拟增生算子方程广义最速下降法的收敛性特征条件

本文给出了广义最速下降法强收敛于定义在一致光滑实Banach空间的真子集上的局部有界拟增生算子的零点的一特征条件.所得的结果推广和统一如徐宗本和蒋耀林等人的相应结果.     

Lipschitz条件下的加权梯形不等式和中点不等式

利用与一阶导数有关的积分恒等式,并通过引入参数求最值,在一 阶导函数满足Lipschitz条件的情况下,给出加权梯形不等式和中点不等式.     

The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

In this paper, we consider the stochastic differential equations with piecewise continuous arguments (SDEPCAs) in which the drift coefficient satisfies the generalized one-sided Lipschitz condition and the diffusion coefficient satisfies the linear growth condition. Since the delay term $t-[t]$ of SDEPCAs is not continuous and differentiable, the variable substitution method is not suitable. To overcome this difficulty, we adopt new techniques to prove the boundedness of the exact solution and the numerical solution. It is proved that the truncated Euler-Maruyama method is strongly convergent to SDEPCAs in the sense of $L^{\bar{q}}(\bar{q}\ge 2)$. We obtain the convergence order with some additional conditions. An example is presented to illustrate the analytical theory.     

Newton’s Method for Solving Generalized Equations Without Lipschitz Condition

Journal of Optimization Theory and Applications - This paper aims to establish higher order convergence of the (inexact) Newton’s method for solving generalized equations composed of the sum...     

广义最速下降法逼近拟增生算子方程解的一个特征条件

倪仁兴最近的文章研究了广义最速下降法强收敛于拟增生算子方程解的一特征条件．本文对此进行了修正和改进，给出了一个新的特征条件．所得结果同时改进和推广了一些已有的结果．     

非李普希茨条件下无穷维随机微分方程的适度解

通过构造收敛的逼近列的方法给出了非李普希茨条件下无穷维随机微分方程dX=[AX+f(X)]dt+[BX+g(X)]dW的适度解的存在唯一性定理.文章推广了[1]和[2]的结论.     

A Comparison Theorem for Stochastic Differential Equations Under the Novikov Condition

We consider a system of stochastic differential equations driven by a standard n-dimensional Brownian motion where the drift function b is bounded and the diffusion coefficient is the identity matrix. We define via a duality relation a vector Z (which depends on b) of square integrable stochastic processes which is shown to coincide with the unique strong solution of the previously mentioned equation. We show that the process Z is well defined independently of the boundedness of b and that it makes sense under the more general Novikov condition, which is known to guarantee only the existence of a weak solution. We then prove that under this mild assumption the process Z solves in the strong sense a related stochastic differential inequality. This fact together with an additional assumption will provide a comparison result similar to well known theorems obtained in the presence of strong solutions. Our framework is also suitable to treat path-dependent stochastic differential equations and an application to the famous Tsirelson equation is presented.     

Generalized Iteration Method for First-Kind Integral Equations

An iteration method is described to solve one-dimensional, first-kind integral equations with finite integration limits and difference kernel, K ( x − x '), that decays exponentially. The method relies on deriving via the Wiener–Hopf factorization and solving by suitable iterations in the Fourier complex plane a pair of integral relations, where each iteration accounts for all end point singularities in x of the exact solution. For even and odd kernels, this pair reduces to decoupled, 2nd-kind Fredholm equations, and the iteration yields Neumann series subject to known convergence criteria. This formulation is applied to a classic problem of steady advection-diffusion in the two-dimensional (2D) potential flow of concentrated fluid. The remarkable overlap of recently derived asymptotic expansions for the flux in this case is shown to be intimately related to the analyticity of the kernel Fourier transform.     

Generalized Alternating Direction Implicit Method for Projected Generalized Lyapunov Equations

We generalize an alternating direction implicit method for projected generalized Lyapunov equations. Low rank versions of this method is also presented that can be used to compute a low rank approximation of the solution of Lyapunov equations with symmetric, positive semidefinite right‐hand side. Numerical example is given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence Rate of The Trust Region Method for Nonlinear Equations Under Local Error Bound Condition

In this paper, we present the new trust region method for nonlinear equations with the trust region converging to zero. The new method preserves the global convergence of the traditional trust region methods in which the trust region radius will be larger than a positive constant. We study the convergence rate of the new method under the local error bound condition which is weaker than the nonsingularity. An example given by Y.X. Yuan shows that the convergence rate can not be quadratic. Finally, some numerical results are given. This work is supported by Chinese NSFC grants 10401023 and 10371076, Research Grants for Young Teachers of Shanghai Jiao Tong University, and E-Institute of Computational Sciences of Shanghai Universities. An erratum to this article is available at .     

局部凸空间P-自反下的中点局部k-一致凸性(光滑性)

在局部凸空间已有的中点局部kk-一致凸性和中点局部k-一致光滑性这一对对偶概念的基础上,证明了中点局部kk-一致凸性与中点局部(k+1)-一致凸性的关系,给出了在P-自反的条件下它们之间的等价对偶定理.     

Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

We study nonlinear elliptic equations in divergence form

$$\text {div }{\mathcal A}(x,Du)=\text {div } G.$$

When \({\mathcal A}\) has linear growth in D u, and assuming that \(x\mapsto {\mathcal A}(x,\xi )\) enjoys \(B^{\alpha }_{\frac {n}\alpha , q}\) smoothness, local well-posedness is found in \(B^{\alpha }_{p,q}\) for certain values of \(p\in [2,\frac {n}{\alpha })\) and \(q\in [1,\infty ]\). In the particular case \({\mathcal A}(x,\xi )=A(x)\xi \), G = 0 and \(A\in B^{\alpha }_{\frac {n}\alpha ,q}\), \(1\leq q\leq \infty \), we obtain \(Du\in B^{\alpha }_{p,q}\) for each \(p<\frac {n}\alpha \). Our main tool in the proof is a more general result, that holds also if \({\mathcal A}\) has growth s?1 in D u, 2 ≤ s ≤ n, and asserts local well-posedness in L ^q for each q > s, provided that \(x\mapsto {\mathcal A}(x,\xi )\) satisfies a locally uniform VMO condition.

     

A rapid Generalized Method of Bisection for solving Systems of Non-linear Equations

Summary A rapid Generalized Method of Bisection for solving Systems of Non-linear Equations is presented in this paper, based on the non-zero value of the topological degree. Further, while the method does not compute the topological degree, it takes care of keeping its non-zero value during the bisections and thus results in a fast bisection algorithm.     

A Generalized Chebyshev Method for Non-linear Parabolic Equations in One Space Variable

The derivation and implementation of a generalized Chebyshevmethod is described for the numerical solution of non-linearparabolic equations in one space dimension. The solution isobtained by using the method of lines and is approximated inthe space variable by piecewise Chebyshev polynomial expansions.These expansions are normally few in number and of high order.It is shown that the method can be derived from a perturbedform of the original equation. A numerical example is givento illustrate its performance compared with the finite elementand finite difference method. A comparison of various Chebyshev methods is made by applyingthem to two-point eigenproblems. It is shown by analysis andnumerical examples that the approach used to derive the generalizedChebyshev method is comparable, in terms of the accuracy obtained,with existing Chebyshev methods.     

Generalized Directional Gradients, Backward Stochastic Differential Equations and Mild Solutions of Semilinear Parabolic Equations

We study a forward-backward system of stochastic differential equations in an infinite-dimensional framework and its relationships with a semilinear parabolic differential equation on a Hilbert space, in the spirit of the approach of Pardoux-Peng. We prove that the stochastic system allows us to construct a unique solution of the parabolic equation in a suitable class of locally Lipschitz real functions. The parabolic equation is understood in a mild sense which requires the notion of a generalized directional gradient, that we introduce by a probabilistic approach and prove to exist for locally Lipschitz functions. The use of the generalized directional gradient allows us to cover various applications to option pricing problems and to optimal stochastic control problems (including control of delay equations and reaction--diffusion equations), where the lack of differentiability of the coefficients precludes differentiability of solutions to the associated parabolic equations of Black--Scholes or Hamilton-Jacobi-Bellman type.     

二阶积分微分方程的广义拟线性化方法

运用广义拟线性化方法研究了正规锥上的二阶积分微分方程初值问题,获得了逼近解序列一致且平方收敛的结果.     

Stability of a Cubically Convergent Method for Generalized Equations

In Geoffroy et al, Acceleration of convergence in Dontchev's iterative method for solving variational inclusions Serdica Math. J. 29 (2003), pp. 45–54] we showed the convergence of a cubic method for solving generalized equations of the form 0 ∈ f(x) +- G(x) where f is a function and G stands for a set-valued map. We investigate here the stability of such a method with respect to some perturbations. More precisely, we consider the perturbed equation y ∈ f(x) +- G(x) and we show that the pseudo-Lipschitzness of the map (f +- G)⁻¹ is closely tied to the uniformity of our method in the sense that the attraction region does not depend on small perturbations of the parameter y. Finally, we provide an enhanced version of the convergence theorem established by Geoffroy, et al.     

高维广义BBM方程的Chebyshev拟谱方法

在非线性长波的研究中[1],提出并研究了BBM方程.由于这类方程在很多数学物理问题中出现,如双温热传导的冷却过程,液体在碎石中的渗流问题等,因而引起了人们的关注.这类方程的数值方法,已有许多工作,但主要是采用差分法和有限元法.[2]使用.Fourier谱方法讨论了一维广义BBM方程,我们在[3]中也用Fourier谱和拟谱方法讨论了高维广义BBM方程.然而对于非周期情况,Fourier谱方法无法使用.在     

A Semilocal Convergence of a Secant–Type Method for Solving Generalized Equations

In this paper we present a study of the existence and the convergence of a secant–type method for solving abstract generalized equations in Banach spaces. With different assumptions for divided differences, we obtain a procedure that have superlinear convergence. This study follows the recent results of semilocal convergence related to the resolution of nonlinear equations (see [11])