specific heuristic properties for the proportional action are defined based on the intuitions in general pid controller applications the new controller is designed to be compatible with these properties the nonlinearity variation index is used as a process independent measure for evaluation of different designs the system has been shown to improve the conventional fuzzy pid controllers on three aspects these include a high degree of transparency with respect to nonlinear tuning parameters versatility to cover various nonlinear functions and simplicity of nonlinear mapping expressions

a compact multigrid solver for convection diffusion equations we introduce a multigrid algorithm to solve the convection diffusion equations using a nine point compact difference scheme we test the efficiency of the algorithm with various smoothers and intergrid transfer operators the algorithm displays a grid independent convergence rate and produces solutions with high accuracy numerical results are presented to validate the conclusions

coaxing a planar curve to comply a long standing problem in computer graphics is to find a planar curve that is shaped the way you want it to be shaped a selection of various methods for achieving this goal is presented the focus is on mathematical conditions that we can use to control curves while still allowing the curves some freedom we start with methods invented by newton 1643 1727 and lagrange 1736 1813 and proceed to recent methods that are the subject of current research we illustrate almost all the methods discussed with diagrams three methods of control that are of special interest are interpolation methods global minimization methods such as least squares and bezier control points we concentrate on the first of these interpolation methods

a theory of multiscale curvature based shape representation for planar curves a shape representation technique suitable for tasks that call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation is presented the method rests on the describing a curve at varying levels of detail using features that are invariant with respect to transformations that do not change the shape of the curve three different ways of computing the representation are described they result in three different representations the curvature scale space image the renormalized curvature scale space image and the resampled curvature scale space image the process of describing a curve at increasing levels of abstraction is referred to as the evolution or arc length evolution of that curve several evolution and arc length evolution properties of planar curves are discussed