Approximation,solution operators and quantale-valued metrics

A generalized solution operator is a mapping abstractly describing a computational problem and its approximate solutions. It assigns a set of \(\varepsilon \)-approximations of a solution to the problem instance f and accuracy of approximation \(\varepsilon \). In this paper we study generalized solution operators for which the accuracy of approximation is described by elements of a complete lattice equipped with a compatible monoid structure, namely, a quantale. We provide examples of computational problems for which the accuracy of approximation of a solution is measured by such objects. We show that the sets of \(\varepsilon \)-approximations are, roughly, closed balls with radii \(\varepsilon \) with respect to a certain family of quantale-valued generalized metrics induced by a generalized solution operator.     

多维广义SRLW方程的Chebyshev拟谱方法分析

考虑了一类多维的广义对称正则长波(SRLW)方程的齐次初边值问题Chebyshev拟谱逼近,构造了全离散的Chebyshev拟谱格式,给出了这种格式近似解的收敛性和最优误差估计。     

Higher order asymptotic expansions to the solutions for a nonlinear damped wave equation

We study the Cauchy problem for a nonlinear damped wave equation. Under suitable assumptions for the nonlinearity and the initial data, we obtain the global solution which satisfies weighted L ¹ and \({L^\infty}\) estimates. Furthermore, we establish the higher order asymptotic expansion of the solution. This means that we construct the nonlinear approximation of the global solution with respect to the weight of the data. Our proof is based on the approximation formula of the linear solution, which is given by Takeda (Asymptot Anal 94:1–31, 2015), and the nonlinear approximation theory for a nonlinear parabolic equation developed by Ishige et al. (J Evol Equ 14:749–777, 2014).     

Approximation of a class of optimal control problems with order of convergence estimates

An approximation scheme for a class of optimal control problems is presented. An order of convergence estimate is then developed for the error in the approximation of both the optimal control and the solution of the control equation.     

Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(τ+h^l+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(τ²+h^l+1).     

线性流形上中心对称矩阵的最佳逼近

1 引 言令Rn×m表示所有n×m阶实矩阵集合;ORn×n表示所有n×n阶正交矩阵之集;A+表示矩阵A的Moore-Penrose广义逆;Iκ表示κ阶单位阵;||·||表示矩阵的Frobenius范数;rank（A）表示矩阵A的秩.设ei为n阶单位矩阵In的第i列（i=1,2,…,n）,记Sn=（en,en-1,…,e1）,易知     

线性流形上一类对称广义中心对称矩阵的最佳逼近问题

<正>1问题的提出为叙述方便起见,首先介绍本文中出现的符号.设SR~(n×n)和OR~(n×n)分别表示所有n阶实对称矩阵和正交矩阵的集合,A~+和‖A‖_F分别表示A的Moore-Penrose广义逆和     

Hopf bifurcation in numerical approximation for delay differential equations

In this paper we investigate the qualitative behaviour of numerical approximation to a class delay differential equation. We consider the numerical solution of the delay differential equations undergoing a Hopf bifurcation. We prove the numerical approximation of delay differential equation had a Hopf bifurcation point if the true solution does.     

Banach空间发展包含周期解的存在性

该文使用Galerkin逼近方法建立了一类发展包含的存在性定理.同时作为应用,给出了一类带有集值右端的偏微分方程的周期解存在的充分条件.     

A CLASS OF FACTORIZED QUASI-NEWTON METHODS FOR NONLINEAR LEAST SQUARES PROBLEMS

1.IntroductionThispaPerdealswiththeproblemofndniedingasumofsquaresofnonlinearfuntions-mwhereri(x),i==1,2,',maretwicecontinuouslyfferentiable,m2n'r(x)=(rl(x),rz(x),'jbe(x))"and"T"denotestranspose.NoIilinearleastSquaresproblemisakindofAnportan0ptiedationprobletnsandisaPpearedinmanyfield8suchasscientilicexperiments,mbomumlikelihoodestimation,solutionofnonlinearequaions'patternrecoghtionandetc.ThederiVativesofthefUnctionj(x)aregivenbywhereAEppxnistheJacobianmatrisofr(x)anditselementsare~=f…     

Convergence of numerical solutions to stochastic pantograph equations with Markovian switching

In this paper, a class of stochastic pantograph equations with Markovian switching is considered. The main purpose is to investigate the convergence of the Euler method of the equations. It is proved that the Euler approximation solution converge to the analytic solution in probability under weaker conditions. An example is provided to illustrate our theory.     

Cubic spline for a class of singular two-point boundary value problems

In this paper we have presented a method based on cubic splines for solving a class of singular two-point boundary value problems. The original differential equation is modified at the singular point then the boundary value problem is treated by using cubic spline approximation. The tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm. Some model problems are solved, and the numerical results are compared with exact solution.     

Exponential stability for a class of state-dependent delay equations via the Crandall-Liggett approach

In this paper we are concerned with the exponential asymptotic stability of the solution of a class of differential equations with state dependent delays. Our approach is based on the Crandall-Liggett approximation and the properties of semigroups.     

A new method to build confidence regions for solutions of stochastic variational inequalities

Stochastic variational inequalities model a large class of equilibrium problems subject to data uncertainty. The true solution to such a problem is usually estimated by a solution to its sample average approximation (SAA) problem. This article proposed a new method to build asymptotically exact confidence regions for the true solution that are computable from the SAA solution.     

一类随机泛函微分方程带随机步长的EM逼近的渐近稳定

研究了一类带有限延迟的随机泛函微分方程的Euler-Maruyama(EM)逼近，给出了该方程的带随机步长的EM算法，得到了随机步长的两个特点:首先，有限个步长求和是停时；其次，可列无限多个步长求和是发散的.最终，由离散形式的非负半鞅收敛定理，得到了在系数满足局部Lipschitz条件和单调条件下，带随机步长的EM数值解几乎处处收敛到0.该文拓展了2017年毛学荣关于无延迟的随机微分方程带随机步长EM数值解的结果.     

Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise

The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order elliptic differential operator. The solution is approximated by means of a finite element discretization in space and a quadrature approximation of an integral representation of the fractional inverse from the Dunford–Taylor calculus. For the resulting approximation, a concise analysis of the weak error is performed. Specifically, for the class of twice continuously Fréchet differentiable functionals with second derivatives of polynomial growth, an explicit rate of weak convergence is derived, and it is shown that the component of the convergence rate stemming from the stochasticity is doubled compared to the corresponding strong rate. Numerical experiments for different functionals validate the theoretical results.     

Two-sided deconvolution for bandlimited kernels

The two-sided deconvolution problem, for a certain class of a bandlimited kernels, is reduced to a discrete deconvolution problem by the sampling theorem, yielding a bandlimited solution. For this solution, in addition, a Galerkin type approximation is given. In general, the solution of the convolutioon equation for bandlimited kernels is not bandlimited. This follows from a characterization of the general solution     

Solution of a periodic optimal control problem by asymptotic series

In order to understand the numerical behavior of a certain class of periodic optimal control problems, a relatively simple problem is posed. The complexity of the extremal paths is uncovered by determining an analytic approximation to the solution by using the Lindstedt-Poincaré asymptotic series expansion. The key to obtaining this series is in the proper choice of the expansion parameter. The resulting expansion is essentially a harmonic series in which, for small values of the expansion parameter and a few terms of the series, excellent agreement with the numerical solution is obtained. A reasonable approximation of the solution is achieved for a relatively large value of the expansion parameter.This work was sponsored partially by the National Science Foundation, Grant No. ECS-84-13745.     

A nonlinear iteration method for solving a two-dimensional nonlinear coupled system of parabolic and hyperbolic equations

A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes.     

Regularized modified α-processes for nonlinear equations with monotone operators

For a stable approximation of the solution to a nonlinear irregular equation with a monotone operator, a two-step method based on Lavrent’ev scheme and nonlinear regularized α-processes is constructed. These processes are shown to have a linear convergence rate when used to approximate the solution of a regularized equation. The error of the regularized solution is estimated, and the two-step method is shown to be order optimal in the well-posedness class of sourcewise representable solutions.