In this paper,continuous Runge-Kutta methods are applied to solve general nonlinear delay integro-differential equations,and a class of numerical algorithms is suggested.The stability of the numerical algorithms is studied,and it is proved that the numerical algorithms are asymptotically stable when the Runge-Kutta methods are (k,l)-algebraically stable and 0 < k < 1.Numerical experiments are used to validate the theoretical results and the effectiveness of the numerical algorithms.

The study of the fractional differential equations has a very long history, and is attracting increasing attention in recent years. As compared to the very limit theoretical work, signi cant progress has been made on numerical investigations. Several research groups have contributed to this progress. This paper has the objective to review the recent progress made in the theoretical and numerical studies of the fractional differential equations. We particularly focus on the development of high order numerical methods. The main content of the paper is to discuss the progress made in recent ten years on theoretical and numerical investigation of the three basic fractional equations: time fractional di usion equation, space fractional di usion equation, and time-space fractional di usion equation. We also provide some illustrative numerical examples to verify the accuracy and effciency of some selected numerical methods.

In this paper,we present a new iterative scheme with the convergence order 3p for solving the systems of nonlinear equations by using a modified two-step iterative algorithm as a predictor and Gauss-Legendre quadrature as a corrector.Numerical examples are given to show that the presented method outperforms the other ones.

In this paper, the numerical methods for the Riesz space fractional convection-dispersion equation is considered. By applying the weighted shifted Grünwald difference scheme to discretize the spatial fractional derivatives and the Crank-Nicolson difference scheme to discretize the temporal derivative in the Riesz space fractional convection-dispersion equation, respectively, we get the discrete results as a system of linear equations whose coefficient matrix is the sum of an identity matrix and two symmetric positive definite Toeplitz matrices. A $\tau$ preconditioner is constructed and the preconditioned conjugate gradient method is applied to solve the discrete system of linear equations. The spectral distribution of the preconditioned matrix is analyzed and the condition number of the preconditioned matrix is estimated. Numerical experiments show that the constructed preconditioner is very effective when it is combined in the preconditioned conjugate gradient method to solve the discrete system of linear equations.

Let p > 1 be an even number.In this paper,based on solving the equation x^p-1=0 we present the Newton iteration and Halley iteration to compute the polar decomposition of a nonsingular matrix,and prove their convergence properties.Numerical examples show that these algorithms are valid.

We live in the digital age, and data has become an essential part of our lives. Images are undoubtedly one of the most important types of data. Image inverse problems, including image denoising, deblurring, restoration, biomedical imaging, etc., are important areas in imaging science. The rapid development of computer technology has enabled us to use sophisticated mathematics and machine learning tools to design effective algorithms for image inverse problems. This paper mainly reviews three types of methods in image inverse problem, namely, applied and computational harmonic analysis method (represented by wavelets and wavelet frames), partial differential equation (PDE) method and deep learning method. We will review the modeling philosophies of these methods, explore the connections and differences among them, their advantages and disadvantages, and further discuss the feasibility and benefit of the integration of these methods.

In order to solve the Jacobi singular problem in the process of the iteration,in this paper,a new numerical continuation method is proposed.The Jacobi singularity is overcome by constructing the double-parameter homotopy operator,using controlled conditions and selecting appropriate parameter,and the convergence of this method is analyzed.Finally,the feasibility and superiority of this method is validated by numerical comparison,especially with the advantages of crossing the Jacobi singular problem (points,lines,surfaces).Thus,to an extent,this method can also solve the problem of being heavily dependent on the initial value,which is the shortcoming of the numerical continuation method.

For the special 3-by-3 block linear equations arising from the Galerkin finite element discretizations of elliptic PDE-constrained optimization problems,a preconditioner is proposed and the explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrix are derived.Numerical results show that the preconditioner is effectively used to accelerate the convergence rate of Krylov subspace methods and match well with the theoretical results as well.

We study the semidiscrete finite element approximation of the linear stochastic nonselfadjoint wave equation forced by additive noise.The results here are more general since the linear operator A does not need to be self-adjoint and we do not need information about eigenvalues and eigenfunctions of the linear operator A.In order to obtain the strong convergence error estimates,a standard finite element method for the spatial discretisation and the properties of a strongly continuous operator cosine function are used.The error estimates are applicable in the multi-dimensional case.

In this paper,a new family of (2n-1)-point binary non-stationary approximating subdivision schemes with shape parameter ω is presented with the help of the sine function.With the changing of n and ω,the theoretical analysis of support length and continuities of the schemes are also given.The corresponding stationary schemes include the methods given by Chaikin,Hormann,Dyn,Daniel and Hassan.With the same control points and the same continuities for the limit curves,comparisons with other methods are given.It shows that the new family of schemes can generate limit curves with better representability than the others.

In this paper,we study Jacobi spectral collocation method for solving the initial value problem (IVP) of a class of fractional multi-delay differential equations.The convergence of the method for this problem is obtained.Some illustrative examples verify our theoretical results successfully.The results of this paper may provide a new good choice for solving fractional delay differential equations.It is believed that these results will be helpful for the further researches on numerical solutions of fractional functional differential equations.

A triangular spectral element approximation scheme using generalized Koornwinder polynomials of index (-1,-1,-1) is proposed and analyzed for the biharmonic eigenvalue problem based on its mixed variational formulation.Further,on the basis of approximation theories of the H¹-and H₀¹-orthogonal spectral element projections oriented to the secondorder equations,error estimates are eventually established for our mixed triangular spectral element method (TSEM),which are optimal with respect to the mesh size h and sub-optimal with respected to the polynomial degree M.However,under certain assumption for the H₀²-orthogonal spectral element projection,we can also obtain an optimal estimate with respect to M.The approximation results of piecewise Koornwinder polynomials show that,under the measurement in some weighted Besov spaces,TSEM converges twice as fast as the hversion finite element method if the eigenfunction of the biharmonic operator has corner singularity.Finally,numerical results show the effectivity of our mixed TSEM and illustrate our theories as well.

In this paper we will considers the finite element method for variable coefficients parabolic equations of one space dimension using projection interpolation and Ritz-Volterra projection. A ultraconvergent corrected scheme for the derivative and displacement is obtained and proved directly. For finite element solution, we can obtain global hk+2 and hk+3 order ultraconvergent results for stress and displacement through correcting, respectively.

Phase retrieval is raised in many areas, such as imaging, optics and quantum tomography etc, which attracts many attentions of experts from different areas, such as computational mathematics and data sciences etc. The aim of this paper is to introduce the basic theoretical problems in phase retrieval and also introduce many algorithms for solving phase retrieval.

Let R∈Cn×n satisfying R = RH=R-1≠±In be a nontrivial generalized reflexive matrix. A∈Cn×n is said to be generalized centrosymmetric if RAR = A. The set of all n×n generalized centrosymmetric matrices is denoted by GCSCn×n. Let X1,Z1∈Cn×k1,Y1,W1∈Cn×l1,S = {A|‖AX1-Z1‖2+‖Y1HA-W1H‖2= min, A∈GCSCn×n}. The following problems are considered. Problem Ⅰ. Given Z2,X2∈ Cn×k2;Y2,W2 ∈Cn×l2, find A∈S such that where ‖·‖ is the Frobenius norm. Problem Ⅱ. Given A∈Cn×n, find A ∈ SE such that where SE is the solution set of Problem Ⅰ. The general form of the solution set SE of Problem Ⅰ is given. Sufficient and necessary conditions for matrix equations AX2=Z2,Y2HA = W2H having a solution A∈S are derived, and the general solutions are given. The expression of the solution to Problem Ⅱ is presented. A numerical example is provided.

In this paper, we prove that if the smoothness of the. solution of.a 2-order elliptic problem is raised, then the Lnh factor in pointwise error estimate of Wilson's rectangular element can be eliminated.

The explicit a posterior error estimators of upwind finite element for two dimensional evolution convection-diffusion equations are formulated and the above and below bound of estimator compared with true error are proved. An adaptive method based on these estimators is then proposed and numerical results show the efficiency of our method.

A class of rational cubic trigonometric Hermite interpolating splines with parameters is presented in this paper, which shares the same properties of standard cubic Hermite interpolating splines. The shape of the interpolation curves not only can be adjusted, but also more approximates the interpolated curves than standard cubic Hermite interpolating splines with taking different values of parameters. Moreover, by selecting proper control points, the spline curves can represent transcendantal curves exactly, such as tetracuspid and quadrifolium.

Parareal algorithm is a very efficient parallel in time computation methods. Compared with traditional parallel methods, this algorithm has the advantages of faster convergence, higher parallel performance and easy coding. This algorithm was first proposed by Lions, Maday and Turinici in 2001 and has attracted many researchers over the past few years. Recently, the application and theoretical analysis of this algorithm for stochastic computation have been investigated by some researchers. In this paper, we analyze the Mean-square stability of the Parareal algorithm in stochastic computation. The sufficient conditions under which the Parareal algorithm is stable are obtained and it is shown that: a) the algorithm converges superlinearly on any bounded time interval and b) the convergence speed is only linear on unbounded time intervals. Finally, numerical results are given to validate our theoretical conclusions.

In 1990, Pardoux and Peng obtained the existence and uniqueness result of the adapted solution for nonlinear backward stochastic differential equations. This result lays the foundation of the theory of forward backward stochastic differential equations. Since then, FBSDEs have been extensively studied, and have been found applications in many important fields, such as stochastic optimal control, partial differential equations, mathematical finance, risk measure, nonlinear mathematical expectation and so on. In this paper, we will review recent progresses for numerical methods for FBSDEs. We shall mainly introduce the integral and differential based numerical approximation methods, including both one-step and multi-step methods, and the corresponding numerical analysis and theoretical analysis will also be presented. It is worth to note that, by using the differential approximation method, one can propose strongly stable, highly accurate, and highly parallelized methods for solving fully coupled FBSDEs with the forward SDE solved by the Euler scheme. At the end of the paper, we briefly introduce some challenging problems on solving FBSDEs and some possible related applications.

In this paper, a traffic flow Aw-Rascle-Zhang(ARZ) model is studied with a proper orthogonal decomposition (POD) technique. A extrapolation reduced-order finite difference scheme (FDS) based on POD method with lower dimension is established. And a numerical example is used to verify that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the extrapolation reduced-order FDS based on POD method is feasible and efficient for finding numerical solutions for traffic flow equation.

A class of general inexact Uzawa methods for the solution of large and sparse saddle point problems are presented, which can not only cover many existing approaches, but also imply many new iteration scheme. Theoretical analyses give the convergence condition for new methods, as well as the choice of the optimal parameter matrices. Numerical results from discrete stokes problems by finite element method show that the new algorithm is efficient, and much faster than existing algorithms.

Three algorithms based on the bifurcation method are applied to computing the D₄(3) symmetric positive solutions to the boundary value problem of Henon equation. Taking r in Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation points are found via the extended systems on the branch of the D₄(3) symmetric positive solutions. Finally, other symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.

A alternating explicit-implicit difference scheme for solving the initial-boundary problem of the nonlinear wave equations at one dimension, two dimensions and three dimensions is given. The convergence of the difference solution is obtained.

A branch-and-bound reduced method is proposed for globally solving a class of linear multiplicative programming problems and the convergence of the algorithm is proved. In this algorithm, the lower bound functions on the multiplications in the constraints and the objective functions are given by using the convex envelopes technique of the two-vector multiplications so as to determine a relaxed convex programming of the original problem. We solve the relaxed convex programming to find the global optimization value's lower bound the feasible solutions of the original problem.In order to improve convergence rate, a rectangular reduction strategy is used. Numerical experiments show that the proposed algorithm is feasible.

 

In this paper, a fully discrete format of nonlinear Galerkin mixed element method with backward one-step Euler discretization of time for the non stationary conduction-convection problems is presented. The scheme is based on two finite element spaces XH and Xh for the approximation of the velocity, defined respectively on a coarse grid with grids size H and another fine grid with grid size h<< H, a finite element space Mh for the approximation of the pressure and two finite element spaces AH and Wh, for the approximation of the temperature,also defined respectivply on the coarse grid with grid size H and another fine grid with grid size h. The existence and the convergence of the fully discrete mixed element solution are shown. The scheme consists in using standard backward one step Euler-Galerkin fully discrete format at first L0 steps (L0 2) on fine grid with grid size h, but using nonlinear Galerkin mixed element method of backward one step Euler-Galerkin fully discrete format through L0 + 1 step to end step. We have proved that the fully discrete nonlinear Galerkin mixed element procedure with respect to the coarse grid spaces with grid size H holds superconvergence.

In this paper, a new unconventional Hermite-type rectangular element for the second order elliptic problem is constructed. The anisotropic character is proved by using anisotropic interpolate basic theorem, thus this element can be applied to arbitrary rectangular subdivision. At the same time, the superclose and super-convergence properties and extrapolation are obtained, which are independent of the regular assumption and quasi-uniform assumption of the meshes. Numerical results which coincide with our theoretical analysis show that this element indeed has very good convergence behavior.

To solve the interpolation problem of Hermit-Birkhoff type for scattered data of 4D, under the condition of minimizing the given functional, a new trivatiate polynomial spline interpolation with natural conditions have been constructed. The characterization, existence, uniqueness, convergence and error estimation of the solution of the interpolation problem have been studied carefully. Some numerical examples have been presented at last to illustrate the method.

In this paper, a second-order viscoelastic equation is studied with a splitting positive definite mixed finite element method. A new splitting positive definite mixed variational form and a semi-discrete formulation with respect to time based on the splitting positive definite mixed variational form are established first. And then, a fully discrete splitting positive definite mixed finite element formulation is established directly from the semi-discrete formulation with respect to time. Finally, the error estimates of the splitting positive definite mixed finite element solutions are provided. The studied approaches could make theoretical argumentation simpler and more convenient, which is a new study attempt for second-order viscoelastic equation.

In this paper, a new DL-type conjugate gradient method is proposed for solving nonconvex unconstrained optimization problems. Different from the existent ones, a new modified Armijo-type line search rule is constructed to give both the steplength and the conjugated parameter being used to determine a search direction in the mean time at each iteration. Under weak conditions, the global convergence of the developed algorithm is established. Numerical experiments show the efficiency of the algorithm, particularly in comparison with the similar ones available in the literature.

Multilevel fast multipole method (MLFMM) can be used to accelerate the iterative solution of integral equation deduced from Maxwell equations or Helmholtz-type equation, with a theoretical complexity of O(N logN), where N is the number of unknowns. MLFMM depends on fast calculating the translation term at each level, and interpolations between levels during upward and downward pass phases. The one-dimentional fast multipole method (FMM1D) similar to which used in N-body problem is introduced in this paper.
Fast Lagrange interpolation algorithm based on FMM1D can reduce the computing time of translation operator from O(N^1.5) to O(N). Fast spectrum interpolation with a hybrid FFT-FMM1D method can also reduce the computing complexity of interpolations between levels from O(K²) to O(K logK), where K is the number of sample points. The numerical results of MLFMM based on these two fast inperpolation methods have near linear time performances and accurate solutions.

Based on Laplace eigen-structure over three special triangle domains (regular triangle, isoceles triangle and triangle with (30°, 60°, 90°)), we propose a unified basis to compute all Laplace eigenvalues over an arbitrary triangle with mixed numerical and symbolic computation. And a class of approximate formulas for evaluating all eigenvalues over an arbitrary triangle as λ_m,n≈π²/24S²(h₁²(7m²-12mn+7n²)+h₂²(3m²-4mn+3n²)-2h₃²(m²-4mn+n²)),Especially, for the smallest eigenvalue λ_min≈π²/S² 11h₁²+7h₂²+6h₃²/24,where S is the area of the triangle with three lengths h₁≤h₂≤h₃.And it can be as a new quality of 2-D triangle grid for 2-nd PDE problems as q(T):=3h₃²/16S² 11h₁²+7h₂²+6h₃²/24.To reflect the influence of the three side-lengths on the eigenvalues over an arbitrary triangle, we put the above three basis together and use numerical computation with some symbolic. This hybrid algorithm may a way to raise the accuracy of eigenvalues in computing.

In this paper,a modified trust-region method for solving symmetric nonlinear equations is proposed.We establish the global convergence of the presented method under favorable conditions.Some preliminary numerical results show that this method is effective for the given problems.

In this paper, a Crouzeix-Raviart type nonconforming linear triangular finite element is applied to the parabolic equation and a new mixed element formulation is established. By utilizing the properties of the interpolation on the element and derivative delivery techniques instead of the Ritz projection operator, which is an indispensable tool in the traditional finite element analysis, the optimal order error estimates for the primitive solution u in broken H¹−norm and L²-norm with integral and the flux p=-▽u in L²-norm are obtained on anisotropic meshes, respectively. The numerical results show the validity of the theoretical analysis.

In this paper, a finite volume element method for non-stationary Stokes equation is studied and a stabilized fully discrete finite volume element formulation based on on two local Gauss integrals for non-stationary Stokes equation is derived. The errors of solution for this formulation is analyzed.

The (generalized) periodic Sylvester equation arises from linear periodic discrete-time systems. This paper is devoted to use the matrix sign function to solve (generalized) periodic Sylvester equations, which eigenvalues are contained in the open left half complex plane and the open right half complex plane or inside the unit circle and outside the unit circle. A numerical example illustrates our results.

In this paper, we present and analyze three new three-step iterative methods for solving the system of nonlinear equations using quadrature formulas. We prove that these new methods are of the convergence of fifth order. Some numerical examples are given to show that the new methods outperform the other existing methods.

In this paper general approximation scheme of low order nonconforming finite elements for Sine-Gordon equation is discussed, the optimal order error estimations of the corresponding unknown functions are derived based on interpolation technique and special properties of the elements.