Reduced Functions, Gradients and Hessians from Fixed-Point Iterations for State Equations

In design optimization and parameter identification, the objective, or response function(s) are typically linked to the actually independent variables through equality constraints, which we will refer to as state equations. Our key assumption is that it is impossible to form and factor the corresponding constraint Jacobian, but one has instead some fixed-point algorithm for computing a feasible state, given any reasonable value of the independent variables. Assuming that this iteration is eventually contractive, we will show how reduced gradients (Jacobians) and Hessians (in other words, the total derivatives) of the response(s) with respect to the independent variables can be obtained via algorithmic, or automatic, differentiation (AD). In our approach the actual application of the so-called reverse, or adjoint differentiation mode is kept local to each iteration step. Consequently, the memory requirement is typically not unduly enlarged. The resulting approximating Lagrange multipliers are used to compute estimates of the reduced function values that can be shown to converge twice as fast as the underlying state space iteration. By a combination with the forward mode of AD, one can also obtain extra-accurate directional derivatives of the reduced functions as well as feasible state space directions and the corresponding reduced or projected Hessians of the Lagrangian. Our approach is verified by test calculations on an aircraft wing with two responses, namely, the lift and drag coefficient, and two variables, namely, the angle of attack and the Mach number. The state is a 2-dimensional flow field defined as solution of the discretized Euler equation under transonic conditions.     

Quadratic Störmer‐type methods for the solution of the Boussinesq equation by the method of lines

We study the numerical treatment of Boussinesq PDE equation using the method of lines. For the space discretization, we choose either classical finite differences or Fourier pseudospectral methods. Both cases result in a system of second‐order ordinary differential equations (ODEs) that is quadratic. In order to take advantage of this special feature, we choose to solve the ODE system using a new type of hybrid Numerov method specially constructed for such problems. Other efficient ODE solvers taken from the literature are used to solve the system of ODEs as well. By taking all the combinations of space discretization methods and ODE solvers, we discuss the stability and accuracy features revealed from the numerical tests. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Necessary and sufficient conditions for linear suboptimality in constrained optimization

This article is devoted to the study of some extremality and optimality notions that are different from conventional concepts of optimal solutions to optimization-related problems. These notions reflect certain amounts of linear subextremality for set systems and linear suboptimality for feasible solutions to multiobjective and scalar optimization problems. In contrast to standard notions of optimality, it is possible to derive necessary and sufficient conditions for linear subextremality and suboptimality in general nonconvex settings, which is done in this article via robust generalized differential constructions of variational analysis in finite-dimensional and infinite-dimensional spaces.      

Symbolic computing the exact distributions of L-statistics from a uniform distribution

The exact probability density function of linear combinations of k=k(n) order statistics selected from the whole order statistics (L-statistic) based on a random sample of size n from the uniform distribution on [0, 1] was derived by Matsunawa (1985, Ann. Inst. Statist. Math., 37, 1–16). As the main expression for the density function given by Matsunawa is not complete for the general situation, we first provide the corrections for this formula. Second, we propose a simple scheme involving symbolic computing for evaluating the corrected version of the density function. The cumulative distribution function and the r-th mean of his L-statistic are also derived.     

A Numerical Approach of the sentinel method for distributed parameter systems

In this paper we consider the problem of detecting pollution in some non linear parabolic systems using the sentinel method. For this purpose we develop and analyze a new approach to the discretization which pays careful attention to the stability of the solution. To illustrate convergence properties we give some numerical results that present good properties and show new ways for building discrete sentinels.      

An alternative approach to nonlinear filtering

The structure of a nonlinear filter with observation process having continuous and discontinuous components is considered. The approach is based on the so-called “Bayes” formula for conditional expectations. “Fubini” type theorems for stochastic integrals are given and used to obtain the representations of an optimal estimate and of the conditional likelihood ratio. A linear unnormalized filtering equation for controlled system process is derived.     

大花无柱兰遗传多样性的SRAP标记分析

大花无柱兰是一种珍稀兰科植物,具有一定的观赏和药用价值,但数量十分稀少,该物种亟待保护。本研究采用SRAP分子标记技术,对10个居群的115份DNA样品进行PCR扩增,并开展遗传多样性分析。从81对引物中筛选出9个条带清晰、多态性好、重复性高的引物组合,共扩增得到305条谱带。在物种水平上,多态性比率（PPB）为100%,Nei’s基因多样性指数（H）为0.209 8,Shannon’s指数（I）为0.340 2;在居群水平上,PPB为24.59%～52.13%,H为0.079 6～0.165 5,I为0.120 9～0.252 3。居群水平上,基因分化度（Gst）为0.520 9,基因流（Nm）为0.459 9,遗传距离为0.091 9～0.198 4。UPGMA聚类结果表明,10个居群可分为3大类,地理距离相近的居群优先聚集。大花无柱兰的遗传多样性较为丰富,居群间存在一定的遗传分化和基因交流,可采用就地保护和人工栽培等方式加以保护。     

一种气象观测数据求导的新方法

气象观测数据不可避免地带有观测误差,而对带有误差的离散数据求导是一个不适定的反问题.为了解决这一长期困扰气象工作者的问题,利用吉洪诺夫正则化思想,提出了矩形区域内观测数据的二维一阶偏导数重构算法.通过系列模拟观测数据试验检验了该算法的性能,表明该算法有效且计算精度高.另外,应用该算法对气象观测资料进行客观分析,结果表明：该算法作为一种新的客观分析方法是可行的,并且还能提高小尺度天气系统的识别能力. 关键词： 数值微分 吉洪诺夫正则化 客观分析     

Products of weighted composition and differentiation operators into weighted Zygmund and Bloch spaces

We characterize boundedness and compactness of products of differentiation operators and weighted composition operators between weighted Banach spaces of analytic functions and weighted Zygmund spaces or weighted Bloch spaces with general weights.