捕食模型高精度参数的估计

研究了捕食者模型在多种观测值条件下的非线性微分方程组参数拟合问题.首先利用龙格-库塔法进行微分方程数值计算,通过首次积分项变形建立线性回归方程,进行最小二乘拟合;其次,考虑到实验数据包含随机误差的扰动,引进正规方程组对模型进行误差分析;最后针对时间变量也出现误差,采用拉依达准则筛选,然后提出了一种较为简单的参数分段动态估计算法.     

一类余维3的鞍-焦点异宿环分支

对余维3系统Xμ(x)具有包含一个双曲鞍-焦点O1和一个非双曲鞍-焦点O2的异宿环￡进行了研究.证明了在￡的邻域内有可数无穷条周期轨线和异宿轨线,当非粗糙异宿轨线ΓO破裂时Xμ(x)会产生同宿轨分支,并给出了相应的分支曲线和两种同宿环共存的参数值.在3参数扰动下ΓO破裂和O2点产生Hopf分支的情况下,在￡的邻域内有一条含O1点同宿环,可数无数多条的轨线同宿于O2点分支出的闭轨HO,一条或无穷多条(可数或连续统的)异宿轨线等.     

一类余维3的鞍-焦点异宿环分支

对余维3系统X_μ(x)具有包含一个双曲鞍-焦点O_1和一个非双曲鞍-焦点O_2的异宿环f进行了研究.证明了在f的邻域内有可数无穷条周期轨线和异宿轨线,当非粗糙异宿轨线Γ~0破裂时X_μ(x)会产生同宿轨分支,并给出了相应的分支曲线和两种同宿环共存的参数值.在3参数扰动下Γ~0破裂和O_2点产生Hopf分支的情况下,在f的邻域内有一条含O_1点同宿环,可数无效多条的轨线同宿于O_2点分支出的闭轨H_0,一条或无穷多条(可数或连续统的)异宿轨线等.     

一种修正的CD共轭梯度法及其全局收敛性

求解无约束优化问题的共轭梯度法,其搜索方向的下降性往往依赖于所采用的线性搜索.将提出一种修正的CD算法,其搜索方向d_k始终满足1-1/u≤(-g_k~Td_k)/(‖g_k‖~2)≤1+1/u(u1),即算法在不依赖任何线性搜索的情况下能始终产生充分下降方向.同时,当采用精确线性搜索时,该修正的CD算法就是标准的CD共轭梯度法.在适当条件下,还证明了修正的CD算法在强Wolfe线性搜索下具有全局收敛性.最后,我们给出了相应的数值结果,说明了算法是一种有效的算法.     

有界噪声扰动下平面微分系统周期轨的分支

本文研究有界噪声扰动下平面微分系统周期轨的分支问题.首先介绍平面微分系统周期轨的三类经典分支:叉型分支、鞍结分支和跨临界分支,并给出其分支方向以及分支出的周期轨的稳定性.然后讨论有界噪声扰动下平面微分系统周期轨的分支现象,考虑该扰动系统极小正向不变集的个数随参数值变化而变化的规律,证明该系统在一定条件下发生所谓的硬分支现象,并给出分支参数值的阶数估计.最后给出具体的例子并进行数值模拟,形象直观地说明本文结果.     

一类捕食者-食饵系统的全局结构

本文中我们证明了关于一般捕食者-食饵系统不存在闭轨线的定理,即文中定理2.应用这一定理和关于捕食者-食饵系统极限环的存在唯一性定理[1],我们完成了在各种参数条件对一个具体的捕食者-食饵系统模型[2]的研究.     

一类Fermi气体光晶格非线性机制的轨线研究

研究了一类Fermi气体光晶格轨线的非线性扰动模型.首先求得了Fermi气体光晶格在无扰动情形下模型轨线的精确解.然后引入一组广义泛函分析同伦映射,构造一组迭代系统,得到了Fermi气体光晶格非线性扰动模型轨线的任意次渐近解.最后讨论了一个微扰系统.该文在方法上可较方便地得到轨线的渐近表示式.     

一类3维反转系统中的异维环分支

研究了一类3维反转系统中包含2个鞍点的对称异维环分支问题, 且仅限于研究系统的线性对合R的不变集维数为1的情形. 给出了R-对称异宿环与R-对称周期轨线存在和共存的条件, 同时也得到了R-对称的重周期轨线存在性. 其 次, 给出了异宿环、 同宿轨线、 重同宿轨线和单参数族周期轨线的存在性、 唯一性和共存性等结论, 并且发现不可数无穷条周期轨线聚集在某一同宿轨线的小邻域内. 最后给出了相应的分支图.     

两个谱共轭梯度法的全局收敛性及数值效果

谱共轭梯度法是求解无约束优化的一种有效算法.该文首先对JJSL共轭参数[Jiang et al.Computational and Applied Mathematics,2021,40（174）]进行投影修正,再通过选取合适谱参数以保证其搜索方向有下降性,从而得到两个有效的谱共轭梯度法.一般假设下,分别使用常规非精确线搜索计算步长,获得这两个新算法的全局收敛性.数值试验结果以及相应性能图进一步说明其数值有效性.     

求解无约束优化问题的新谱共轭梯度法及其收敛性

强Wolfe条件不能保证标准CD共轭梯度法全局收敛．本文通过建立新的共轭参数,提出无约束优化问题的一个新谱共轭梯度法,该方法在精确线搜索下与标准CD共轭梯度法等价,在标准wolfe线搜索下具有下降性和全局收敛性．初步的数值实验结果表明新方法是有效的,适合于求解非线性无约束优化问题．     

On the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer

The problem of parabolic spline interpolation, according to Subbotin, of functions with large gradients in the boundary layer is considered. In the case of a uniform grid it is proved and in the case of the Shishkin mesh it is experimentally shown that with a parabolic spline interpolation of functions with large gradients in the exponential boundary layer, the error can unrestrictedly increase when the small parameter tends to zero and the number of grid nodes is fixed. An approximation process using parabolic splines with defect 1 is proposed; the error estimates are found to be uniform in the small parameter.     

Central limit theorems for realized volatility under hitting times of an irregular grid

We consider a continuous semi-martingale sampled at hitting times of an irregular grid. The goal of this work is to analyze the asymptotic behavior of the realized volatility under this rather natural observation scheme. This framework strongly differs from the well understood situations when the sampling times are deterministic or when the grid is regular. Indeed, neither Gaussian approximations nor symmetry properties can be used. In this setting, as the distance between two consecutive barriers tends to zero, we establish central limit theorems for the normalized error of the realized volatility. In particular, we show that there is no bias in the limiting process.     

Data-driven modeling of the temporal evolution of breakers’ states in the French electrical transmission grid

In electrical transmission grids, it is common to observe the states of circuit breakers. While they are known at irregular times, system modeling and grid state estimation are of the highest importance to ensure secure operations. This paper proposes a richer method to estimate the grid state over its reference configurations based on the temporal evolution of its breakers’ states. The first contribution consists in developing a general multi-observation continuous-time finite-state Hidden Markov Model with filter-based parameter estimation to infer the hidden state (e.g., the grid reference configuration) handling multiple observed processes with irregular “jump” times (e.g., the breakers’ states). As a second contribution, we build a numerical scheme with no discretization error adapted to all state jumps generated by the observed processes. Finally, we apply our model to simulated and real data to illustrate the approach’s performance. The available data consists of historical records of breakers’ states during the electrical transmission grid operated normally. For this real-data-driven application, we also present a clustering approach to identify the set of grid reference configurations.     

高精度参数估计问题

对高精度参数的估计问题进行了研究.在观测数据无误差的情况下,将微分方程组转化为线性方程组,利用矩阵的奇异值分解给出了参数的最优解.在有观测数据误差的情况下,采用高斯-牛顿迭代法进行求解,给出了改进的高斯-牛顿法和阻尼最小二乘算法;通过灰色估计法给出了模型的初始解,通过微分方程数值解法计算模型迭代过程中误差和偏导数.最后,通过对迭代过程中的状态变量引入误差项,导出了基于总体最小二乘的高斯-牛顿迭代法,从系统的角度解决了观测时间有误差下的参数估计问题.     

Robust and Accurate Inference via a Mixture of Gaussian and Student’s t Errors

A Gaussian measurement error assumption, that is, an assumption that the data are observed up to Gaussian noise, can bias any parameter estimation in the presence of outliers. A heavy tailed error assumption based on Student’s t distribution helps reduce the bias. However, it may be less efficient in estimating parameters if the heavy tailed assumption is uniformly applied to all of the data when most of them are normally observed. We propose a mixture error assumption that selectively converts Gaussian errors into Student’s t errors according to latent outlier indicators, leveraging the best of the Gaussian and Student’s t errors; a parameter estimation can be not only robust but also accurate. Using simulated hospital profiling data and astronomical time series of brightness data, we demonstrate the potential for the proposed mixture error assumption to estimate parameters accurately in the presence of outliers. Supplemental materials for this article are available online.     

A polycycle and limit cycles in a non-differentiable predator-prey model

For a non-differentiable predator-prey model, we establish conditions for the existence of a heteroclinic orbit which is part of one contractive polycycle and for some values of the parameters, we prove that the heteroclinic orbit is broken and generates a stable limit cycle. In addition, in the parameter space, we prove that there exists a curve such that the unique singularity in the realistic quadrant of the predator-prey model is a weak focus of order two and by Hopf bifurcations we can have at most two small amplitude limit cycles.     

卫星运动预测及规避的建模与求解

近年来,随着各国侦查卫星系统的不断完善,有效根据观测数据预测卫星的轨迹,同时建立适当的规避防范措施变得尤为重要·通过研究地面监测站监测的卫星数据及地理信息,提出了一种基于三维坐标系变换的卫星轨迹求解模型,具体建立模型的方法为:建立4种不同的坐标系,通过卫星坐标在4种坐标系中的变换,将地面检测站检测到的卫星数据进行处理,拟合出卫星的六个运动参数,得出卫星运动轨迹方程,并较为精准的预测卫星被观测站观测到的情况,以及任意时刻星下点的经纬度坐标.针对卫星的运动情况,为满足我方军事目标的运动需求,根据新疆地图建立军队行军路线的有向图,然后,在考虑卫星规避的情况下对每条路线进行模拟分析,最终,通过每条路线模拟分析的结果进行比对,得出最优的行军路线.     

A characterization of mapping unstructured grids onto structured grids and using multigrid as a preconditioner

Many problems based on unstructured grids provide a natural multigrid framework due to using an adaptive gridding procedure. When the grids are saved, even starting from just a fine grid problem poses no serious theoretical difficulties in applying multigrid. A more difficult case occurs when a highly unstructured grid problem is to be solved with no hints how the grid was produced. Here, there may be no natural multigrid structure and applying such a solver may be quite difficult to do. Since unstructured grids play a vital role in scientific computing, many modifications have been proposed in order to apply a fast, robust multigrid solver. One suggested solution is to map the unstructured grid onto a structured grid and then apply multigrid to a sequence of structured grids as a preconditioner. In this paper, we derive both general upper and lower bounds on the condition number of this procedure in terms of computable grid parameters. We provide examples to illuminate when this preconditioner is a useful (e. g.,p orh-p formulated finite element problems on semi-structured grids) or should be avoided (e.g., typical computational fluid dynamics (CFD) or boundary layer problems). We show that unless great care is taken, this mapping can lead to a system with a high condition number which eliminates the advantage of the multigrid method. This work was partially supported by ONR Grant # N0014-91-J-1576.     

Parameters estimation,mixed synchronization,and antisynchronization in chaotic systems

Mixed synchronization between two Hindmarsh–Rose neuron models is realized by optimizing the scheme of Lyapunov function with two selectable gain coefficients. Based on the Lyapunov stability theory, the distribution of synchronization region and the nonsynchronization region in the two‐parameter phase space is calculated, respectively. And then the optimized parameter observers and controllers are approached analytically. All unknown parameters with different orders of magnitude are identified accurately, and the error function for corresponding variables decreases to stable value when the two gain coefficients are given values in the synchronization region. Otherwise, only the four larger unknown parameters are estimated exactly and the error function of corresponding variables decreases stably to certain minimal value with an order about 1 × 10^?6, whereas the smallest unknown parameter is approached greatly although the error of corresponding variables are stabilized within certain transient period. © 2014 Wiley Periodicals, Inc. Complexity 20: 64–73, 2014     

Two optimization problems in linear regression with interval data

We consider a linear regression model where neither regressors nor the dependent variable is observable; only intervals are available which are assumed to cover the unobservable data points. Our task is to compute tight bounds for the residual errors of minimum-norm estimators of regression parameters with various norms (corresponding to least absolute deviations (LAD), ordinary least squares (OLS), generalized least squares (GLS) and Chebyshev approximation). The computation of the error bounds can be formulated as a pair of max–min and min–min box-constrained optimization problems. We give a detailed complexity-theoretic analysis of them. First, we prove that they are NP-hard in general. Then, further analysis explains the sources of NP-hardness. We investigate three restrictions when the problem is solvable in polynomial time: the case when the parameter space is known apriori to be restricted into a particular orthant, the case when the regression model has a fixed number of regression parameters, and the case when only the dependent variable is observed with errors. We propose a method, called orthant decomposition of the parameter space, which is the main tool for obtaining polynomial-time computability results.