Data fitting with geometric-programming-compatible softmax functions

Motivated by practical applications in engineering, this article considers the problem of approximating a set of data with a function that is compatible with geometric programming (GP). Starting with well-established methods for fitting max-affine functions, it is shown that improved fits can be obtained using an extended function class based on the softmax of a set of affine functions. The softmax is generalized in two steps, with the most expressive function class using an implicit representation that allows fitting algorithms to locally tune softness. Each of the proposed function classes is directly compatible with the posynomial constraint forms in GP. Max-monomial fitting and posynomial fitting are shown to correspond to fitting special cases of the proposed implicit softmax function class. The fitting problem is formulated as a nonlinear least squares regression, solved locally using a Levenberg–Marquardt algorithm. Practical implementation considerations are discussed. The article concludes with numerical examples from aerospace engineering and electrical engineering.     

非均匀背景下的红外图像曲面拟合分割

红外图像往往存在着背景不一致的特点,因此在分割时无法有效地提取出目标。本文针对这一问题提出了一种基于曲面拟合的图像分割方法,以分割非均匀背景下的红外目标。这种方法首先对背景进行光顺限制的曲面拟合,再通过设置一个偏移量来形成阈值曲面。通过研究发现,曲面拟合时的偏离项和光顺项的权重系数比是由图像背景的双拉普拉斯变换以及噪声均方差共同决定的,从而在估计噪声均方差的基础上实现了对权重系数的自适应选取。从仿真结果可以看出,本文提出的曲面拟合分割法在背景去除和目标提取上要优于传统的Ostu法和局部阈值法。     

基于椭球约束的径向基函数隐式曲面重建

对三维点云进行隐式曲面重建是解决虚拟现实等方面所存在问题的关键.本文提出了一种基于椭球约束的径向基函数隐式曲面建模的算法,该方法在仅有点云信息的前提下仍能够非常精确地拟合点云数据.当点云稀疏时拟合后的模型可以非常好地保证模型的主要特征,但对于拟合大规模数据点集时,模型会出现冗余现象,保特征效果不理想且效率低下.需将点云进行适当分割,然后并行拟合被分割点云并将它们进行光滑拼接处理.实验效果表明该算法保特征效果非常好且效率明显提高.     

Response Surface Optimization in the Presence of Internal Noise With Application to Optimal Alignment of Carbon Nanotubes

Internal noise, which means fluctuation of input factors around their set values, is common in many experiments in the physical and engineering sciences. Existing methods for response surface optimization in the presence of internal noise typically adopt a two-step approach: (a) fitting a response model as a function of the set value and (b) using Monte Carlo methods to account for internal noise while optimizing the response. In this article, motivated by a problem in optimizing alignment of carbon nanotubes (CNT), we propose a Bayesian approach for response surface optimization in the presence of internal noise. A unit-free and interpretable measure to quantify the strength of internal noise is proposed. Suitable objective functions or performance measures consistent with the overall goal of optimizing the response function are identified, methods for estimating them from available experimental data are suggested, and simulations are conducted to compare them with respect to their ability to account for internal noise in the optimization problem. The loss accrued by ignoring the internal noise in the optimization problem is quantified and studied via simulation. The proposed method is demonstrated through its application in the CNT alignment problem.     

Calibration of probe volume in fluorescence correlation spectroscopy

In fluorescence correlation spectroscopy (FCS), an accurate evaluation of the probe volume is the basis of correct interpretation of experimental data and solution of an appropriate diffusion model. Poor fitting convergence has been a problem in the determination of the dimensional parameters, the beam radius, omega, and the distance along the optical axis of the probe volume, l. In this work, the instability of fitting during the calibration process is investigated by examining the chi(2) surfaces. We demonstrate that the minimum of chi(2) in the omega dimension is well defined for both converging and diverging data. The difficulty of fitting comes from the l dimension. The uncertainty in l could be significantly larger than that in omega, as determined by F-statistics. A modified calibration process is recommended based on examining two data treatment methods, combining several short data sets into a single long run and averaging the correlation functions of several short data sets. It is found that by using the mean of several converging correlation functions from short data sets instead of a long time correlation, more stable and consistent dimensional parameters are extracted to define the probe volume.     

Convex piecewise-linear fitting

We consider the problem of fitting a convex piecewise-linear function, with some specified form, to given multi-dimensional data. Except for a few special cases, this problem is hard to solve exactly, so we focus on heuristic methods that find locally optimal fits. The method we describe, which is a variation on the K-means algorithm for clustering, seems to work well in practice, at least on data that can be fit well by a convex function. We focus on the simplest function form, a maximum of a fixed number of affine functions, and then show how the methods extend to a more general form.     

Thermocapillary Convection Problem for Two Compressible Immiscible Fluids

A study is made of the motion of two compressible barotropic fluids separated by a closed interface where the surface tension depending on the temperature is taken into account. We have the so-called Marangoni effect which plays the leading role in thermodynamics under low gravity. Local (in time) unique solvability of the problem is obtained in Hölder spaces of functions with power-like decay at infinity. After the passage to Lagrangian coordinates, we arrive at a nonlinear, noncoercive initial boundary—value problem which is equivalent to the original one for small time. We establish the existence theorem for this problem on the basis of the solvability of the problem for two compressible fluids with constant surface tension coefficient on the interface. All results are obtained under some restrictions on fluid densities and viscosities which mean that the fluids are not so different from each other.     

基于分数阶傅里叶变换的NURBS曲面拟合

为了实现对自由曲面模型的高精度非均匀有理B样条(NURBS)曲面拟合,提出了一种基于分数阶傅里叶变换的NURBS曲面拟合方法.首先,借助分数阶傅里叶变换对自由曲面模型点云数据的高程图像展开分析,从中提取出表征自由曲面表面三维结构的特征点;然后,利用外切圆取点法结合提取的特征点选取用于NURBS曲面拟合的数据点;最后,利...     

风险决策的情绪数理模型

已有风险决策模型只能对部分个体的决策结果作出解释。其根本原因在于3个方面：没有将个体情绪包含在内,所采用的单一决策模式及单向数据拟合方法不合理。这些问题需要采用4项措施来解决。依据情绪函数的一般特性,为结果及其实现概率分别设计情绪函数。根据信息的控制加工逻辑,确定情绪函数之间的加和关系,用多模式决策替代单一模式决策。建立双向拟合方法,替代单向拟合方法。从而建立决策情绪模型,使所有个体的选择行为都获得解释。利用前景理论经典案例及14个异常行为案例,对所建模型进行了验证。所建模型为风险决策问题的解决提供了一个新的途径.     

An adjoint method in inverse problems of chromatography

How to determine adsorption isotherms is an issue of significant importance in chromatography. A modern technique of obtaining adsorption isotherms is to solve an inverse problem so that the simulated batch separation coincides with actual experimental results. In this work, as well as the natural least-square approach, we consider a Kohn–Vogelius type formulation for the reconstruction of adsorption isotherms in chromatography, which converts the original boundary fitting problem into a domain fitting problem. Moreover, using the first momentum regularizing strategy, a new regularization algorithm for both the Equilibrium-Dispersive model and the Transport-Dispersive model is developed. The mass transfer resistance coefficients in the Transport-Dispersive model are also estimated by the proposed inverse method. The computation of the gradients of objective functions for both of the two models is derived by the adjoint method. Finally, numerical simulations for both a synthetic problem and a real-world problem are given to show the robustness of the proposed algorithm.     

光纤模斑谱在应变检测中的应用研究

在本文中介绍了一种以光纤模斑谱为基础的表面应变测量方法。首先分析了相关的理论基础并且介绍了这种检测方法的原理,然后给出了实验结果以及相关的曲线拟合算法,最后利用LabVIEW对实验数据进行了拟合运算,并给出得到的拟合曲线。结果表明采用这种方法能够实现对于表面应变的检测。     

Distributed dynamic load identification based on shape function method and polynomial selection technique

A new approach based on shape function method of moving least square fitting (SFM_MLSF) and polynomial selection technique is proposed in this paper for distributed dynamic load identification. The distributed dynamic load is represented as a product of spatial distribution function and time history, and the two parts are assumed to be independent. The modal transformation of structural dynamic equation throws light on the fact that all modal loads share the same form with the distributed dynamic load in time domain. As the structural modal parameters are known, the time history of dynamic load can be precisely identified through SFM_MLSF method which approximates the local dynamic load with shape functions in the moving supporting time domain. Then, the spatial distribution function of the load is substituted as a series of basic functions, and the identification of distribution function is transformed into a linear fitting problem. Through polynomial selection technique based on error reduction ratio, the significant components are picked out from the basis function responses, which greatly improves stability and precision of the load distribution function. During the inverse analyses of both the time history and distribution function, appropriate regularization methods are still applied to overcome the unavoidable ill-conditioned problem. Numerical examples demonstrate the validity and accuracy of the proposed method.     

Modeling of surface and subsurface crack behavior under contact load in the presence of lubricant

A new mathematical model for lubricated elastic solids weakened by cracks is proposed. Surface and subsurface cracks are taken into account, and the interaction of lubricant with elastic solids within cavities of surface cracks is regarded as the most interesting aspect of the problem. The boundary conditions characterizing the behavior of lubricant within crack cavities such as pressure rise in crack cavities fully filled with lubricant as well as other boundary and additional conditions are derived. The problem is reduced to a system of integro-differential equations with nonlinear boundary conditions in the form of alternating equations and inequalities. A new iterative numerical method is developed for solution of the proposed problem. The method guarantees conservation of lubricant volumes trapped within closed crack cavities and allows for all three functions (normal and tangential displacement jumps and normal stress applied to crack faces) characterizing the problem solution to be determined simultaneously. Examples of numerical results for surface and subsurface cracks are presented and numerical and asymptotic results for small subsurface cracks are compared to each other. The numerical analysis indicates that depending on a surface crack orientation its normal stress intensity factor may be two or more orders of magnitude higher than the one for a similar subsurface one.     

自适应变长线性拟合的多阵方位平滑同步方法

在多阵方位轨迹关联融合过程中,由于单阵输出方位的节拍和时刻不同需要进行同步并且在拟合过程中定长、定阶拟合函数与任意方位曲线存在失配问题,为此文章提出了一种自适应变长线性拟合方法。该方法首次采用分段线性拟合对方位轨迹正切值进行拟合,通过实时监测跳变点及平滑后的拟合效果(即连续在拟合线段同一侧的平滑前的轨迹点占平滑前总轨迹点数的比例),实时改变拟合线段长度。试验数据处理结果表明,该方法可以解决多阵方位同步问题,对各种方位轨迹曲线有较好的拟合精度且计算速度快,为后续多阵方位轨迹关联融合处理提供支撑。     

A surface fitting method for three dimensional scattered data

A method is presented that may be used to empirically establish the type of relationship that is present between a response variable and its influencing factors, by fitting a mathematical model to three dimensional scattered data. The generated response surface is composed of continuous triangular planes that are fitted to the corresponding data in the least squares sense. The method may be easily implemented. It requires some fairly large number of scattered data, two initial boundary conditions and a desired accuracy for the band-wise partitioning of the data. The proposed surface fitting technique has been successfully applied to solar radiation modelling for a number of different data combinations.     

Parratt-based and model-independent X-ray reflectivity fitting procedure for nanoscale thin film characterization

A general-purpose fitting procedure is presented for X-ray reflectivity data. The Parratt formula was used to fit the low-angle region of the reflectivity data and the resulting electron density profile (continuous base EDP or cbEDP) was then divided into a series of electron density slabs of width 1 angstroms (discrete base EDP or dbEDP), which is then easily incorporated into the Distorted Wave Born Approximation (DWBA). An additional series of density slabs of resolution-limited width are overlapped to the dbEDP, and the density value of the each additional slab is allowed to vary to further fit the data model-independently using DWBA. Because this procedure combines the Parratt formula and the model-independent DWBA fitting, each fitting method can always be employed depending on the type of thin film. Moreover, it provides a way to overcome the difficulties when both fitting methods do not work well for certain types of thin films. Simulations show that this procedure is suitable for nanoscale thin film characterization.     

Application of the Method of Neural Networks to Solution of the Inverse Problem of Heat Transfer

The possibility of application of the method of neural networks to solution of inverse problems is considered using a coefficient inverse problem of heat transfer as an example. A method of fitting a neural network is suggested. The errors of recovery of sought parameters as functions of the properties of the medium and the number of neurons in the internal layer are investigated. The results of recovery of parameters are compared with the results obtained by the fitting method.