A note on estimation of the Jacobian elliptic parameter in cnoidal wave theory

A new and simple calculating technique for the Jacobian elliptic parameter is presented in this study. The technique is very useful in generating a train of cnoidal waves in both laboratory and numerical wave tanks. Upon specification of water depth, the wave height and either the wave period or the wavelength, the proposed technique uses the Newton–Raphson method to estimate the Jacobian elliptic parameter directly, without trial and error procedures or look-up in tables. It is shown that the technique provides equally accurate results as the ad hoc methods previously used.     

A combined numerical–empirical method to calculate finite-time Lyapunov exponents from experimental time series with application to vessel capsizing

This paper presents a method to calculate finite-time Lyapunov exponents (FTLEs) for experimental time series using numerical simulation to approximate the local Jacobian of the system at each time step. This combined numerical–experimental approach to the calculation of FTLE is applicable to any physical system which can be numerically approximated. By way of example, the method is applied to the problem of vessel capsize.     

Geostatistical inversion of coupled problems: dealing with computational burden and different types of data

Practical application of geostatistical inversion to coupled problems is hampered by a number of difficulties. In this paper, we address two of them: first, the computational cost of sensitivity (Jacobian) matrices and, second, the evaluation of the relative weights of different types of data. Regarding the first, we revise the adjoint state equations to propose a form whose cost is independent of the number of unknown parameters and only grows with the number of observation wells. Regarding the second, we derive expressions for the relative weights of different types of data. These expressions are based on minimizing the expected likelihood, rather than the likelihood itself. The efficiency of both improvements is tested on a synthetic example. The example analyzes a wide range of groundwater flow and solute transport conditions. Yet, the expected likelihood consistently yields the optimal weights. The proposed form of the adjoint state equations leads to one order of magnitude reduction in CPU time with respect to the conventional sensitivity equations.     

Approximation of Gaussian spatial autoregressive models for massive regular square tessellation data

Most of the literature to date proposes approximations to the determinant of a positive definite n × n spatial covariance matrix (the Jacobian term) for Gaussian spatial autoregressive models that fail to support the analysis of massive georeferenced data sets. This paper briefly surveys this literature, recalls and refines much simpler Jacobian approximations, presents selected eigenvalue estimation techniques, summarizes validation results (for estimated eigenvalues, Jacobian approximations, and estimation of a spatial autocorrelation parameter), and illustrates the estimation of the spatial autocorrelation parameter in a spatial autoregressive model specification for cases as large as n = 37,214,101. The principal contribution of this paper is to the implementation of spatial autoregressive model specifications for any size of georeferenced data set. Its specific additions to the literature include (1) new, more efficient estimation algorithms; (2) an approximation of the Jacobian term for remotely sensed data forming incomplete rectangular regions; (3) issues of inference; and (4) timing results.     

An algorithm for fast 3D inversion of surface electrical resistivity tomography data: application on imaging buried antiquities

In this work a new algorithm for the fast and efficient 3D inversion of conventional 2D surface electrical resistivity tomography lines is presented. The proposed approach lies on the assumption that for every surface measurement there is a large number of 3D parameters with very small absolute Jacobian matrix values, which can be excluded in advance from the Jacobian matrix calculation, as they do not contribute significant information in the inversion procedure. A sensitivity analysis for both homogeneous and inhomogeneous earth models showed that each measurement has a specific region of influence, which can be limited to parameters in a critical rectangular prism volume. Application of the proposed algorithm accelerated almost three times the Jacobian (sensitivity) matrix calculation for the data sets tested in this work. Moreover, application of the least squares regression iterative inversion technique, resulted in a new 3D resistivity inversion algorithm more than 2.7 times faster and with computer memory requirements less than half compared to the original algorithm. The efficiency and accuracy of the algorithm was verified using synthetic models representing typical archaeological structures, as well as field data collected from two archaeological sites in Greece, employing different electrode configurations. The applicability of the presented approach is demonstrated for archaeological investigations and the basic idea of the proposed algorithm can be easily extended for the inversion of other geophysical data.     

具有任意阶非线性薛定谔方程的新行波解

对具有任意阶非线性薛定谔方程的行波解进行了研究。给出具有任意阶非线性薛定谔方程的精确解。利用行波变换和辅助函数法把具有任意阶非线性薛定谔方程最终转化为一个非线性常微分方程的解,通过对这个微分方程的研究可以得到具有任意阶非线性薛定谔方程的行波解。具有任意阶非线性薛定谔方程存在的孤立波解、三角孤立波解、扭孤立波解、Jacobi椭圆函数解。得到了具有任意阶非线性薛定谔方程新的Jacobi椭圆函数解。     

基于解析雅克比矩阵的E-E_x广域电磁法一维并行约束反演

对于野外实测数据的反演解释,一维反演仍然占据着重要地位。提出一种使用探测深度和等对数域剖分的方式来剖分层厚、只进行电阻率单参数反演的方案。反演时,先使用MT反演来建立初始模型,之后采用MATLAB自带的parfor循环和最优化工具箱来进行最优化计算,使用解析法来计算雅克比矩阵,从而大大提高了计算速度。对于实际地电模型,充分考虑到已有的先验信息,通过使用井震约束反演有效降低了反演多解性,使反演结果更加接近真实的地层情况。     

Stability analysis of the heave motion of marine cable-body systems

The dynamic stability of the heave motion of marine cable-body systems operating in alternating taut–slack conditions is considered, based upon a single-degree-of-freedom model. In this model the fluid damping is linearised and the cable is replaced by a spring of bi-linear stiffness. The period-one Poincare map is derived, and its stability is analysed by examining its Jacobian matrix. Numerical simulations are also carried out to show the transition from a periodic response to a chaotic one through period doubling.     

集合平均所构成的动力系统及其特性研究

以Lorenz系统为例,推导出集合平均所定义的完整动力方程(均值方程),将初值的集合平均问题作为一个广义的动力系统问题来进行研究;对于双初值和多初值的均值方程,利用定性理论分析了其吸引中心的位置和个数,并使用数值试验进行了验证,结果表明平均值的吸引子的结构与原解的吸引子位置、数量和结构均有不同。对均值方程的特征矩阵分析表明,定点附近的稳定性与原方程相同,而且特征方程所对应的特征值也与原方程相同。均值方程对应的相流散度为负值且数值上与原系统相同,因此其在相空间中的体积收缩速度和原系统相同,最终趋向一个低纬曲面,均值方程的这个性质使得Lorenz系统的集合平均解趋于一个吸引子。均值方程可以保持原方程的耗散特性、吸引子特性,但稳定点位置和个数发生了变化,非定点处的Jacobian矩阵特征值与原系统也有不同。简而言之,一旦使用了集合平均方法,那么集合数值解并不是原系统的解,仅保持了原系统的部分特征,因而集合平均是否有效需要根据具体问题和其他外部限定条件才能确定。     

大气遥感辐射传输模式的一种有效的Jacobian算法

介绍并验证了一种快速、精确地计算Jacobian的解析方法(辐射传输方程的线性化,简称LR方法),将其结果与NAST-I真正的线性模式及数值方法(差分方法)的Jacobian值进行比较。结果表明,这种解析算法不仅快速、精确,而且独立于辐射传输方程,在有效Jacobian计算中只需要输入大气透过率,可适用于任何垂直探测器权重函数的计算。