on the convergence of additive and multiplicative splitting iterations for systems of linear equations we study convergence conditions for the additive and the multiplicative splitting iteration methods i e two generalizations of the additive and the multiplicative schwarz iterations for hermitian and non hermitian systems of linear equations under an algebraic setting theoretical analyses show that when the coefficient and the splitting matrices are hermitian or non hermitian but diagonalizable satisfying mild conditions both additive and multiplicative splitting iteration methods are convergent even if the coefficient matrix is indefinite

implicit finite difference approximation for time fractional diffusion equations time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical brownian motion model in this paper we develop an implicit unconditionally stable numerical method to solve the one dimensional linear time fractional diffusion equation formulated with caputo s fractional derivative on a finite slab several numerical examples of interest are also included

self similar transport in incomplete chaos particle chaotic dynamics along a stochastic web is studied for three dimensional hamiltonian flow with hexagonal symmetry in a plane two different classes of dynamical motion obtained by different values of a control parameter and corresponding to normal and anomalous diffusion have been considered and compared it is shown that the anomalous transport can be characterized by powerlike wings of the distribution function of displacement flights which are similar to l evy flights approximate trappings of orbits near the boundary layer of islands and anomalous behavior of the moments of a distribution function considered as a function of the number of the moment the main result is related to the self similar properties of different topological and dynamical characteristics of the particle motion this self similarity appears in the weierstrass like random walk process that is responsible for the anomalous transport exponent in the mean moment dependence on t this exponent can be expressed as a ratio of fractal dimensions of space and time sets in the weierstrass like process an explicit form for the expression of the anomalous transport exponent through the local topological properties of orbits has been given

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

preconditioned iterative methods for fractional diffusion equation in this paper we are concerned with numerical methods for the solution of initial boundary value problems of anomalous diffusion equations of order 1 2 the classical crank nicholson method is used to discretize the fractional diffusion equation and then the spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates two preconditioned iterative methods namely the preconditioned generalized minimal residual preconditioned gmres method and the preconditioned conjugate gradient for normal residual preconditioned cgnr method are proposed to solve relevant linear systems numerical experiments are given to illustrate the efficiency of the methods

leave one out bounds for kernel methods in this article we study leave one out style cross validation bounds for kernel methods the essential element in our analysis is a bound on the parameter estimation stability for regularized kernel formulations using this result we derive bounds on expected leave one out cross validation errors which lead to expected generalization bounds for various kernel algorithms in addition we also obtain variance bounds for leave one out errors we apply our analysis to some classification and regression problems and compare them with previous results

improving the sample complexity using global data we study the sample complexity of proper and improper learning problems with respect to different q loss functions we improve the known estimates for classes which have relatively small covering numbers in empirical l sub 2 spaces e g log covering numbers which are polynomial with exponent p 2 we present several examples of relevant classes which have a small fat shattering dimension and hence fit our setup the most important of which are kernel machines

numerical approximation of a time dependent nonlinear space fractional diffusion equation in this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity the analysis is performed for a general fractional order diffusion operator the nonlinear term studied is a product of the unknown function and a convolution operator of order 0 convergence of the approximation and a priori error estimates are given numerical computations are included which confirm the theoretical predictions

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

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

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

circulant and skew circulant splitting methods for toeplitz systems we study efficient iterative methods for toeplitz systems based on the circulant and skew circulant splitting cscs of the toeplitz matrix theoretical analysis show that if the circulant and the skew circulant splitting matrices are positive definite then the cscs method converges to the unique solution of the system of linear equations moreover we derive an upper bound of the contraction factor of the cscs iteration which is dependent solely on the spectra of the circulant and the skew circulant matrices involved numerical examples are presented to demonstrate the method

existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods given a nonsingular matrix a and a matrix t of the same order under certain very mild conditions there is a unique splitting a b c such that t b 1 c moreover all properties of the splitting are derived directly from the iteration matrix t these results do not hold when the matrix a is singular in this case given a matrix t and a splitting a b c such that t b 1 c there are infinitely many other splittings corresponding to the same matrices a and t and different splittings can have different properties for instance when t is nonnegative some of these splittings can be regular splittings while others can be only weak splittings analogous results hold in the symmetric positive semidefinite case given a singular matrix a not for all iteration matrices t there is a splitting corresponding to them necessary and sufficient conditions for the existence of such splittings are examined as an illustration of the theory developed the convergence of certain alternating iterations is analyzed different cases where the matrix is monotone singular and positive semi definite are studied

learning bounds for kernel regression using effective data dimensionality kernel methods can embed finite dimensional data into infinite dimensional feature spaces in spite of the large underlying feature dimensionality kernel methods can achieve good generalization ability this observation is often wrongly interpreted and it has been used to argue that kernel learning can magically avoid the curse of dimensionality phenomenon encountered in statistical estimation problems this letter shows that although using kernel representation one can embed data into an infinite dimensional feature space the effective dimensionality of this embedding which determines the learning complexity of the underlying kernel machine is usually small in particular we introduce an algebraic definition of a scale sensitive effective dimension associated with a kernel representation based on this quantity we derive upper bounds on the generalization performance of some kernel regression methods moreover we show that the resulting convergent rates are optimal under various circumstances

optimization of extrapolated cayley transform with non hermitian positive definite matrix abstract for the extrapolated cayley transform we give necessary and sufficient conditions for guaranteeing its convergence and contraction in the euclidean norm we derive upper bounds for the convergence and the contraction factors and compute the optimal parameters minimizing these upper bounds and the corresponding optimal values of these upper bounds numerical computations show that these upper bounds are reasonably sharp compared with the exact convergence and the exact contraction factors of the extrapolated cayley transform respectively

fractional order anisotropic diffusion for image denoising this paper introduces a new class of fractional order anisotropic diffusion equations for noise removal these equations are euler lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function so the proposed equations can be seen as generalizations of second order and fourth order anisotropic diffusion equations we use the discrete fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain it is one important aspect of the algorithm that it considers the input image as a periodic image to overcome this problem we use a folded algorithm by extending the image symmetrically about its borders finally we list various numerical results on denoising real images experiments show that the proposed fractional order anisotropic diffusion equations yield good visual effects and better signal to noise ratio

preconditioning techniques for diagonal times toeplitz matrices in fractional diffusion equations the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal times toeplitz matrices standard circulant preconditioners may not work for such toeplitz like linear systems the main aim of this paper is to propose and develop approximate inverse preconditioners for such toeplitz like matrices an approximate inverse preconditioner is constructed to approximate the inverses of weighted toeplitz matrices by circulant matrices and then combine them together row by row because of toeplitz structure both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast fourier transforms theoretically we show that the spectra of the resulting preconditioned matrices are clustered around one thus krylov subspace methods with the proposed preconditione

on adi like iteration method for fractional diffusion equations abstract the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of the identity matrix with two diagonal times toeplitz matrices in this paper the alternating direction implicit adi like method based on the classical adi iteration is proposed to solve the discretized linear system theoretical analyses show that the adi like iteration method is convergent each iteration of this method requires the solutions of two linear systems with hessenberg coefficient matrices these two systems can be solved effectively by hessenberg lu factorization numerical examples are presented to illustrate the effectiveness of the adi like iteration method

a new two step gradient type method for large scale unconstrained optimization in this paper we propose some improvements on a new gradient type method for solving large scale unconstrained optimization problems in which we use data from two previous steps to revise the current approximate hessian the new method which we considered resembles to that of barzilai and borwein bb method the innovation features of this approach consist in using approximation of the hessian in diagonal matrix form based on the modified weak secant equation rather than the multiple of the identity matrix in the bb method using this approach we can obtain a higher order accuracy of hessian approximation when compares to other existing bb type method by incorporating a simple monotone strategy the global convergence of the new method is achieved practical insights into the effectiveness of the proposed method are given by numerical comparison with the bb method and its variant

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

on inexact hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems we study theoretical properties of two inexact hermitian skew hermitian splitting ihss iteration methods for the large sparse non hermitian positive definite system of linear equations in the inner iteration processes we employ the conjugate gradient cg method to solve the linear systems associated with the hermitian part and the lanczos or conjugate gradient for normal equations cgne method to solve the linear systems associated with the skew hermitian part respectively resulting in ihss cg lanczos and ihss cg cgne iteration methods correspondingly theoretical analyses show that both ihss cg lanczos and ihss cg cgne converge unconditionally to the exact solution of the non hermitian positive definite linear system moreover their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the hermitian part but are less dependent on the spectrum of the skew hermitian part and are independent of the eigenvectors of the matrices involved optimal choices of the inner iteration steps in the ihss cg lanczos and ihss cg cgne iterations are discussed in detail by considering both global convergence speed and overall computation workload and computational efficiencies of both inexact iterations are analyzed and compared deliberately

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

a circulant preconditioner for fractional diffusion equations the implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable is employed to discretize fractional diffusion equations the resulting systems are toeplitz like and then the fast fourier transform can be used to reduce the computational cost of the matrix vector multiplication the preconditioned conjugate gradient normal residual method with a circulant preconditioner is proposed to solve the discretized linear systems the spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant hence the convergence rate of the proposed iterative algorithm is superlinear numerical experiments are carried out to demonstrate that our circulant preconditioner works very well even though for cases of variable diffusion coefficients