圆柱土样Biot固结模型的Green函数及其土体体积应变规律

通过建立横观各向同性圆柱土样轴对称Biot固结问题的Green·函数,得到了用Green函数表示的径向位移分布表达式和相应的空隙水压力的精确解析解.该方法不仅避免了将问题的解分解为弹性静力学解和渗流拟动态解的叠加的过程和复杂的积分变换,而且问题的级数解形式简洁,收敛速度较快,便于数值计算和圆柱土样的全场渗流固结规律的分析.最后以此结果具体分析了Mandel-Cryer效应在圆柱土样不同位置的强弱程度和土的泊松比对Mandel-Cryer效应的影响,表明本文方法的正确性.     

The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems

In this paper, we present an efficient numerical algorithm for solving a general class of nonlinear singular boundary value problems. This present algorithm is based on the Adomian decomposition method (ADM) and Green’s function. The method depends on constructing Green’s function before establishing the recursive scheme. In contrast to the existing recursive schemes based on ADM, the proposed numerical algorithm avoids solving a sequence of transcendental equations for the undetermined coefficients. The approximate series solution is calculated in the form of series with easily computable components. Moreover, the convergence analysis and error estimation of the proposed method is given. Furthermore, the numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The numerical results reveal that the proposed method is very effective.     

Near field imaging of small isotropic and extended anisotropic scatterers

In this paper, we consider two time-harmonic inverse scattering problems of reconstructing penetrable inhomogeneous obstacles from near field measurements. First, we appeal to the Born approximation for reconstructing small isotropic scatterers via the MUSIC algorithm. Some numerical reconstructions using the MUSIC algorithm are provided for reconstructing the scatterer and piecewise constant refractive index using a Bayesian method. We then consider the reconstruction of an anisotropic extended scatterer by a modified linear sampling method and the factorization method applied to the near field operator. This provides a rigorous characterization of the support of the anisotropic obstacle in terms of a range test derived from the measured data. Under appropriate assumptions on the material parameters, the derived factorization can be used to determine the support of the medium without a priori knowledge of the material properties.     

Potential kernels,probabilities of hitting a ball,harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. We also prove a supremum estimate and a regularity result for functions harmonic with respect to a general isotropic unimodal Lévy process.In the second part we apply the recent results on the boundary Harnack inequality and Martin representation of harmonic functions for the class of isotropic unimodal Lévy processes. As a sample application, we provide sharp two-sided estimates of the Green function of a half-space.     

Green functions for the Dirac operator under local boundary conditions and applications

In this article, we define the Green function for the Dirac operator under two local boundary conditions: the condition associated with a chirality operator (also called the chiral bag boundary condition) and the MIT bag boundary condition. Then we give some applications of these constructions for each Green function. From the existence of the chiral Green function, we derive an inequality on a spin conformal invariant which, in some cases, solves the Yamabe problem on manifolds with boundary. Finally, using the MIT Green function, we give a simple proof of a positive mass theorem previously proved by Escobar.     

基于粒子群优化Kriging模型的边坡可靠度分析

采用传统极限平衡法进行边坡可靠度分析时,不可避免会遇到一个问题,即边坡功能函数形式的高度非线性以及隐含性.对于隐式功能函数,传统的求解方法是通过对功能函数进行多次迭代,从而得到安全系数值.但是由于功能函数的形式较为复杂,导致迭代计算的过程变得尤为繁琐且效率低下.鉴于传统边坡可靠度分析中存在的安全系数计算繁琐耗时的问题,提出一种基于粒子群优化(PSO)算法的自动采样Kriging代理模型方法,该方法可以代替功能函数的作用进行安全系数的求解.首先用拉丁超立方抽样方法(LHS)选取少量土体参数组,并通过极限平衡法求出对应的安全系数,将土体参数组和安全系数作为初始样本建立Kriging模型;其次由粒子群优化算法将最有期望改善模型拟合精度的样本点添加到样本集合中,以逐步迭代提升Kriging模型的计算精度;最后集合经典蒙特卡洛模拟(MCS)获得边坡的破坏概率.通过一个双层的土质边坡算例分析,证明了该方法可以实现准确高效的安全系数计算,尤其是在安全系数计算量十分庞大的情况下可以大大节省计算时间,是一种有效的边坡工程稳定可靠度分析方法.     

On the Green function in visco‐elastic media obeying a frequency power‐law

In this work, we present an explicit expression for the Green function in a visco‐elastic medium. We choose Szabo and Wu's frequency power law model to describe the visco‐elastic properties and derive a generalized visco‐elastic wave equation. We express the ideal Green function (without any viscous effect) in terms of the viscous Green function using an attenuation operator. By means of an approximation of the ideal Green function, we address the problem of reconstructing a small anomaly in a visco‐elastic medium from wavefield measurements. Copyright © 2011 John Wiley & Sons, Ltd.     

A derivative-free optimization algorithm based on conditional moments

In this paper we propose a derivative-free optimization algorithm based on conditional moments for finding the maximizer of an objective function. The proposed algorithm does not require calculation or approximation of any order derivative of the objective function. The step size in iteration is determined adaptively according to the local geometrical feature of the objective function and a pre-specified quantity representing the desired precision. The theoretical properties including convergence of the method are presented. Numerical experiments comparing with the Newton, Quasi-Newton and trust region methods are given to illustrate the effectiveness of the algorithm.     

The Green’s Function in the Problem of Charge Dynamics on A One-Dimensional Lattice with An Impurity Center

In the “tight-binding” approximation (the Hückel model), we consider the evolution of the charge wave function on a semi-infinite one-dimensional lattice with an additional energy U at a single impurity site. In the case of the continuous spectrum (for |U| < 1) where there is no localized state, we construct the Green’s function using the expansion in terms of eigenfunctions of the continuous spectrum and obtain an expression for the time Green’s function in the form of a power series in U. It unexpectedly turns out that this series converges absolutely even in the case where the localized state is added to the continuous spectrum. We can therefore say that the Green’s function constructed using the states of the continuous spectrum also contains an implicit contribution from the localized state.     

Regions of positivity for polyharmonic Green functions in arbitrary domains

The Green function for the biharmonic operator on bounded domains with zero Dirichlet boundary conditions is in general not of fixed sign. However, by extending an idea of Z. Nehari, we are able to identify regions of positivity for Green functions of polyharmonic operators. In particular, the biharmonic Green function is considered in all space dimensions. As a consequence we see that the negative part of any such Green function is somehow small compared with the singular positive part.

     

Des statistiques liées à la fonction de Green du laplacien

In this Note, we show that an invariant test of uniformity for a sample from a compact 2-point homogeneous space can be based on the Green function of the Laplacian. The three celebrated Watson, Cramér–von-Mises and Anderson–Darling statistics are shown to be particular cases of this family of statistics. To cite this article: J.-R. Pycke, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

The Green’s function of a five-point discretization of a two-dimensional finite-gap Schrödinger operator: The case of four singular points on the spectral curve

We consider a regular Riemann surface of finite genus and “generalized spectral data,” a special set of distinguished points on it. From them we construct a discrete analog of the Baker-Akhiezer function with a discrete operator that annihilates it. Under some extra conditions on the generalized spectral data, the operator takes the form of the discrete Cauchy-Riemann operator, and its restriction to the even lattice is annihilated by the corresponding Schrödinger operator. In this article we construct an explicit formula for the Green’s function of the indicated operator. The formula expresses the Green’s function in terms of the integral along a special contour of a differential constructed from the wave function and the extra spectral data. In result, the Green’s function with known asymptotics at infinity can be obtained at almost every point of the spectral curve.     

基本初等函数的高精度快速计算的加速算法

在现有的基本初等函数的高精度快速算法基础上,进一步研究基本初等函数的加速算法.现有的基本初等函数的高精度快速算法是通过对函数进行幂级数展开的方式来实现函数的任意精度快速计算.而其加速算法则是在幂级数展开之前,先利用函数的多种性质来缩减函数的参数,减少函数在进行幂级数展开时的计算难度,提高函数的计算速度.给出了加速算法,并从计算误差和算法复杂性两方面对该算法进行了分析,给出了误差最小,算法复杂性最低的最优加速算法.然后,对于三角函数、双曲函数、指数函数以及它们的反函数,在实数域上给出了的具体的加速过程和计算结果.     

Regular two-point boundary value problems for the Schrödinger operator on a path

In this work we study the different type of regular boundary value problems on a path associated with the Schrödinger operator. In particular, we obtain the Green function for each problem and we emphasize the case of Sturm-Liouville boundary conditions. In any case, the Green function is given in terms of second kind Chebyshev polynomials since they verify a recurrence law similar to the one verified by the Schödinger operator on a path.     

Optimality functions in stochastic programming

Optimality functions define stationarity in nonlinear programming, semi-infinite optimization, and optimal control in some sense. In this paper, we consider optimality functions for stochastic programs with nonlinear, possibly nonconvex, expected value objective and constraint functions. We show that an optimality function directly relates to the difference in function values at a candidate point and a local minimizer. We construct confidence intervals for the value of the optimality function at a candidate point and, hence, provide a quantitative measure of solution quality. Based on sample average approximations, we develop an algorithm for classes of stochastic programs that include CVaR-problems and utilize optimality functions to select sample sizes.     

Mathematical simulation of acoustic wave refraction near a caustic

Issues related to the computation of wave fields in an acoustic medium near caustics are considered. A boundary condition on a caustic is established, and the Green’s function of a boundary value problem for the general case of a varying speed of sound is constructed. For this purpose, an auxiliary Goursat problem is considered and a system of its particular solutions is constructed using hyper-geometric functions. A Volterra integral equation for the Green’s function is obtained, and an algorithm for its expansion with respect to smoothness is described. A finite difference scheme approximating the solution of the differential problem with an unbounded coefficient is proposed. Numerical results are presented.     

An algorithm for approximating piecewise linear concave functions from sample gradients

An effective algorithm for solving stochastic resource allocation problems is to build piecewise linear, concave approximations of the recourse function based on sample gradient information. Algorithms based on this approach are proving useful in application areas such as the newsvendor problem, physical distribution and fleet management. These algorithms require the adaptive estimation of the approximations of the recourse function that maintain concavity at every iteration. In this paper, we prove convergence for a particular version of an algorithm that produces approximations from stochastic gradient information while maintaining concavity.     

The factorization method for cracks in inhomogeneous media

We consider the inverse scattering problem of determining the shape and location of a crack surrounded by a known inhomogeneous media. Both the Dirichlet boundary condition and a mixed type boundary conditions are considered. In order to avoid using the background Green function in the inversion process, a reciprocity relationship between the Green function and the solution of an auxiliary scattering problem is proved. Then we focus on extending the factorization method to our inverse shape reconstruction problems by using far field measurements at fixed wave number. We remark that this is done in a non intuitive space for the mixed type boundary condition as we indicate in the sequel.     

Fast calculation of coefficients in the Smolyak algorithm

For many numerical problems involving smooth multivariate functions on d-cubes, the so-called Smolyak algorithm (or Boolean method, sparse grid method, etc.) has proved to be very useful. The final form of the algorithm (see equation (12) below) requires functional evaluation as well as the computation of coefficients. The latter can be done in different ways that may have considerable influence on the total cost of the algorithm. In this paper, we try to diminish this influence as far as possible. For example, we present an algorithm for the integration problem that reduces the time for the calculation and exposition of the coefficients in such a way that for increasing dimension, this time is small compared to dn, where n is the number of involved function values.     

Existence and stability of planar diffusion waves for 2-D Euler equations with damping

To study the non-linear stability of a non-trivial profile for a multi-dimensional systems of gas dynamics, the combination of the Green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential. In this paper, we study the stability of a planar diffusion wave for the isentropic Euler equations with damping in two-dimensional space. By introducing an approximate Green function for the linearized equations around the planar diffusion wave and by applying the energy method, we prove the global existence and the L₂ convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave. The decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the L₂ norm. Furthermore, the constructed approximate Green function in this paper can be used for the pointwise and the Lp estimates of the solutions concerned. In fact, the approach by combining of the Green function and energy method can be applied to other system especially when the derivatives of the coefficients in the system have certain time decay properties.