A hybrid projective method for solving system of equilibrium problems with demicontractive mappings applicable in image restoration problems

In this paper, we propose a general iterative scheme based on CQ projection method for finding a common solution of system of equilibrium problems and the fixed point set of a finite family of demicontractive mappings. We also prove strong convergence of the scheme to a common element of the two above-described sets. We then give a numerical example to justify our main result. An example is given in an infinite dimensional space for supporting our main result. Moreover, we apply our main result to solve the unconstrained image restoration problems with a finite family of blurring operators. Our results extend and improve some existing results in the literature.     

Study of convergence of the projection-difference method for hyperbolic equations

We consider the Cauchy problem for an abstract quasilinear hyperbolic equation with variable operator coefficients and a nonsmooth but Bochner integrable free term in a Hilbert space. Under study is the scheme for approximate solution of this problem which is a combination of the Galerkin scheme in space variables and the three-layer difference scheme with time weights. We establish an a priori energy error estimate without any special conditions on the projection subspaces. We give a concrete form of this estimate in the case when discretization in the space variables is carried out by the finite element method (for a partial differential equation) and by the Galerkin method in Mikhlin form.     

Extragradient method for solving quasivariational inequalities

We study methods for solving a class of the quasivariational inequalities in Hilbert space when the changeable set is described by translation of a fixed, closed and convex set. We consider one variant of the gradient-type projection method and an extragradient method. The possibilities of the choice of parameters of the gradient projection method in this case are wider than in the general case of a changeable set. The extragradient method on each iteration makes one trial step along the gradient, and the value of the gradient at the obtained point is used at the first point as the iteration direction. In the paper, we establish sufficient conditions for the convergence of the proposed methods and derive a new estimate of the rates of the convergence. The main result of this paper is contained in the convergence analysis of the extragradient method.     

Stokes方程的投影／方向分裂谱方法

我们提出和分析了一种求解Stokes方程的数值方法．新方法基于空间上的Legendre谱离散,时间上则采用投影／方向分裂格式．更确切地说,时间离散的出发点是旋度形式的压力校正投影法,在此基础上进一步应用方向分裂法,把速度和压力方程分裂为一系列一维的椭圆型子问题．然后生成的这些一维子问题用Legendre谱方法进行空间离散．另外,我们证明了全离散格式的稳定性．一些数值实验验证了收敛性和方法的有效性．     

Discrete Quantum Scattering Theory

We formulate quantum scattering theory in terms of a discrete L ₂-basis of eigen differentials. Using projection operators in the Hilbert space, we develop a universal method for constructing finite-dimensional analogues of the basic operators of the scattering theory: S- and T-matrices, resolvent operators, and Möller wave operators as well as the analogues of resolvent identities and the Lippmann–Schwinger equations for the T-matrix. The developed general formalism of the discrete scattering theory results in a very simple calculation scheme for a broad class of interaction operators.     

On the approximation of fractional resolution families

We study the approximation of fractional resolution families with respect to space and time. In the approach, we use the notions of general approximation scheme including the finite-difference method, the finite-element method, and projection methods.     

Convergence of a finite difference method for combustion model problems

We study a finite difference scheme for a combustion model problem. A projection scheme near the combustion wave, and the standard upwind finite difference scheme away from the combustion wave are applied. Convergence to weak solutions with a combustion wave is proved under the normal Courant-Friedrichs-Lewy condition. Some con-     

对流占优的Sobolev方程的投影稳定化有限元方法

对于对流占优的Sobolev方程,提出了一种新的投影稳定化有限元方法,建立了半离散和全离散的投影稳定化格式,给出了解的稳定性和收敛性分析.该方法能够有效克服对流占优,与内罚方法相比,投影格式更简单,计算量更小,且得到的C—N格式是无条件稳定的,时间精度达到了二阶.最后,通过实验证明,数值结果与理论结果完全一致.     

Landweber iterative methods for angle-limited image reconstruction

We introduce a general iterative scheme for angle-limited image reconstruction based on Landwebet＇s method. We derive a representation formula for this scheme and consequently establish its convergence conditions. Our results suggest certain relaxation strategies for an accelerated convergence for angle-limited image reconstruction in L^2-norm comparing with alternative projection methods. The convolution-backprojection algorithm is given for this iterative process.     

Stochastic Theta Method for a Reflected Stochastic Differential Equation

In this article, a stochastic theta method for a reflected stochastic differential equation is proposed. When the parameter θ = 0, this method coincides with the projection Euler scheme; while when the parameter θ = 1, it is called an implicit projection Euler scheme which is first proposed in this article. Under some conditions, the strong convergence and the A-stability of this numerical scheme are proved.     

New a priori analysis of first‐order system least‐squares finite element methods for parabolic problems

We provide new insights into the a priori theory for a time‐stepping scheme based on least‐squares finite element methods for parabolic first‐order systems. The elliptic part of the problem is of general reaction‐convection‐diffusion type. The new ingredient in the analysis is an elliptic projection operator defined via a nonsymmetric bilinear form, although the main bilinear form corresponding to the least‐squares functional is symmetric. This new operator allows to prove optimal error estimates in the natural norm associated to the problem and, under additional regularity assumptions, in the L² norm. Numerical experiments are presented which confirm our theoretical findings.     

A two-level finite element method for the Navier–Stokes equations based on a new projection

We consider a fully discrete two-level approximation for the time-dependent Navier–Stokes equations in two dimension based on a time-dependent projection. By defining this new projection, the iteration between large and small eddy components can be reflected by its associated space splitting. Hence, we can get a weakly coupled system of large and small eddy components. This two-level method applies the finite element method in space and Crank–Nicolson scheme in time. Moreover,the analysis and some numerical examples are shown that the proposed two-level scheme can reach the same accuracy as the classical one-level Crank–Nicolson method with a very fine mesh size h by choosing a proper coarse mesh size H. However, the two-level method will involve much less work.     

A second order scheme for the navier-stokes equations: stability and convergence

In this paper, a fully discretized projection method is introduced. It contains a parameter operator. Depending on this operator, we can obtain a first-order scheme, which is appropriate for theoretical analysis, and a second-order scheme, which is more suitable for actual computations. In this method, the boundary conditions of the intermediate velocity field and pressure are not needed. We give the proof of the stability and convergence for the first-order case. For the higher order cases, the proof were different, and we will present it elsewhere.

In a forthcoming article [7], we apply this scheme to the driven-cavity problem and compare it with other schemes     

Strong Convergence of an Iterative Scheme by a New Type of Projection Method for a Family of Quasinonexpansive Mappings

We deal with a common fixed point problem for a family of quasinonexpansive mappings defined on a Hilbert space with a certain closedness assumption and obtain strongly convergent iterative sequences to a solution to this problem. We propose a new type of iterative scheme for this problem. A feature of this scheme is that we do not use any projections, which in general creates some difficulties in practical calculation of the iterative sequence. We also prove a strong convergence theorem by the shrinking projection method for a family of such mappings. These results can be applied to common zero point problems for families of monotone operators.     

Unconstrained Optimization Techniques for the Acceleration of Alternating Projection Methods

Alternating projection methods have been extensively used to find the closest point, to a given point, in the intersection of several given sets that belong to a Hilbert space. One of the characteristics of these schemes is the slow convergence that can be observed in practical applications. To overcome this difficulty, several techniques, based on different ideas, have been developed to accelerate their convergence. Recently, a successful acceleration scheme was developed specially for Cimmino's method when applied to the solution of large-scale saddle point problems. This specialized acceleration scheme is based on the use of the well-known conjugate gradient method for minimizing a related convex quadratic map. In this work, we extend and further analyze this optimization approach for several alternating projection methods on different scenarios. In particular, we include a specialized analysis and treatment for the acceleration of von Neumann-Halperin's method and Cimmino's method on subspaces, and Kaczmarz method on linear varieties. For some specific applications we illustrate the advantages of our acceleration schemes with encouraging numerical experiments.     

Behavior of different ansätze in the generalized projection operator method

Using the recently reported generalized projection operator method for the nonlinear Schrödinger equation, we derive the generalized pulse parameters equations for ansätze like hyperbolic secant and raised cosine functions. In general, each choice of the phase factor θ in the projection operator gives a different set of ordinary differential equations. For θ = 0 or θ = π/2, the corresponding projection operator scheme is equivalent to the Lagrangian variation method or the bare approximation of the collective variable theory. We prove that because of the inherent symmetric property between the pulse parameters of a Gaussian ansätz results the same set of pulse parameters equations for any value of the generalized projection operator parameter θ. Finally we prove that after the substitution of the ansätze function, the Lagrange function simplifies to the same functional form irrespective of the ansätze used because of a special property shared by all the anätze chosen in this work.     

Error analysis of a fully discrete projection method for magnetohydrodynamic system

In this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi-implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H ( curl ) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.     

Multilevel augmentation methods for solving the sine-Gordon equation

In this paper we develop the multilevel augmentation method for solving nonlinear operator equations of the second kind and apply it to solving the one-dimensional sine-Gordon equation. We first give a general setting of the multilevel augmentation method for solving the second kind nonlinear operator equations and prove that the multilevel augmentation method preserves the optimal convergence order of the projection method while reducing computational cost significantly. Then we describe the semi-discrete scheme and the fully-discrete scheme based on multiscale methods for solving the sine-Gordon equation, and apply the multilevel augmentation method to solving the discrete equation. A complete analysis for convergence order is proposed. Finally numerical experiments are presented to confirm the theoretical results and illustrate the efficiency of the method.     

关于两个非线性算子的一个分裂方法(英文)

<正>This paper generalizes a class of projection type methods for monotone variational inequalities to general monotone inclusion.It is shown that when the normal cone operator in projection is replaced by any maximal monotone operator,the resulting method inherits all attractive convergence properties of projection type methods,and allows an adjusting step size rule.Weaker convergence assumption entails an extra projection at each iteration.Moreover,this paper also addresses applications of the resulting method to convex programs and monotone variational inequalities.     

Weak solvability of interior transmission problems via mixed finite elements and Dirichlet-to-Neumann mappings

We study the weak solvability of an interior linear-nonlinear transmission problem arising in steady heat transfer and potential theory. For the variational formulation, we use a Dirichlet-to-Neumann mapping on the interface, which is obtained from the application of the boundary integral method to the linear domain, and we utilize a mixed finite element method in the nonlinear region. Existence and uniqueness of solution for the continuous formulation are provided and general approximation results for a fully discrete Galerkin method are derived. In particular, a compatibility condition between the mesh sizes involved is deduced in order to conclude the solvability and stability of this Galerkin scheme.