Euler schemes and large deviations for stochastic Volterra equations with singular kernels

In this paper, an Euler type approximation is constructed for stochastic Volterra equation with singular kernels, which provides an algorithm for numerical calculation. Then, the large deviation estimates of small perturbation to equations of this type are obtained. We finally apply them to SDEs with the kernel of fractional Brownian motion with Hurst parameter H∈(0,1).     

Building nonredundant adaptive wavelets by update lifting

In this paper, we propose a technique for building adaptive wavelets by means of an extension of the lifting scheme. Our scheme comprises an adaptive update lifting step and a fixed prediction lifting step. The adaptivity consists hereof that the system can choose between two different update filters, and that this choice is triggered by the local gradient of the original signal. If the gradient is large (in some seminorm sense) it chooses one filter, if it is small the other. We derive necessary and sufficient conditions for the invertibility of such an adaptive system for various scenarios. Furthermore, we present some examples to illustrate our theoretical results.     

RCM‐TVD hybrid scheme for hyperbolic conservation laws

We describe a hybrid method for the solution of hyperbolic conservation laws. A third‐order total variation diminishing (TVD) finite difference scheme is conjugated with a random choice method (RCM) in a grid‐based adaptive way. An efficient multi‐resolution technique is used to detect the high gradient regions of the numerical solution in order to capture the shock with RCM while the smooth regions are computed with the more efficient TVD scheme. The hybrid scheme captures correctly the discontinuities of the solution and saves CPU time. Numerical experiments with one‐ and two‐dimensional problems are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

美式期权定价问题的变网格差分方法

本文提出一种求解美式期权定价自由边值问题的变网格差分方法.通过建立一个自由边界所满足的方程,利用变网格技术可同时求出期权的差分解和最佳执行边界.本文分别讨论了显式和隐式变网格差分格式,并给出了差分解的收敛性和稳定性分析.数值实验表明本文算法是一个非常有效的期权定价算法.     

Banach空间中渐近非扩张映象具误差的强收敛定理

设E是一实的Banach空间,其范数是一致Gteaux可微的;D是E的一非空闭凸子集,设T:D→D是具有序列{k_n}[1,∞),lim_(n→∞) k_n=1的渐近非扩张映象.本文证明了,在一定条件下,由(1.3)和(1.5)式定义的具误差的迭代序列{x_n}强收敛于T的不动点.本文结果也推广和改进了最近一些人的最新结果.     

High-Order Leap-Frog Based Discontinuous Galerkin Method for the Time-Domain Maxwell Equations on Non-Conforming Simplicial Meshes

A high-order leap-flog based non-dissipative discontinuous Galerkin timedomain method for solving Maxwell's equations is introduced and analyzed. The proposed method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements, with a Nth-order leap-frog time scheme. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes. The method is proved to be stable under some CFL-like condition on the time step. The convergence of the semi-discrete approximation to Maxwell's equations is established rigorously and bounds on the global divergence error are provided. Numerical experiments with highorder elements show the potential of the method.     

On association schemes all elements of which have valency 1 or 2

In the present note, we investigate schemes S in which each element s satisfies n_s2 and n_s^*s≠2. We show that such a scheme is schurian. More precisely, we show that it is isomorphic to G//t, where G is a finite group and t an involution of G weakly closed in C_G(t).

Groups G with an involution t weakly closed in C_G(t) have been described in Glauberman's Z^*-Theorem [G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966) 403–420] with the help of the largest normal subgroup of G having odd order.     

CONVERGENCE OF AN IMMERSED INTERFACE UPWIND SCHEME FOR LINEAR ADVECTION EQUATIONS WITH PIECEWISE CONSTANT COEFFICIENTS I： L^1-ERROR ESTIMATES

We study the L^l-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in L^l-norm to the differential equation with the corresponding interface condition. We derive the one-halfth order L^l-error bounds with explicit coefficients following a technique used in [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].     

Locally one-dimensional difference schemes for the fractional order diffusion equation

Locally-one-dimensional difference schemes for the fractional diffusion equation in multidimensional domains are considered. Stability and convergence of locally one-dimensional schemes for this equation are proved.     

A discrete eigenfunctions method for computing mixed hyperbolic problems based on an implicit difference scheme

In this paper numerical solutions of mixed hyperbolic problems are computed using a discrete eigenfunctions method combined with an implicit difference scheme. This new numerical technique preserves the qualitative properties of the analytic solution due to the Sturm-Liouville structure of the underlying discrete linear boundary-value problem and has computational stability advantages vs other methods. Illustrative examples are included.