energy levels of a two dimensional anharmonic oscillator characteristic function approach the eigenlevels of a two dimensional quartic anharmonic oscillator are identified as the zeros of a characteristic function derived from the resolvent numerical values are obtained by means of pade approximants

a difference scheme for a nonlinear partial integrodifferential equation a difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in proving consistency the verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature a stability result is derived that is applicable to equations and numerical methods far more general than those treated in this paper

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth for planar landmark based shapes taking into account the non euclidean geometry of the shape space a statistical test for a common mean first geodesic principal component gpc is devised which rests on one of two asymptotic scenarios for both scenarios strong consistency and central limit theorems are established along with an algorithm for the computation of a ziezold mean geodesic in application this allows to verify the geodesic hypothesis for leaf growth of canadian black poplars and to discriminate genetically different trees by observations of leaf shape growth over brief time intervals with a test based on procrustes tangent space coordinates not involving the shape space s curvature neither can be achieved

shape curves and geodesic modelling a family of shape curves is introduced that is useful for modelling the changes in shape in a series of geometrical objects the relationship between the preshape sphere and the shape space is used to define a general family of curves based on horizontal geodesics on the preshape sphere methods for fitting geodesics and more general curves in the non euclidean shape space of point sets are discussed based on minimizing sums of squares of procrustes distances likelihood based inference is considered we illustrate the ideas by carrying out statistical analysis of two dimensional landmarks on rats skulls at various times in their development and three dimensional landmarks on lumbar vertebrae from three primate species

geodesic regression and the theory of least squares on riemannian manifolds this paper develops the theory of geodesic regression and least squares estimation on riemannian manifolds geodesic regression is a method for finding the relationship between a real valued independent variable and a manifold valued dependent random variable where this relationship is modeled as a geodesic curve on the manifold least squares estimation is formulated intrinsically as a minimization of the sum of squared geodesic distances of the data to the estimated model geodesic regression is a direct generalization of linear regression to the manifold setting and it provides a simple parameterization of the estimated relationship as an initial point and velocity analogous to the intercept and slope a nonparametric permutation test for determining the significance of the trend is also given for the case of symmetric spaces two main theoretical results are established first conditions for existence and uniqueness of the least squares problem are provided second a maximum likelihood criteria is developed for a suitable definition of gaussian errors on the manifold while the method can be generally applied to data on any manifold specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age

numerical methods for the solution of partial differential equations of fractional order anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas to model this transport mechanism fractional order space derivative operators can be used here the numerical properties of partial differential equations of fractional order 1 2 are studied two numerical schemes an explicit and a semi implicit one are used in solving these equations two different discretization methods of the fractional derivative operator have also been used the accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order

a new fractional numerical differentiation formula to approximate the caputo fractional derivative and its applications in the present work first a new fractional numerical differentiation formula called the l1 2 formula to approximate the caputo fractional derivative of order 0 1 is developed it is established by means of the quadratic interpolation approximation using three points t j 2 f t j 2 t j 1 f t j 1 and t j f t j for the integrand f t on each small interval t j 1 t j j 2 while the linear interpolation approximation is applied on the first small interval t 0 t 1 as a result the new formula can be formally viewed as a modification of the classical l1 formula which is obtained by the piecewise linear approximation for f t both the computational efficiency and numerical accuracy of the new formula are superior to that of the l1 formula the coefficients and truncation errors of this formula are discussed in detail two test examples show the numerical accuracy of l1 2 formula second by the new formula two improved finite difference schemes with high order accuracy in time for solving the time fractional sub diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed respectively in addition the application of the new formula into solving fractional ordinary differential equations is also presented several numerical examples are computed the comparison with the corresponding results of finite difference methods by the l1 formula demonstrates that the new l1 2 formula is much more effective and more accurate than the l1 formula when solving time fractional differential equations numerically

compact difference scheme for the fractional sub diffusion equation with neumann boundary conditions an effective finite difference scheme is considered for solving the time fractional sub diffusion equation with neumann boundary conditions a difference scheme combining the compact difference approach the spatial discretization and l 1 approximation for the caputo fractional derivative is proposed and analyzed although the spatial approximation order at the neumann boundary is one order lower than that for interior mesh points the unconditional stability and the global convergence order o t 2 a h 4 in discrete l 2 norm of the compact difference scheme are proved rigorously where t is the temporal grid size and h is the spatial grid size numerical experiments are included to support the theoretical results and comparison with the related works are presented to show the effectiveness of our method

numerical methods for solving a two dimensional variable order anomalous subdiffusion equation australian research council dp0559807 dp0986766 national natural science foundation of china 10271098 natural science foundation of fujian province 2009j01014

a compact finite difference scheme for the fractional sub diffusion equations in this paper a compact finite difference scheme for the fractional sub diffusion equations is derived after a transformation of the original problem the l1 discretization is applied for the time fractional part and fourth order accuracy compact approximation for the second order space derivative the unique solvability of the difference solution is discussed the stability and convergence of the finite difference scheme in maximum norm are proved using the energy method where a new inner product is introduced for the theoretical analysis the technique is quite novel and different from previous analytical methods finally a numerical example is provided to show the effectiveness and accuracy of the method

finite difference methods for two dimensional fractional dispersion equation fractional order partial differential equations as generalizations of classical integer order partial differential equations are increasingly used to model problems in fluid flow finance and other areas of application in this paper we discuss a practical alternating directions implicit method to solve a class of two dimensional initial boundary value fractional partial differential equations with variable coefficients on a finite domain first order consistency unconditional stability and therefore first order convergence of the method are proven using a novel shifted version of the classical grunwald finite difference approximation for the fractional derivatives a numerical example with known exact solution is also presented and the behavior of the error is examined to verify the order of convergence

weighted average finite difference methods for fractional diffusion equations a class of finite difference methods for solving fractional diffusion equations is considered these methods are an extension of the weighted average methods for ordinary non fractional diffusion equations their accuracy is of order x 2 and t except for the fractional version of the crank nicholson method where the accuracy with respect to the timestep is of order t 2 if a second order approximation to the fractional time derivative is used their stability is analyzed by means of a recently proposed procedure akin to the standard von neumann stability analysis a simple and accurate stability criterion valid for different discretization schemes of the fractional derivative arbitrary weight factor and arbitrary order of the fractional derivative is found and checked numerically some examples are provided in which the new methods numerical solutions are obtained and compared against exact solutions

a compact integrated rbf method for time fractional convection diffusion reaction equations abstract in this paper a local compact integrated radial basis function cirbf method is proposed to solve the time fractional convection diffusion reaction equations the proposed cirbf scheme derives high accuracy and fast convergence rate with grid refinement the spatial derivatives are discretized by combining with the integrated rbf approximation and compact approximation in a stencil the nodal function values and the nodal second derivatives are used to establish the relation between the physical space and the rbf weight space the time discretization is replaced by the second order shifted grunwald scheme moreover the unconditional stability and convergence of the proposed method are shown finally some numerical examples on 2d and 3d domains are carried out to verify the accuracy and efficiency of the proposed algorithm for the fractional differential equations

numerical simulations of 2d fractional subdiffusion problems the growing number of applications of fractional derivatives in various fields of science and engineering indicates that there is a significant demand for better mathematical algorithms for models with real objects and processes currently most algorithms are designed for 1d problems due to the memory effect in fractional derivatives in this work the 2d fractional subdiffusion problems are solved by an algorithm that couples an adaptive time stepping and adaptive spatial basis selection approach the proposed algorithm is also used to simulate a subdiffusion convection equation

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

two high order numerical algorithms for solving the multi term time fractional diffusion wave equations in this paper we apply a high order difference scheme and galerkin spectral technique for the numerical solution of multi term time fractional partial differential equations the proposed methods are based on a finite difference scheme in time the time fractional derivatives which have been described in caputo s sense are approximated by a scheme of order o 3 1 2 and the space derivative is discretized with a fourth order compact finite difference procedure and galerkin spectral method we prove the unconditional stability of the compact procedure by coefficient matrix property the l convergence of the compact finite difference method has been proved by the energy method also we obtain an error estimate for galerkin spectral method numerical results are provided to verify the accuracy and efficiency of the proposed schemes

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

combined compact difference scheme for the time fractional convection diffusion equation with variable coefficients combined compact finite difference scheme for solving the time fractional convection diffusion reaction equation is given the primary unknown and its derivative are solved simultaneously the coefficients of the fractional partial differential equation are variable both dirichlet and robin boundary conditions are discussed and the algorithm gives high order convergence results fourth order combined compact finite difference scheme is given for solving the time fractional convection diffusion reaction equation with variable coefficients we introduce the flux as a new variable and transform the original equation into a system of two equations compact difference is used as a high order approximation for spatial derivatives of integer order in the coupled partial differential equations the caputo fractional derivative is discretized by a high order approximation both dirichlet and robin boundary conditions are discussed convergence analysis is given for the problem of integer order with constant coefficients under some assumption numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

estimating multiple pathways of object growth using nonlongitudinal image data this article presents an infinite bayesian mixture of monotonic regression models for analyzing multiple growth pathways of star shaped objects growing over time by using nonlongitudinal data a motivating example is the analysis of nanocrystal growth processes a radius function representation used for the outlines of star shaped objects allows us to represent an object growth i e expansion of outlines by an increasing sequence of random radius functions we propose a monotonic regression model to fit the increasing sequence to nonlongitudinal data ensuring that the fitted radius functions always monotonically increase in the sequence to model the multiple pathways of object growth we propose the dirichlet infinite location mixture of multiple monotonic regression models and use a block gibbs sampler as a numerical solver we apply the proposed model to nonlongitudinal sets of microscopic image data for inferring multiple growth pathways of nanocrystals comparing the inference result with real nan

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

shape procrustes tangent projections and bilateral symmetry summary in shape analysis with concentrated data the procrustes tangent projection of shapes plays a fundamental role since this projection can be used to convert shape analysis to standard multivariate analysis in view of its importance a systematic and rigorous treatment of the procrustes tangent projection is developed here to facilitate its use in applications one important application involves an analytical description of bilateral symmetry of objects together with some related distributional results an explicit expression for the tangent projection is also derived

the complex watson distribution and shape analysis the complex watson distribution is an important simple distribution on the complex sphere which is used in statistical shape analysis we describe the density obtain the integrating constant and provide large sample approximations maximum likelihood estimation and hypothesis testing procedures for one and two samples are described the particular connection with shape analysis is discussed and we consider an application examining shape differences between normal and schizophrenic brains we make some observations about bayesian shape inference and finally we describe a more general rotationally symmetric family of distributions

intrinsic manova for riemannian manifolds with an application to kendall s space of planar shapes we propose an intrinsic multifactorial model for data on riemannian manifolds that typically occur in the statistical analysis of shape due to the lack of a linear structure linear models cannot be defined in general to date only one way manova is available for a general multifactorial model we assume that variation not explained by the model is concentrated near elements defining the effects by determining the asymptotic distributions of respective sample covariances under parallel transport we show that they can be compared by standard manova often in applications manifolds are only implicitly given as quotients where the bottom space parallel transport can be expressed through a differential equation for kendall s space of planar shapes we provide an explicit solution we illustrate our method by an intrinsic two way manova for a set of leaf shapes while biologists can identify genotype effects by sight we can detect height effects that are otherwise not identifiable

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

a box type scheme for fractional sub diffusion equation with neumann boundary conditions combining order reduction approach and l1 discretization a box type scheme is presented for solving a class of fractional sub diffusion equation with neumann boundary conditions a new inner product and corresponding norm with a sobolev embedding inequality are introduced a novel technique is applied in the proof of both stability and convergence the global convergence order in maximum norm is o t 2 a h 2 the accuracy and efficiency of the scheme are checked by two numerical tests

studies on algebraic methods to solve linear eigenvalue problems generalised anharmonic oscillators a detailed presentation of the recently introduced integration free method with applications to determine the energy levels of the generalised quantum anharmonic oscillators are given numerical calculations are realised for the quartic and the sextic oscillators energy eigenvalues obtained for the ground state as well as for the first few excited states accurate to thirty digits are very impressive and demonstrate the efficiency of the method certain remarks about the selection of the basis functions and a convergence discussion on the presented simple approximation scheme are also included in this paper

a new high order adi numerical difference formula for time fractional convection diffusion equation abstract based on exponential transformation quadratic interpolation polynomial and pade approximation a new high order finite difference scheme is proposed for solving the two dimensional 2d time fractional convection dominated diffusion equation of order 0 1 the resulting scheme is of 3 order accuracy in time and fourth order accuracy in space in order to reduce the amount of computation the alternating direction implicit adi scheme is further developed numerical experiments are given to demonstrate the high accuracy and robustness of our new scheme

high order finite element methods for time fractional partial differential equations the aim of this paper is to develop high order methods for solving time fractional partial differential equations the proposed high order method is based on high order finite element method for space and finite difference method for time optimal convergence rate o dt 2 a n r is proved for the r 1 th order finite element method r 2

statistical shape analysis clustering learning and testing using a differential geometric treatment of planar shapes we present tools for 1 hierarchical clustering of imaged objects according to the shapes of their boundaries 2 learning of probability models for clusters of shapes and 3 testing of newly observed shapes under competing probability models clustering at any level of hierarchy is performed using a minimum variance type criterion and a markov process statistical means of clusters provide shapes to be clustered at the next higher level thus building a hierarchy of shapes using finite dimensional approximations of spaces tangent to the shape space at sample means we implicitly impose probability models on the shape space and results are illustrated via random sampling and classification hypothesis testing together hierarchical clustering and hypothesis testing provide an efficient framework for shape retrieval examples are presented using shapes and images from eth surrey and amcom databases

high order local discontinuous galerkin method combined with wsgd approximation for a fractional subdiffusion equation abstract in this paper a high order local discontinuous galerkin ldg method combined with weighted and shifted grunwald difference wsgd approximation is developed and discussed for a caputo time fractional subdiffusion equation the time fractional derivative of order 0 1 is approximated by a third order method based on the idea of wsgd operator while the spatial operator is approximated by the ldg method some useful lemmas are first introduced and proved then the analysis of stability and optimal error estimate o t 3 h k 1 are obtained for the ldg method extensive numerical results using p k k 0 1 2 3 elements are presented to validate our theoretical analysis

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

an o n log2n alternating direction finite difference method for two dimensional fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices consequently the numerical solution of these methods often require computational work of o n 3 per time step and memory of o n 2 for where n is the number of grid points in this paper we develop a fast alternating direction implicit finite difference method for space fractional diffusion equations in two space dimensions the method only requires computational work of o n log 2n per time step and memory of o n while retaining the same accuracy and approximation property as the regular finite difference method with gaussian elimination our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations to achieve the same accuracy the new method has a significant reduction of the cpu time from more than 2 months and 1 week consumed by a traditional finite difference method to 1 5h using less than one thousandth of memory the standard method does this demonstrates the utility of the method

fourth order finite difference schemes for time space fractional sub diffusion equations in this paper we devote to the study of high order finite difference schemes for one and two dimensional time space fractional sub diffusion equations a fourth order finite difference scheme is invoked for the spatial fractional derivatives and the l 1 approximation is applied to the temporal fractional parts for the two dimensional case an alternating direction implicit scheme based on l 1 approximation is proposed the stability and convergence of the proposed methods are studied numerical experiments are performed to verify the effectiveness and accuracy of the proposed difference schemes

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

a novel approach of high accuracy analysis of anisotropic bilinear finite element for time fractional diffusion equations with variable coefficient abstract in this paper by using bilinear finite element and l 1 approximation a fully discrete scheme is established for time fractional diffusion equation with variable coefficient on anisotropic meshes unconditionally stable analysis of the proposed scheme are presented in both l 2 norm and h 1 norm moreover convergence superclose and superconvergence results are derived by combining interpolation with projection which is the key technique for the numerical analysis specifically by defining a novel projection operator the error estimate between the projection and the exact solution is obtained on anisotropic meshes furthermore high accuracy analysis on interpolation of bilinear finite element and projection is gained by means of some known results about the interpolation and mean value technique based on the related results about projection and interpolation the optimal error estimate in l 2 norm and superclose of interpolation in h 1 norm are deduced by skillfully dealing with fractional derivative at the same time the global superconvergence is presented by employing interpolation postprocessing operator finally numerical results are provided to demonstrate the validity of the theoretical analysis

a fourier method for the fractional diffusion equation describing sub diffusion in this paper a fractional partial differential equation fpde describing sub diffusion is considered an implicit difference approximation scheme idas for solving a fpde is presented we propose a fourier method for analyzing the stability and convergence of the idas derive the global accuracy of the idas and discuss the solvability finally numerical examples are given to compare with the exact solution for the order of convergence and simulate the fractional dynamical systems

a galerkin finite element method for time fractional stochastic heat equation abstract in this study a galerkin finite element method is presented for time fractional stochastic heat equation driven by multiplicative noise which arises from the consideration of heat transport in porous media with thermal memory with random effects the spatial and temporal regularity properties of mild solution to the given problem under certain sufficient conditions are obtained numerical techniques are developed by the standard galerkin finite element method in spatial direction and gorenflo mainardi moretti paradisi scheme is applied in temporal direction the convergence error estimates for both semi discrete and fully discrete schemes are established finally numerical example is provided to verify the theoretical results

intrinsic regression models for medial representation of subcortical structures the aim of this article is to develop a semiparametric model to describe the variability of the medial representation of subcortical structures which belongs to a riemannian manifold and establish its association with covariates of interest such as diagnostic status age and gender we develop a two stage estimation procedure to calculate the parameter estimates the first stage is to calculate an intrinsic least squares estimator of the parameter vector using the annealing evolutionary stochastic approximation monte carlo algorithm and then the second stage is to construct a set of estimating equations to obtain a more efficient estimate with the intrinsic least squares estimate as the starting point we use wald statistics to test linear hypotheses of unknown parameters and establish their limiting distributions simulation studies are used to evaluate the accuracy of our parameter estimates and the finite sample performance of the wald statistics we apply our methods to the detection of the diffe

principal boundary on riemannian manifolds we consider the classification problem and focus on nonlinear methods for classification on manifolds for multivariate datasets lying on an embedded nonlinear riemannian manifold within the higher

high order compact alternating direction implicit method for the generalized sine gordon equation high order compact alternating direction implicit scheme is given for solving the generalized sine gordon equation in a two dimensional rectangular domain we apply the compact finite difference operators to obtain a fourth order discretization for the second order space derivatives and we give a linearized three time level algorithm for solving the original nonlinear equation error estimate is given by the energy method numerical results are provided to verify the accuracy and efficiency of this algorithm

analysis of principal nested spheres a general framework for a novel non geodesic decomposition of high dimensional spheres or high dimensional shape spaces for planar landmarks is discussed the decomposition principal nested spheres leads to a sequence of submanifolds with decreasing intrinsic dimensions which can be interpreted as an analogue of principal component analysis in a number of real datasets an apparent one dimensional mode of variation curving through more than one geodesic component is captured in the one dimensional component of principal nested spheres while analysis of principal nested spheres provides an intuitive and flexible decomposition of the high dimensional sphere an interesting special case of the analysis results in finding principal geodesics similar to those from previous approaches to manifold principal component analysis an adaptation of our method to kendall s shape space is discussed and a computational algorithm for fitting principal nested spheres is proposed the result provides a coordinate system to visualize the data structure and an intuitive summary of principal modes of variation as exemplified by several datasets

finite element approximation of fractional order elliptic boundary value problems a finite element numerical method is investigated for fractional order elliptic boundary value problems with homogeneous dirichlet type boundary conditions it is pointed out that an appropriate stiffness matrix can be obtained by taking the prescribed fractional power of the stiffness matrix corresponding to the non fractional elliptic operators it is proved that this approach which is also called the matrix transformation or matrix transfer method delivers optimal rate of convergence in the l 2 norm

shape space smoothing splines for planar landmark data a method is developed for fitting smooth curves through a series of shapes of landmarks in two dimensions using unrolling and unwrapping procedures in riemannian manifolds an explicit method of calculation is given which is analogous to that of jupp kent 1987 for spherical data the resulting splines are called shape space smoothing splines the method resembles that of fitting smoothing splines in real spaces in that if the smoothing parameter is zero the resulting curve interpolates the data points and if it is infinitely large the curve is a geodesic line the fitted path to the data is defined such that its unrolled version at the tangent space of the starting point is a cubic spline fitted to the unwrapped data with respect to that path computation of the fitted path consists of an iterative procedure which converges quickly and the resulting path is given in a discretised form in terms of a piecewise geodesic path the procedure is applied to the analysis of some human movement data and a test for the appropriateness of a mean geodesic curve is given

analysis of mixed finite element method mfem for solving the generalized fractional reaction diffusion equation on nonrectangular domains abstract in the current manuscript we consider a generalized fractional reaction diffusion equation the considered model is based on the time fractional derivative the developed scheme is based on two procedures at first we obtain a semi discrete scheme for the temporal direction the time fractional derivative is discretized by using a difference scheme with second order accuracy thus the final time discrete formula has second order accuracy in the temporal direction then we obtain a fully discrete scheme by applying the mixed finite element method mfem for the constructed numerical technique we prove the unconditional stability and also obtain an error bound for the full discrete scheme by using the energy method we employ some test problems to show the accuracy of the proposed technique

numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term in this paper we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term a new implicit difference method is constructed the stability and convergence are discussed using a new energy method finally some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis

a direct o nlog2n finite difference method for fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 where n is the number of grid points in this paper we develop a fast finite difference method for fractional diffusion equations which only requires storage of o n and computational cost of o nlog 2n while retaining the same accuracy and approximation property as the regular finite difference method numerical experiments are presented to show the utility of the method

new solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation a physical mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order in this paper an anomalous subdiffusion equation asub de is considered a new implicit numerical method inm and two solution techniques for improving the order of convergence of the inm for solving the asub de are proposed the stability and convergence of the inm are investigated by the energy method some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis these methods and supporting theoretical results can also be applied to other fractional integro differential equations and higher dimensional problems

local polynomial regression for symmetric positive definite matrices local polynomial regression has received extensive attention for the nonparametric estimation of regression functions when both the response and the covariate are in euclidean space however little has been done when the response is in a riemannian manifold we develop an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite spd matrices as responses that lie in a riemannian manifold with covariate in euclidean space the primary motivation and application of the proposed methodology is in computer vision and medical imaging we examine two commonly used metrics including the trace metric and the log euclidean metric on the space of spd matrices for each metric we develop a cross validation bandwidth selection method derive the asymptotic bias variance and normality of the intrinsic local constant and local linear estimators and compare their asymptotic mean square errors simulation studies are further used to compare the estimators under the two metrics and to examine their finite sample performance we use our method to detect diagnostic differences between diffusion tensors along fiber tracts in a study of human immunodeficiency virus

a level set based variational principal flow method for nonparametric dimension reduction on riemannian manifolds we propose a variational formulation for dimension reduction on riemannian manifolds the algorithm is developed based on the level set method together with a recently developed principal flow algorithm the original principal flow algorithm is a lagrangian technique which extends the principal component analysis pca to dimension reduction on riemannian manifolds we propose to incorporate the level set method to obtain a fully implicit formulation so that the overall algorithm can naturally handle various topological changes in the curve evolution the variational formulation consists of two terms which try to balance the contributions from both the dataset itself and the principal direction by the pca we will demonstrate that the method is insensitive to the initial guess and is robust enough for noisy data

implicit compact linearized θ methods with factorization for multidimensional reaction diffusion equations an iterative predictor corrector technique for the elimination of the approximate factorization errors which result from the factorization of implicit three point compact linearized q methods in multidimensional reaction diffusion equations is proposed and its convergence and linear stability are analyzed four compact approximate factorization techniques which do not account for the approximate factorization errors and which involve three point stencils for each one dimensional operator are developed the first technique uses the full jacobian matrix of the reaction terms requires the inversion of in general dense matrices and its approximate factorization errors are second order accurate in time the second and third methods approximate the jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are however first order accurate in time the fourth approximately factorized compact implicit method has approximate factorization errors which are second order accurate in time and requires the inversion of lower and upper triangular matrices the techniques are applied to a nonlinear two species two dimensional system of reaction diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the jacobian matrix as functions of the allocation of the reaction terms space and time

the riemannian structure of euclidean shape spaces a novel environment for statistics the riemannian metric structure of the shape space sum k m for k labelled points in mathbb r m was given by kendall for the atypically simple situations in which m 1 or 2 and k geq 2 here we deal with the general case m geq 1 k geq 2 by using the properties of riemannian submersions and warped products as studied by o neill the approach is via the associated size and shape space that is the warped product of the shape space and the half line mathbb r carrying size the warping function being equal to the square of the size when combined with parallel studies by le of the corresponding global geodesic geometry the results obtained here determine the environment in which shape statistical calculations have to be acted out finally three different applications are discussed that illustrate the theory and its use in practice

an unconditionally stable compact adi method for three dimensional time fractional convection diffusion equation abstract a high order compact finite difference method is presented for solving the three dimensional 3d time fractional convection diffusion equation of order 1 2 the original equation is first transformed to a fractional diffusion wave equation then using fourth order pade approximation for spatial derivatives and the center difference method for time derivative respectively a fully discrete implicit compact scheme is obtained furthermore based on different splitting terms three unconditionally stable adi compact schemes with optimal convergence order are developed respectively the resulting schemes in each adi solution step corresponding to a strictly diagonally dominant matrix equation can be solved using the 1d tridiagonal thomas algorithm with a considerable saving in computing time numerical experiments show that these schemes can significantly improve the time accuracy

compact exponential scheme for the time fractional convection diffusion reaction equation with variable coefficients high order compact exponential finite difference scheme for solving the time fractional convection diffusion reaction equation with variable coefficients is considered in this paper the convection diffusion and reaction coefficients can depend on both the spatial and temporal variables we begin with the one dimensional problem and after transforming the original equation to one with diffusion coefficient unity the new equation is discretized by a compact exponential finite difference scheme with a high order approximation for the caputo time derivative we prove the solvability of this fully discrete implicit scheme and analyze its local truncation error for the fractional equation with constant coefficients we use fourier method to prove the stability and utilize matrix analysis as a tool for the error estimate then we discuss the two dimensional problem give the compact adi scheme with the restriction that besides the time variable the convection coefficients can only depend on the corresponding spatial variables respectively numerical results are provided to verify the accuracy and efficiency of the proposed algorithm the time fractional convection diffusion reaction equation is numerically solved compact exponential scheme for the variable coefficients equation is given fourier method and matrix analysis are used for the analysis the compact exponential adi scheme is given for the two dimensional problem high accuracy and good efficiency of the proposed algorithm is demonstrated

compact alternating direction implicit method for two dimensional time fractional diffusion equation high order compact finite difference scheme with operator splitting technique for solving two dimensional time fractional diffusion equation is considered in this paper a grunwald letnikov approximation is used for the riemann liouville time derivative and the second order spatial derivatives are approximated by the compact finite differences to obtain a fully discrete implicit scheme alternating direction implicit adi method is used to split the original problem into two separate one dimensional problems the local truncation error is analyzed and the stability is discussed by the fourier method the proposed scheme is suitable when the order of the time fractional derivative c lies in the interval 12 1 a correction term is added to maintain high accuracy when c 0 12 numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

the tps direct transport a new method for transporting deformations in the size and shape space modern shape analysis allows the fine comparison of shape changes occurring between different objects very often the classic machineries of generalized procrustes analysis and principal component analysis are used in order to contrast the shape change occurring among configurations represented by homologous landmarks however if size and shape data are structured in different groups thus constituting different morphological trajectories a data centering is needed if one wants to compare solely the deformation representing the trajectories to do that inter individual variation must be filtered out this maneuver is rarely applied in studies using simulated or real data a geometrical procedure named parallel transport that can be based on various connection types is necessary to perform such kind of data centering usually the levi civita connection is used for interpolation of curves in a riemannian space it can also be used to transport a deformation we demonstrate that this procedure does not preserve some important characters of the deformation even in the affine case we propose a novel procedure called tps direct transport which is able to perfectly transport deformation in the affine case and to better approximate non affine deformation in comparison to existing tools we recommend to center shape data using the methods described here when the differences in deformation rather than in shape are under study

a characteristic function approach to the discrete spectrum of electrically charged particles the purpose of this article is to obtain a scalar function the zeros of which give the discrete energy values for a system of electrically charged particles the relation between the serial expansion of the characteristic function in powers of the system s eigenvalue and the stieltjes series has been revealed not only for the electrically charged particles but also generally for any positive or negative definite hermitian operator which has only a discrete spectrum the use of pade approximants to express the characteristic function has offered a rapidly convergent scheme to evaluate the system s eigenvalues the first few elements of the pade table for the reciprocal of the characteristic function of certain systems have been given to verify the presented idea numerically the determination of these elements needs the values of certain complicated integrals which we name zeroth order hyperspherical spectral coefficients hsc 0 the first two of these coefficients are investigated and their evaluation is realized analytically

finite difference methods and a fourier analysis for the fractional reaction subdiffusion equation various fields of science and engineering deal with dynamical systems that can be described by fractional partial differential equations fpde for example systems biology chemistry and biochemistry applications due to anomalous diffusion effects in constrained environments however effective numerical methods and numerical analysis for fpde are still in their infancy in this paper we consider a fractional reaction subdiffusion equation fr subde in which both the motion and the reaction terms are affected by the subdiffusive character of the process using the relationship between the riemann liouville and grunwald letnikov definitions of fractional derivatives an implicit and an explicit difference methods for the fr subde are presented the stability and the convergence of the two numerical methods are investigated by a fourier analysis the solvability of the implicit finite difference method is also proved the high accuracy algorithm is structured using richardson extrapolation finally a comparison between the exact solution and the two numerical solutions is given the numerical results are in excellent agreement with our theoretical analysis