benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

discontinuous spectral element methods for time and space fractional advection equations we develop spectral element methods for a time and space fractional advection equation of the form prescript 0 mathcal d tau t u x t theta prescript 0 mathcal d nu x u x t f x t of order tau in 0 1 nu in 0 1 subject to dirichlet initial boundary conditions we present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms instead of employing traditional low order time integration methods to this end we first develop a petrov galerkin in time and discontinuous galerkin in space pg dg method where we carry out the time integration using a single time domain spectral method sm and we perform the spatial discretization using the discontinuous spectral hp element method dsem this scheme also leads to a more efficient time integration when the time derivative is integer order i e tau 1 we develop the sm dsem scheme based on a new spectral theory for fractional sturm liouville problems

finite element method for nonlinear riesz space fractional diffusion equations on irregular domains in this paper we consider two dimensional riesz space fractional diffusion equations with nonlinear source term on convex domains applying galerkin finite element method in space and backward difference method in time we present a fully discrete scheme to solve riesz space fractional diffusion equations our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes which can help us to deal with space fractional terms on any convex domain the stability and convergence of the scheme are also discussed numerical examples are given to verify accuracy and stability of our scheme a finite element method is developed to solve 2d space fractional diffusion equations with nonlinear source term we develop a finite element method for fpdes on irregular domain we obtain explicit expressions for fractional derivatives of shape functions we give the details on how to compute fractional stiffness matrix fractional fitzhugh nagumo model is solved on circular domain

finite difference scheme for a fractional telegraph equation with generalized fractional derivative terms abstract in this paper a finite difference scheme is presented for the generalized time fractional telegraph equation gtfte defined using generalized fractional derivative gfd terms introduced recently the generalization of fractional derivatives is done by introducing scale and weight functions and for their particular choices gfd reduces to caputo and riemann liouville derivatives we present the solution behaviour of the gtfte by changing the weight and scale functions in gfd the convergence and the stability of the finite difference scheme fds are also presented and for the numerical simulation of the fds we consider examples which validate our numerical method

jacobi collocation method for the approximate solution of some fractional order riccati differential equations with variable coefficients abstract this paper presents a computational method for the approximate solution of arbitrary order non linear fractional riccati differential equations with variable coefficients proposed computational method is a combination of the operational matrix of integration method and the collocation method associated with the jacobi polynomials convergence analysis of the proposed method is provided numerical results for different fractional orders of the riccati differential equations are discussed figures and tables are used to show the numerical results derived from the proposed computational method for particular cases of jacobi polynomials such as the legendre polynomials the chebyshev polynomials of the second kind the chebyshev polynomials of the third kind the chebyshev polynomial of the fourth kind and the gegenbauer or ultraspherical polynomials numerical results from the proposed methods are compared from those derived by using the existing analytical and numerical methods it is observed that the results from the proposed method are more accurate maximum absolute error and the root mean square error tables are given for all five kinds of polynomials for comparison purposes

a globally convergent version of the polak ribiere conjugate gradient method in this paper we propose a new line search algorithm that ensures global convergence of the polak ribiere conjugate gradient method for the unconstrained minimization of nonconvex differentiable functions in particular we show that with this line search every limit point produced by the polak ribiere iteration is a stationary point of the objective function moreover we define adaptive rules for the choice of the parameters in a way that the first stationary point along a search direction can be eventually accepted when the algorithm is converging to a minimum point with positive definite hessian matrix under strong convexity assumptions the known global convergence results can be reobtained as a special case from a computational point of view we may expect that an algorithm incorporating the step size acceptance rules proposed here will retain the same good features of the polak ribiere method while avoiding pathological situations

a nonlinear conjugate gradient method with a strong global convergence property conjugate gradient methods are widely used for unconstrained optimization especially large scale problems the strong wolfe conditions are usually used in the analyses and implementations of conjugate gradient methods this paper presents a new version of the conjugate gradient method which converges globally provided the line search satisfies the standard wolfe conditions the conditions on the objective function are also weak being similar to those required by the zoutendijk condition

on the ground states and dynamics of space fractional nonlinear schrodinger gross pitaevskii equations with rotation term and nonlocal nonlinear interactions in this paper we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional schrodinger equation fse with a rotation term and nonlocal nonlinear interactions in particular a newly developed gaussian sum gausum solver is used for the nonlocal interaction evaluation 31 to compute the ground states we integrate the preconditioned krylov subspace pseudo spectral method 4 and the gausum solver for the dynamics simulation using the rotating lagrangian coordinates transform 14 we first reformulate the fse into a new equation without rotation then a time splitting pseudo spectral scheme incorporated with the gausum solver is proposed to simulate the new fse in parallel to the numerical schemes we also prove some existence and nonexistence results for the ground states dynamical laws of some standard quantities including the mass energy angular momentum and the center of mass are stated the ground states properties with respect to the fractional order and or rotating frequencies dynamics involving decoherence and turbulence together with some interesting phenomena are reported

finite difference finite element method for two dimensional space and time fractional bloch torrey equations in this paper a class of two dimensional space and time fractional bloch torrey equations 2d stfbtes are considered some definitions and properties of fractional derivative spaces are presented by finite difference method and galerkin finite element method a semi discrete variational formulation for 2d stfbtes is obtained the stability and convergence of the semi discrete form are discussed then a fully discrete scheme of 2d stfbtes is derived and the convergence is investigated finally some numerical examples based on linear piecewise polynomials and quadratic piecewise polynomials are given to prove the correctness of our theoretical analysis

numerical methods of solutions of boundary value problems for the multi term variable distributed order diffusion equation solutions of the dirichlet and robin boundary value problems for the multi term variable distributed order diffusion equation are studied a priori estimates for the corresponding differential and difference problems are obtained by using the method of the energy inequalities the stability and convergence of the difference schemes follow from a priory estimates the credibility of the obtained results is verified by performing numerical calculations for test problems

regularization of inverse problems by two point gradient methods in banach spaces in this paper we propose and analyze a two point gradient method for solving inverse problems in banach spaces which is based on the landweber iteration and an extrapolation strategy the method allows to use non smooth penalty terms including the l 1 like and the total variation like penalty functionals which are significant in reconstructing special features of solutions such as sparsity and piecewise constancy in practical applications the design of the method involves the choices of the step sizes and the combination parameters which are carefully discussed numerical simulations are presented to illustrate the effectiveness of the proposed method

a fast finite volume method for conservative space fractional diffusion equations in convex domains we develop a fast finite volume method for variable coefficient conservative space fractional diffusion equations in convex domains via a volume penalization approach the method has an optimal storage and an almost linear computational complexity the method retains second order accuracy without requiring a richardson extrapolation numerical results are presented to show the utility of the method

inexact newton landweber iteration in banach spaces with nonsmooth convex penalty terms by making use of tools from convex analysis we formulate an inexact newton landweber iteration method for solving nonlinear inverse problems in banach spaces the method consists of two components an outer newton iteration and an inner scheme the inner scheme provides increments by applying landweber iteration with nonsmooth uniformly convex penalty terms to local linearized equations the outer iteration is then terminated by the discrepancy principle detailed convergence analysis is present under standard conditions on the nonlinearity finally numerical simulations are reported to test the performance of the method the pdf has been replaced

a new conjugate gradient method with guaranteed descent and an efficient line search a new nonlinear conjugate gradient method and an associated implementation based on an inexact line search are proposed and analyzed with exact line search our method reduces to a nonlinear version of the hestenes stiefel conjugate gradient scheme for any inexact line search our scheme satisfies the descent condition bf g k sf t bf d k le frac 7 8 bf g k 2 moreover a global convergence result is established when the line search fulfills the wolfe conditions a new line search scheme is developed that is efficient and highly accurate efficiency is achieved by exploiting properties of linear interpolants in a neighborhood of a local minimizer high accuracy is achieved by using a convergence criterion which we call the approximate wolfe conditions obtained by replacing the sufficient decrease criterion in the wolfe conditions with an approximation that can be evaluated with greater precision in a neighborhood of a local minimum than the usual sufficient decrease criterion numerical comparisons are given with both l bfgs and conjugate gradient methods using the unconstrained optimization problems in the cute library

compact difference schemes for the modified anomalous fractional sub diffusion equation and the fractional diffusion wave equation in this paper compact finite difference schemes for the modified anomalous fractional sub diffusion equation and fractional diffusion wave equation are studied schemes proposed previously can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two based on the idea of weighted and shifted grunwald difference operator we establish schemes with temporal and spatial accuracy order equal to two and four respectively

an extension of the fletcher reeves method to linear equality constrained optimization problem in this paper combining the feasible direction method with the conjugate gradient method we propose a feasible fletcher reeves conjugate gradient method for solving linear equality constrained optimization problem the directions generated by the method are feasible and descent for the objective function under mild conditions we show that the proposed method with exact line search is globally convergent numerical experiments are also given to show the efficiency of the method

a compact difference scheme for a two dimensional fractional klein gordon equation with neumann boundary conditions in this paper a high order finite difference scheme for a two dimensional fractional klein gordon equation subject to neumann boundary conditions is proposed the difficulty induced by the nonlinear term and the neumann conditions is carefully handled in the proposed scheme the stability and convergence of the finite difference scheme are analyzed using the matrix form of the scheme numerical examples are provided to demonstrate the theoretical results

crank nicolson adi galerkin finite element method for two dimensional fractional fitzhugh nagumo monodomain model in this paper a two dimensional fractional fitzhugh nagumo monodomain model 2d ffhnmm with zero dirichlet boundary condition is considered the model consists of a coupled two dimensional space fractional nonlinear reaction diffusion model 2d sfnrdm and an ordinary differential equation the 2d sfnrdm and ordinary differential equation are decoupled at each time step a new crank nicolson alternating direction implicit adi galerkin finite element method for the 2d sfnrdm is developed the stability and convergence of the numerical method are discussed finally some numerical examples on 2d sfnrdm and 2d ffhnmm are given for verification of our theoretical analysis

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

on the discrepancy principle for some newton type methods for solving nonlinear inverse problems we consider the computation of stable approximations to the exact solution x dagger of nonlinear ill posed inverse problems f x y with nonlinear operators f x y between two hilbert spaces x and y by the newton type methods x k 1 delta x 0 g alpha k left f x k delta f x k delta right f x k delta left f x k delta y delta f x k delta x k delta x 0 right of y satisfying y delta y le delta with a given small noise level delta 0 we terminate the iteration by the discrepancy principle in which the stopping index k delta is determined as the first integer such that f x k delta delta y delta le tau delta f x k delta y delta quad 0 le k k delta converges to x dagger as delta rightarrow 0 and establish various order optimal convergence rate results it is remarkable that we even can show the order optimality under merely the lipschitz condition on the frechet derivative f of f if x 0 x dagger is smooth enough

a general convergence analysis of some newton type methods for nonlinear inverse problems we consider the methods x n 1 delta x n delta g alpha n f x n delta f x n delta f x n delta f x n delta y delta for solving nonlinear ill posed inverse problems f x y using the only available noise data y delta satisfying y delta y leq delta with a given small noise level delta 0 we terminate the iteration by the discrepancy principle f x n delta delta y delta leq tau delta 1 under certain conditions on alpha n and f we prove for a large class of spectral filter functions g alpha the convergence of x n delta delta to a true solution as delta rightarrow0 moreover we derive the order optimal rates of convergence when certain holder source conditions hold numerical examples are given to test the theoretical results

the barzilai and borwein gradient method for the large scale unconstrained minimization problem the barzilai and borwein gradient method for the solution of large scale unconstrained minimization problems is considered this method requires few storage locations and very inexpensive computations furthermore it does not guarantee descent in the objective function and no line search is required recently the global convergence for the convex quadratic case has been established however for the nonquadratic case the method needs to be incorporated in a globalization scheme in this work a nonmonotone line search strategy that guarantees global convergence is combined with the barzilai and borwein method this strategy is based on the nonmonotone line search technique proposed by grippo lampariello and lucidi siam j numer anal 23 1986 pp 707 716 numerical results to compare the behavior of this method with recent implementations of the conjugate gradient method are presented these results indicate that the global barzilai and borwein method may allow some significant reduction in the number of line searches and also in the number of gradient evaluations

inexact newton regularization methods in hilbert scales we consider a class of inexact newton regularization methods for solving nonlinear inverse problems in hilbert scales under certain conditions we obtain the order optimal convergence rate result

fast algorithm based on tt m fe system for space fractional allen cahn equations with smooth and non smooth solutions abstract in this article a fast algorithm based on time two mesh tt m finite element fe scheme which aims at solving nonlinear problems quickly is considered to numerically solve the nonlinear space fractional allen cahn equations with smooth and non smooth solutions the implicit second order scheme containing both implicit crank nicolson scheme and second order backward difference method is applied to time direction a fast tt m method is used to increase the speed of calculation and the fe method is developed to approximate the spacial direction the tt m fe algorithm includes the following main computing steps firstly a nonlinear implicit second order fe scheme on the time coarse mesh c is solved by a nonlinear iterative method secondly based on the chosen initial iterative value a linearized fe system on time fine mesh c is solved where some useful coarse numerical solutions are found by the lagrange s interpolation formula the analysis for both stability and a priori error estimates are made in detail finally three numerical examples with smooth and non smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations from which it is easy to find that the computing time can be saved

the dimension free structure of nonhomogeneous random matrices let x be a symmetric random matrix with independent but non identically distributed centered gaussian entries we show that begin aligned mathbf e vert x vert s p asymp mathbf e bigg bigg sum i bigg sum j x ij 2 bigg p 2 bigg 1 p bigg end aligned for any 2 le p le infty where s p denotes the p schatten class and the constants are universal the right hand side admits an explicit expression in terms of the variances of the matrix entries this settles in the case p infty a conjecture of the first author and provides a complete characterization of the class of infinite matrices with independent gaussian entries that define bounded operators on ell 2 along the way we obtain optimal dimension free bounds on the moments mathbf e vert x vert s p p 1 p that are of independent interest we develop further extensions to non symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non gaussian entries that arise for example in the study of random graphs and in applied mathematics

the convergence rate of a restart mfr conjugate gradient method with inexact line search abstract in this paper we investigate the convergence rate of the modified fletcher reeves mfr method proposed by zhang et al l zhang w j zhou d h li global convergence of a modified fletcher revves conjugate gradient method with armijo type line search numer math 104 2006 561 572 under reasonable conditions we show that the mfr method with some inexact line search will be n step superlinearly and even quadratically convergent if some restart technique is used some numerical results are also reported to verify the theoretical results

global convergence properties of conjugate gradient methods for optimization this paper explores the convergence of nonlinear conjugate gradient methods without restarts and with practical line searches the analysis covers two classes of methods that are globally convergent on smooth nonconvex functions some properties of the fletcher reeves method play an important role in the first family whereas the second family shares an important property with the polak ribiere method numerical experiments are presented

convergence properties of algorithms for nonlinear optimization this paper reviews some of the most successful methods for unconstrained constrained and nondifferentiable optimization calculations particular attention is given to the contribution that theoretical analysis has made to the development of algorithms it seems that practical considerations provide the main new ideas and that subsequent theoretical studies give improvements to algorithms coherence to the subject and better understanding

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

a mixed finite element method for a time fractional fourth order partial differential equation in this paper a numerical theory based on the mixed finite element method for a time fractional fourth order partial differential equation pde is presented and analyzed an auxiliary variable email protected is introduced then the fourth order equation can be split into the coupled system of two second order equations the time caputo fractional derivative is discretized by a finite difference method and the spatial direction is approximated by the mixed finite element method the stabilities based on a priori analysis for two variables are discussed and some a priori error estimates in l 2 norm for the scalar unknown u and the variable email protected are derived respectively moreover an a priori error result in h 1 norm for the scalar unknown u also is proved for verifying the theoretical analysis a numerical test is made by using matlab procedure

finite element method for the space and time fractional fokker planck equation we develop the finite element method for the numerical resolution of the space and time fractional fokker planck equation which is an effective tool for describing a process with both traps and flights the time fractional derivative of the equation is used to characterize the traps and the flights are depicted by the space fractional derivative the stability and error estimates are rigorously established and we prove that the convergent order is o k 2 alpha h mu where k is the time step size and h the space step size numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims

an l p theory of sparse graph convergence ii ld convergence quotients and right convergence we extend the lplp theory of sparse graph limits which was introduced in a companion paper by analyzing different notions of convergence under suitable restrictions on node weights we prove the equivalence of metric convergence quotient convergence microcanonical ground state energy convergence microcanonical free energy convergence and large deviation convergence our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree while the proofs require new techniques based on uniform upper regularity examples to which our theory applies include stochastic block models power law graphs and sparse versions of ww random graphs

a convergence analysis of the landweber iteration for nonlinear ill posed problems in this paper we prove that the landweber iteration is a stable method for solving nonlinear ill posed problems for perturbed data with noise level delta we propose a stopping rule that yields the convergence rate o delta 1 2 under appropriate conditions we illustrate these conditions for a few examples

on a heuristic stopping rule for the regularization of inverse problems by the augmented lagrangian method in this paper we propose a heuristic stopping rule of hanke raus type for the regularization of linear ill posed inverse problems by the augmented lagrangian method this stopping rule requires no information on the noise level under certain source conditions we derive a posteriori error estimates in term of bregman distance by imposing certain conditions on the noise data we establish convergence results numerical results are presented to illustrate the performance

nonstationary iterated tikhonov regularization in banach spaces with uniformly convex penalty terms we consider the nonstationary iterated tikhonov regularization in banach spaces which defines the iterates via minimization problems with uniformly convex penalty term the penalty term is allowed to be non smooth to include l 1 l 1 and total variation tv like penalty functionals which are significant in reconstructing special features of solutions such as sparsity and discontinuities in practical applications we present the detailed convergence analysis and obtain the regularization property when the method is terminated by the discrepancy principle in particular we establish the strong convergence and the convergence in bregman distance which sharply contrast with the known results that only provide weak convergence for a subsequence of the iterative solutions some numerical experiments on linear integral equations of first kind and parameter identification in differential equations are reported

error estimate of finite element finite difference technique for solution of two dimensional weakly singular integro partial differential equation with space and time fractional derivatives abstract in the current investigation an error estimate has been proposed to solve the two dimensional weakly singular integro partial differential equation with space and time fractional derivatives based on the finite element finite difference scheme the time and space derivatives are based on the riemann liouville and riesz fractional derivatives respectively at first the temporal variable has been discretized by a second order difference scheme and then the space variable has been approximated by the finite element method fem the analytical study shows that the presented scheme is unconditionally stable and convergent finally some examples have been introduced to confirm the theoretical results and efficiency of the proposed technique

a barzilai and borwein scaling conjugate gradient method for unconstrained optimization problems in this paper we combine the conjugate gradient method with the barzilai and borwein gradient method and propose a barzilai and borwein scaling conjugate gradient method for nonlinear unconstrained optimization problems the new method does not require to compute and store matrices associated with hessian of the objective functions and has an advantage of less computational efforts moreover the descent direction property and the global convergence are established when the line search fulfills the wolfe conditions the limited numerical experiments and comparisons show that the proposed algorithm is potentially efficient

alternating direction implicit galerkin finite element method for the two dimensional fractional diffusion wave equation new numerical techniques are presented for the solution of the two dimensional fractional diffusion wave equation with a time fractional derivative of order 1 2 in these methods galerkin finite element is used for the spatial discretization and for the time stepping new alternating direction implicit adi method based on the crank nicolson method are considered the unconditional stability and l 2 norm convergence of the adi scheme are proved rigorously numerical results are presented to support our theoretical analysis

alternating direction method of multipliers for linear inverse problems in this paper we propose an iterative method using alternating direction method of multipliers admm strategy to solve linear inverse problems in hilbert spaces with a general convex penalty term when the data is given exactly we give a convergence analysis of our admm algorithm without assuming the existence of a lagrange multiplier in case the data contains noise we show that our method is a regularization method as long as it is terminated by a suitable stopping rule various numerical simulations are performed to test the efficiency of the method

sparse exchangeable graphs and their limits via graphon processes in a recent paper caron and fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in r here we show that by generalizing the classical definition of graphons as functions over probability spaces to functions over finite measure spaces we can model a large family of exchangeable graphs including the caron fox graphs and the traditional exchangeable dense graphs as special cases explicitly modelling the underlying space of features by a finite measure space s s and the connection probabilities by an integrable function w s s 0 1 we construct a random family gt t 0 of growing graphs such that the vertices of gt are given by a poisson point process on s with intensity t with two points x y of the point process connected with probability w x y we call such a random family a graphon process we prove that a graphon process has convergent subgraph frequencies with possibly infinite limits and that in the natural extension of the cut metric to our setting the sequence converges to the generating graphon we also show that the underlying graphon is identifiable only as an equivalence class over graphons with cut distance zero more generally we study metric convergence for arbitrary not necessarily random sequences of graphs and show that a sequence of graphs has a convergent subsequence if and only if it has a subsequence satisfying a property we call uniform regularity of tails finally we prove that every graphon is equivalent to a graphon on r equipped with lebesgue measure

global convergence of a modified fletcher reeves conjugate gradient method with armijo type line search in this paper we are concerned with the conjugate gradient methods for solving unconstrained optimization problems it is well known that the direction generated by a conjugate gradient method may not be a descent direction of the objective function in this paper we take a little modification to the fletcher reeves fr method such that the direction generated by the modified method provides a descent direction for the objective function this property depends neither on the line search used nor on the convexity of the objective function moreover the modified method reduces to the standard fr method if line search is exact under mild conditions we prove that the modified method with armijo type line search is globally convergent even if the objective function is nonconvex we also present some numerical results to show the efficiency of the proposed method

galerkin finite element method for two dimensional riesz space fractional diffusion equations abstract in this article a class of two dimensional riesz space fractional diffusion equations is considered some fractional spaces are established and some equivalences between fractional derivative spaces and fractional sobolev space are presented by the galerkin finite element method and backward difference method a fully discrete scheme is obtained according to lax milgram theorem the existence and uniqueness of the solution to the fully discrete scheme are investigated the stability and convergence of the scheme are also derived finally some numerical examples are given for verification of our theoretical analysis

fractional difference finite element approximations for the time space fractional telegraph equation abstract in this paper we study the numerical solution of the time space fractional order fractional for simplicity telegraph equation which can be used in signal analysis for transmission and propagation of electrical signals also the modeling of the reaction diffusion and the random walk of suspension flows and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the spatial riemann liouville fractional derivative with order 1 1 2 and the finite difference schemes for the temporal caputo derivatives with orders 1 2 1 and 2 are analyzed respectively results on the existence and uniqueness of the solution the numerical stability and the error estimates are displayed in details numerical examples are included to confirm the theoretical analysis

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

error analysis of a finite element method for the space fractional parabolic equation we consider an initial boundary value problem for a one dimensional fractional order parabolic equation with a space fractional derivative of riemann liouville type and order alpha in 1 2 we study a spatial semidiscrete scheme using the standard galerkin finite element method with piecewise linear finite elements as well as fully discrete schemes based on the backward euler method and the crank nicolson method error estimates in the l 2 d and h alpha 2 d norm are derived for the semidiscrete scheme and in the l 2 d norm for the fully discrete schemes these estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data extensive numerical results are presented to illustrate the theoretical results

a two grid mixed finite element method for a nonlinear fourth order reaction diffusion problem with time fractional derivative in this article we develop a two grid algorithm based on the mixed finite element mfe method for a nonlinear fourth order reaction diffusion equation with the time fractional derivative of caputo type we formulate the problem as a nonlinear fully discrete mfe system where the time integer and fractional derivatives are approximated by finite difference methods and the spatial derivatives are approximated by the mfe method to solve the nonlinear mfe system more efficiently we propose a two grid algorithm which is composed of two steps we first solve a nonlinear mfe system on a coarse grid by nonlinear iterations then solve the linearized mfe system on the fine grid by newton iteration numerical stability and optimal error estimate o k 2 h r 1 h 2 r 2 in l 2 norm are proved for our two grid scheme where k h and h are the time step size coarse grid mesh size and fine grid mesh size respectively we implement the two grid algorithm and present the numerical results justifying our theoretical error estimate the numerical tests also show that the two grid method is much more efficient than solving the nonlinear mfe system directly

on the convergence of a regularizing levenberg marquardt scheme for nonlinear ill posed problems in this note we study the convergence of the levenberg marquardt regularization scheme for nonlinear ill posed problems we consider the case that the initial error satisfies a source condition our main result shows that if the regularization parameter does not grow too fast not faster than a geometric sequence then the scheme converges with optimal convergence rates our analysis is based on our recent work on the convergence of the exponential euler regularization scheme hochbruck et al in inverse probl 25 7 075009 2009

nonlinear large deviations abstract we present a general technique for computing large deviations of nonlinear functions of independent bernoulli random variables the method is applied to compute the large deviation rate functions for subgraph counts in sparse random graphs previous technology based on szemeredi s regularity lemma works only for dense graphs applications are also made to exponential random graphs and three term arithmetic progressions in random sets of integers

new properties of a nonlinear conjugate gradient method this paper provides several new properties of the nonlinear conjugate gradient method in 5 firstly the method is proved to have a certain self adjusting property that is independent of the line search and the function convexity secondly under mild assumptions on the objective function the method is shown to be globally convergent with a variety of line searches thirdly we find that instead of the negative gradient direction the search direction defined by the nonlinear conjugate gradient method in 5 can be used to restart any optimization method while guaranteeing the global convergence of the method some numerical results are also presented

on a regularized levenberg marquardt method for solving nonlinear inverse problems we consider a regularized levenberg marquardt method for solving nonlinear ill posed inverse problems we use the discrepancy principle to terminate the iteration under certain conditions we prove the convergence of the method and obtain the order optimal convergence rates when the exact solution satisfies suitable source wise representations

on an a posteriori parameter choice strategy for tikhonov regularization of nonlinear ill posed problems in the study of the choice of the regularization parameter for tikhonov regularization of nonlinear ill posed problems scherzer engl and kunisch proposed an a posteriori strategy in 1993 to prove the optimality of the strategy they imposed many very restrictive conditions on the problem under consideration their results are difficult to apply to concrete problems since one can not make sure whether their assumptions are valid in this paper we give a further study on this strategy and show that tikhonov regularization is order optimal for each 0 nu le 2 with the regularization parameter chosen according to this strategy under some simple and easy checking assumptions this paper weakens the conditions needed in the existing results and provides a theoretical guidance to numerical experiments

a fast second order accurate method for a two sided space fractional diffusion equation with variable coefficients in this paper we consider a type of fractional diffusion equation fde with variable coefficients on a finite domain firstly we utilize a second order scheme to approximate the riemannliouville fractional derivative and present the finite difference scheme specifically we discuss the cranknicolson scheme and solve it in matrix form secondly we prove the stability and convergence of the scheme and conclude that the scheme is unconditionally stable and convergent with the second order accuracy of o 2 h2 furthermore we develop a fast accurate iterative method for the cranknicolson scheme which only requires storage of o m and computational cost of o mlogm while retaining the same accuracy and approximation property as gauss elimination where m 1 h is the partition number in space direction finally several numerical examples are given to show the effectiveness of the numerical method and the results are in excellent agreement with the theoretical analysis

three term conjugate gradient method for the convex optimization problem over the fixed point set of a nonexpansive mapping many constrained sets in problems such as signal processing and optimal control can be represented as a fixed point set of a certain nonexpansive mapping and a number of iterative algorithms have been presented for solving a convex optimization problem over a fixed point set this paper presents a novel gradient method with a three term conjugate gradient direction that is used to accelerate conjugate gradient methods for solving unconstrained optimization problems it is guaranteed that the algorithm strongly converges to the solution to the problem under the standard assumptions numerical comparisons with the existing gradient methods demonstrate the effectiveness and fast convergence of this algorithm

globally convergent modified perry s conjugate gradient method abstract conjugate gradient methods are probably the most famous iterative methods for solving large scale optimization problems in scientific and engineering computation characterized by the simplicity of their iteration and their low memory requirements in this paper we propose a new conjugate gradient method which is based on the mbfgs secant condition by modifying perry s method our proposed method ensures sufficient descent independent of the accuracy of the line search and it is globally convergent under some assumptions numerical experiments are also presented

a nonstationary accelerating alternating direction method for frame based poissonian image deblurring abstract poissonian image deblurring problem has been an attractive research subject in many areas of imaging science and engineering in order to obtain accurate restorations one has to run several times to manually select a good value of a regularization parameter in many models in this paper we first propose an optimization model by combining a new data fitting term with a frame based analysis regularizer and a nonnegative constraint a discrepancy principle based nonstationary accelerating alternating direction method called dp naadm is then proposed to solve the optimization model without estimating the regularization parameter numerical results show that dp naadm provides competitive restoration results compared to the well known poisson image deconvolution by augmented lagrangian method called pidal also dp naadm provides better restorations in terms of both accuracy and visual quality than the discrepancy principle based pidal method

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

a new difference scheme for the time fractional diffusion equation abstract in this paper we construct a new difference analog of the caputo fractional derivative called the l 2 1 formula the basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered stability of the suggested schemes and also their convergence in the grid l 2 norm with the rate equal to the order of the approximation error are proved the obtained results are supported by the numerical calculations carried out for some test problems

some high order difference schemes for the space and time fractional bloch torrey equations in this paper several difference schemes are proposed for both one dimensional and two dimensional space and time fractional bloch torrey equations the spatial second order scheme and the spatial fourth order compact scheme are established respectively the obtained schemes can achieve the global second order numerical accuracy in time the unique solvability unconditional stability and convergence of the proposed schemes are proved by the energy method two adi schemes are also discussed for the two dimensional problem numerical examples are given to verify the numerical accuracy and efficiency of the difference schemes

a class of modified fr conjugate gradient method and applications to non negative matrix factorization conjugate gradient method as an efficient method is used to solve unconstrained optimization problems in this paper we propose a class of modified fletcherreeves conjugate gradient method with armijo type line search which generates the direction is descent for the objective function under mild conditions we give that the proposed method is descent and globally convergent moreover the method is applied to nonnegative matrix factorization and the experimental results demonstrate the validity of the proposed method

on convergence rates of inexact newton regularizations reginn is an algorithm of inexact newton type for the regularization of nonlinear ill posed problems inverse problems 15 1999 pp 309 327 in the present article convergence is shown under weak smoothness assumptions source conditions moreover convergence rates are established some computational illustrations support the theoretical results

θ schemes for finite element discretization of the space time fractional diffusion equations the numerical solution of a space time fractional diffusion equation used to model the anomalous diffusion is considered spatial discretization is effected using a finite element method whereas the scheme is used for temporal discretization the fully discrete scheme is analyzed for all 0 1 to determine conditional and unconditional stability regimes for the scheme and also to obtain error estimates for the approximate solution the analysis is facilitated by making use of a variational formulation of the equations that is based on a recently developed nonlocal calculus one dimensional numerical examples are provided that illustrate the theoretical stability and convergence results

exponentially accurate spectral and spectral element methods for fractional odes current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy here we present different methods for fractional odes that lead to exponentially fast decay of the error first we develop a petrov galerkin pg spectral method for fractional initial value problems fivps of the form d t 0 u t f t and fractional final value problems ffvps d t t u t g t where 0 1 subject to dirichlet initial final conditions these schemes are developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently developed in 1 specifically we obtain solutions to fivps and ffvps in terms of the new fractional non polynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of second kind fslp ii subsequently we develop a discontinuous spectral method dsm of petrov galerkin sense for the aforementioned fivps and ffvps where the basis functions do not satisfy the initial final conditions finally we extend the dsm scheme to a discontinuous spectral element method dsem for efficient longer time integration and adaptive refinement in these discontinuous schemes we employ the asymptotic eigensolutions to fslp i ii which are of jacobi polynomial forms as basis and test functions our numerical tests confirm the exponential algebraic convergence respectively in p and h refinements for various test cases with integer and fractional order solutions

a weak galerkin finite element method for high dimensional time fractional diffusion equation abstract this paper is concerned with the application of a weak galerkin finite element method wg fem to the time fractional diffusion equation the wg fem with l1 formula and the fast evaluation scheme based on wg fem are designed the optimal convergence rates for both semi discrete and fully discrete wg fem schemes are obtained and the stability analysis for the semi discrete wg fem is derived numerical experiments are implemented to verify the theoretical results

a finite difference finite element technique with error estimate for space fractional tempered diffusion wave equation abstract an efficient numerical technique is proposed to solve one and two dimensional space fractional tempered fractional diffusion wave equations the space fractional is based on the riemann liouville fractional derivative at first the temporal direction is discretized using a second order accurate difference scheme then a classic galerkin finite element is employed to obtain a full discrete scheme furthermore for the time discrete and the full discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method finally two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique

numerical algorithm for two dimensional fractional stokes first problem for a heated generalized second grade fluid with smooth and non smooth solution abstract in this paper we consider the numerical solution of the two dimensional fractional stokes first problem for a heated generalized second grade fluid the proposed method is based on the l1 finite difference scheme for the temporal direction while the legendre spectral method for the spatial direction numerical stability and convergence of the method are rigorously established at the mean time the case of non smooth solution is considered by adding some correction terms numerical experiments are included which verify the theoretical predictions

levenberg marquardt method in banach spaces with general convex regularization terms we propose a levenberg marquardt method with general uniformly convex regularization terms to solve nonlinear inverse problems in banach spaces which is an extension of the scheme proposed by hanke in inverse probl 13 79 95 1997 in hilbert space setting the method is so designed that it can be used to detect the features of the sought solutions such as sparsity or piecewise constancy it can also be used to deal with the situation that the data is contaminated by noise containing outliers by using tools from convex analysis in banach spaces we establish the convergence of the method numerical simulations are reported to test the performance of the method

upper tails and independence polynomials in random graphs abstract the upper tail problem in the erd s renyi random graph g g n p asks to estimate the probability that the number of copies of a graph h in g exceeds its expectation by a factor 1 chatterjee and dembo showed that in the sparse regime of p 0 as n with p n for an explicit h 0 this problem reduces to a natural variational problem on weighted graphs which was thereafter asymptotically solved by two of the authors in the case where h is a clique here we extend the latter work to any fixed graph h and determine a function c h such that for p as above and any fixed 0 the upper tail probability is exp c h o 1 n 2 p log 1 p where is the maximum degree of h as it turns out the leading order constant in the large deviation rate function c h is governed by the independence polynomial of h defined as p h x i h k x k where i h k is the number of independent sets of size k in h for instance if h is a regular graph on m vertices then c h is the minimum between 1 2 2 m and the unique positive solution of p h x 1

large deviations of subgraph counts for sparse erdős renyi graphs abstract for any fixed simple graph h v e and any fixed u 0 we establish the leading order of the exponential rate function for the probability that the number of copies of h in the erd s renyi graph g n p exceeds its expectation by a factor 1 u assuming n h p 1 with h 1 2 where 1 is the maximum degree of h this improves on a previous result of chatterjee and the second author who obtained h c e for a constant c 0 moreover for the case of cycle counts we can take as large as 1 2 we additionally obtain the sharp upper tail for schatten norms of the adjacency matrix as well as the sharp lower tail for counts of graphs for which sidorenko s conjecture holds as a key step we establish quantitative versions of szemeredi s regularity lemma and the counting lemma suitable for the analysis of random graphs in the large deviations regime

numerical solution of distributed order fractional differential equations by hybrid functions in this paper a new numerical method for solving the distributed fractional differential equations is presented the method is based upon hybrid functions approximation the properties of hybrid functions consisting of block pulse functions and bernoulli polynomials are presented the riemann liouville fractional integral operator for hybrid functions is introduced this operator is then utilized to reduce the solution of the distributed fractional differential equations to a system of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

finite element method for two dimensional space fractional advection dispersion equations the backward euler and crank nicolson galerkin fully discrete approximate schemes for two dimensional space fractional advection dispersion equations are established firstly we prove that the corresponding variational problem has a unique solution and the proposed fully discrete schemes are unconditionally stable whose solutions are all unique secondly the optimal error estimates are derived by use of properties of projection operator and fractional derivatives finally numerical examples demonstrate effectiveness of numerical schemes and confirm the theoretical analysis

an improved wei yao liu nonlinear conjugate gradient method for optimization computation in this paper we take a little modification to the wei yao liu nonlinear conjugate gradient method proposed by wei et al z wei s yao l liu the convergence properties of some new conjugate gradient methods appl math comput 183 2006 1341 1350 such that the modified method possesses better convergence properties in fact we prove that the modified method satisfies sufficient descent condition with greater parameter email protected 0 12 in the strong wolfe line search and converges globally for nonconvex minimization we also extend these results to the hestenes stiefel method and prove that the modified hs method is globally convergent for nonconvex functions with the standard wolfe conditions numerical results are reported by using some test problems in the cute library

a fourth order compact adi scheme for two dimensional nonlinear space fractional schrodinger equation in this paper a novel compact operator is derived for the approximation of the riesz derivative with order alpha in 1 2 the compact operator is proved with fourth order accuracy combining the compact operator in space discretization a linearized difference scheme is proposed for a two dimensional nonlinear space fractional schrodinger equation it is proved that the difference scheme is uniquely solvable stable and convergent with order o tau 2 h 4 where tau is the time step size h max h 1 h 2 and h 1 h 2 are space grid sizes in the x direction and the y direction respectively based on the linearized difference scheme a compact alternating direction implicit scheme is presented and analyzed numerical results demonstrate that the compact operator does not bring in extra computational cost but improves the accuracy of the scheme greatly

galerkin method with trigonometric basis on stable numerical differentiation abstract this paper considers the p p 1 2 3 order numerical differentiation on function y in 0 2 they are transformed into corresponding fredholm integral equation of the first kind computational schemes with analytic solution formulas are designed using galerkin method on trigonometric basis convergence and divergence are all analysed in corollaries 5 1 5 2 and a priori error estimate is uniformly obtained in theorem 6 1 7 1 7 2 therefore the algorithm achieves the optimal convergence rate o 2 2 1 1 2 or 1 with periodic sobolev source condition of order 2 p besides we indicate a noise independent a priori parameter choice when the function y possesses the form of k 0 p 1 a k t k k 1 n 1 b k cos k t k 1 n 2 c k sin k t b n 1 c n 2 0 in particular in numerical differentiations for functions above good filtering effect error approaches 0 is displayed with corresponding parameter choice in addition several numerical examples are given to show that even derivatives with discontinuity can be recovered well

on the iteratively regularized gauss newton method for solving nonlinear ill posed problems the iteratively regularized gauss newton method is applied to compute the stable solutions to nonlinear ill posed problems f x y when the data y is given approximately by y with y y in this method the iterative sequence x k is defined successively by x k 1 x k k i f x k f x k 1 f x k f x k y k x k x 0 where x 0 x 0 is an initial guess of the exact solution x and k is a given decreasing sequence of positive numbers admitting suitable properties when x k is used to approximate x the stopping index should be designated properly in this paper an a posteriori stopping rule is suggested to choose the stopping index of iteration and with the integer k determined by this rule it is proved that x k x c inf x k x k k 0 1 with a constant c independent of where x k denotes the iterative solution corresponding to the noise free case as a consequence of this result the convergence of x k is obtained and moreover the rate of convergence is derived when x 0 x satisfies a suitable source wise representation the results of this paper suggest that the iteratively regularized gauss newton method combined with our stopping rule defines a regularization method of optimal order for each 0 v 1 numerical examples for parameter estimation of a differential equation are given to test the theoretical results

an l p theory of sparse graph convergence i limits sparse random graph models and power law distributions we introduce and develop a theory of limits for sequences of sparse graphs based on lp graphons which generalizes both the existing l1 theory of dense graph limits and its extension by bollob as and riordan to sparse graphs without dense spots in doing so we replace the no dense spots hypothesis with weaker assumptions which allow us to analyze graphs with power law degree distributions this gives the rst broadly applicable limit theory for sparse graphs with unbounded average degrees in this paper we lay the foundations of the lp theory of graphons characterize convergence and develop corresponding random graph models while we prove the equivalence of several alternative metrics in a companion paper