Phase Retrieval of Phase-Shifting Interferometry with Iterative Least-Squares Fitting Algorithm: Experiments

We report our experimental results of phase-shifting interferometry with an iterative least-squares fitting technique to estimate both the wave front phases and the phase shifts. The method allows phase retrieval from phase-shifting interferograms even though the calibration data of the phase shifter is unknown. The algorithm is used to analyze two sets of experimental interferograms. One records by moving a piezoelectric transducer shifter randomly and therefore has embedded random phase shifter errors, and the other samples the interference movie recorded by a video recorder while driving a stepping motor and therefore has embedded random intensity noises. The results are comparable with that of the conventional M-frame algorithm. Investigation of the effects of the intensity noises and phase shift errors shows the algorithm to perform well in both. Problems such as convergence, unique solution and reliability are also discussed.     

Least-squares collocation meshless approach for radiative heat transfer in absorbing and scattering media

A least-squares collocation meshless method is employed for solving the radiative heat transfer in absorbing, emitting and scattering media. The least-squares collocation meshless method for radiative transfer is based on the discrete ordinates equation. A moving least-squares approximation is applied to construct the trial functions. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted to form the total residuals of the problem. The least-squares technique is used to obtain the solution of the problem by minimizing the summation of residuals of all collocation and auxiliary points. Three numerical examples are studied to illustrate the performance of this new solution method. The numerical results are compared with the other benchmark approximate solutions. By comparison, the results show that the least-squares collocation meshless method is efficient, accurate and stable, and can be used for solving the radiative heat transfer in absorbing, emitting and scattering media.     

调制度分析在等步长相移法相位展开中的应用

推导了两种常用等步长相移算法的调制度表达式,提出一种新的调制度分析方法。该方法用于等步长相移法中基于加权最小二乘法的相位展开,能够充分利用调制度信息,构造二值和小数权重,从而增强相位展开过程对多种干扰因素的免疫力。实验结果说明了该方法的有效性和实用性。最后比较了二值权重和小数权重在加权最小二乘法的相位展开中的性能表现。     

On the roles of minimization and linearization in least-squares finite element models of nonlinear boundary-value problems

In this paper we examine the roles of minimization and linearization in the least-squares finite element formulations of nonlinear boundary-values problems. The least-squares principle is based upon the minimization of the least-squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations (in the present case we consider L₂ norms). Since the least-squares method is independent of the discretization procedure and the solution scheme, the least-squares principle suggests that minimization should be performed prior to linearization, where linearization is employed in the context of either the Picard or Newton iterative solution procedures. However, in the least-squares finite element analysis of nonlinear boundary-value problems, it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations. The main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged. The assessment is performed mathematically, through an examination of the variational setting for the least-squares formulation of an abstract nonlinear boundary-value problem, and also computationally, through the numerical simulation of the least-squares finite element solutions of both a nonlinear form of the Poisson equation and also the incompressible Navier–Stokes equations. The assessment suggests that although the least-squares principle indicates that minimization should be performed prior to linearization, such an approach is often impractical and not necessary.     

An analytical least squares solution applied to spectral series limit determinations

A method is derived to determine spectral series limits analytically. It is assumed that the graphical quantum defect solution (n?n^* vs ν_lim?ν) is approximately linear (for the correct ν_lim) so that a least-squares fit may be obtained. The problem reduces to a non-linear system of three equations in three unknowns (two least-squares coefficients and the series limit) with a unique solution found numerically. Least-squares errors are also obtained for each variable. Results using this method agree well with published values obtained by the traditional graphical method. This procedure provides an operational method for series-limit determinations, independent of the subjective problems involved with the graphical method.     

磁流体数值模拟中一种消去磁场散度的方法

充分利用时空守恒元和解元（CESE）方法的特点（在CESE方法中，守恒变量及它们的空间导数都作为独立的更新量并且求解点在每个控制体边界上）给出一种用最小二乘法求解来消去磁场散度的方法.且我们进一步探究了磁场散度限制方程取不同的权重时对结果的影响.通过比较，我们发现当方程权重取为1时，可以非常有效地消去磁场散度误差.     

带有分数阶热流条件的时间分数阶热波方程及其参数估计问题

研究了Caputo导数定义下带有分数阶热流条件的一维时间分数阶热波方程及其参数估计问题.首先,对正问题给出了解析解;其次,基于参数敏感性分析,利用最小二乘算法同时对分数阶阶数α和热松弛时间τ进行参数估计;最后对不同的热流分布函数所构成的两个初边值问题,分别进行参数估计仿真实验,分析温度真实值和估计值的拟合程度.实验结果表明,最小二乘算法在求解时间分数阶热波方程的两参数估计问题中是有效的.本文为分数阶热波模型的参数估计提供了一种有效的方法.     

Spectral/hp least-squares finite element formulation for the Navier–Stokes equations

We consider the application of least-squares finite element models combined with spectral/hp methods for the numerical solution of viscous flow problems. The paper presents the formulation, validation, and application of a spectral/hp algorithm to the numerical solution of the Navier–Stokes equations governing two- and three-dimensional stationary incompressible and low-speed compressible flows. The Navier–Stokes equations are expressed as an equivalent set of first-order equations by introducing vorticity or velocity gradients as additional independent variables and the least-squares method is used to develop the finite element model. High-order element expansions are used to construct the discrete model. The discrete model thus obtained is linearized by Newton’s method, resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method. Spectral convergence of the L₂ least-squares functional and L₂ error norms is verified using smooth solutions to the two-dimensional stationary Poisson and incompressible Navier–Stokes equations. Numerical results for flow over a backward-facing step, steady flow past a circular cylinder, three-dimensional lid-driven cavity flow, and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation.     

A least-squares/finite element method for the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line

In this article we discuss the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line separating two immiscible incompressible viscous fluids near a solid wall. The method we employ combines a finite element space approximation with a time discretization by operator-splitting. To solve the Cahn–Hilliard part of the problem, we use a least-squares/conjugate gradient method. We also show that the scheme has the total energy decaying in time property under certain conditions. Our numerical experiments indicate that the method discussed here is accurate, stable and efficient.     

Novel temperature modeling and compensation method for bias of ring laser gyroscope based on least-squares support vector machine

Bias of ring-laser-gyroscope (RLG) changes with temperature in a nonlinear way. This is an important restraining factor for improving the accuracy of RLG. Considering the limitations of least-squares regression and neural network, we propose a new method of temperature compensation of RLG bias building function regression model using least-squares support vector machine (LS-SVM). Static and dynamic temperature experiments of RLG bias are carried out to validate the effectiveness of the proposed method. Moreover, the traditional least-squares regression method is compared with the LS-SVM-based method. The results show the maximum error of RLG bias drops by almost two orders of magnitude after static temperature compensation, while bias stability of RLG improves by one order of magnitude after dynamic temperature compensation. Thus, the proposed method reduces the influence of temperature variation on the bias of the RLG effectively and improves the accuracy of the gyro scope considerably.     

Multi-exponential analysis of magnitude MR images using a quantitative multispectral edge-preserving filter

A solution for discrete multi-exponential analysis of T(2) relaxation decay curves obtained in current multi-echo imaging protocol conditions is described. We propose a preprocessing step to improve the signal-to-noise ratio and thus lower the signal-to-noise ratio threshold from which a high percentage of true multi-exponential detection is detected. It consists of a multispectral nonlinear edge-preserving filter that takes into account the signal-dependent Rician distribution of noise affecting magnitude MR images. Discrete multi-exponential decomposition, which requires no a priori knowledge, is performed by a non-linear least-squares procedure initialized with estimates obtained from a total least-squares linear prediction algorithm. This approach was validated and optimized experimentally on simulated data sets of normal human brains.     

Least-squares finite element formulations for one-dimensional radiative transfer

We present least-squares-based finite element formulations for the numerical solution of the radiative transfer equation in its first-order primitive variable form. The use of least-squares principles leads to a variational unconstrained minimization problem in a setting of residual minimization. In addition, the resulting linear algebraic problem will always have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for its solution. We consider space-angle coupled and decoupled formulations. In the coupled formulation, the space-angle dependency is represented by two-dimensional finite element expansions and the least-squares functional minimized in the continuous space-angle domain. In the decoupled formulation the angular domain is represented by discrete ordinates, the spatial dependence represented by one-dimensional finite element expansions, and the least-squares functional minimized continuously in space domain and at discrete locations in the angle domain. Numerical examples are presented to demonstrate the merits of the formulations in slab geometry, for absorbing, emitting, anisotropically scattering mediums, allowing for spatially varying absorption and scattering coefficients. For smooth solutions in space-angle domain, exponentially fast decay of error measures is demonstrated as the p-level of the finite element expansions is increased. The formulations represent attractive alternatives to weak form Galerkin finite element formulations, typically applied to the more complicated second-order even- and odd-parity forms of the radiative transfer equation.     

Simple Least-Squares Estimation of Intensities of Overlapping Signals in 2D NMR Spectra

It is described how a least-squares estimate of a two-dimensional NMR signal intensity can be calculated from two one-dimensional intensities, as estimated from a row and a column, respectively. The result is independent of artifacts associated with the Fourier transform, and it is unaffected by phase distortions and poor base plane definition in the spectrum. Cases of overlapping signals are discussed, and it is shown that because of the involved least-squares procedure, optimum results can be produced even when the overlap is substantial. Results of test calculations on simulated examples are presented, and it is shown that deviations of the calculated two-dimensional intensities from the correct values are small and in agreement with the standard deviations, also provided by the method. Finally, two-dimensional intensities calculated from experimental data are presented.     

三维不规则区域热传导问题无网格方法的数值模拟

配点型无网格法是纯无网格法,它不需要任何背景网格,效率高。本文用加权最小二乘配点方法(Weighted Least-Squares Collocation Method-WLSCM)计算不规则区域热传导问题,形函数采用径向基函数近似。通过二维具有分析解的实例表明WLSCM方法精度高,稳定性好且具有较高的计算效率。此外,将WLSCM方法应用于工程中常见的三维不规则区域热传导问题,结果表明:WLSCM方法的计算结果与FLUENT的计算结果符合很好。     

Adaptive optics correction based on stochastic parallel gradient descent technique under various atmospheric scintillation conditions: numerical simulation

An adaptive optics system utilizing a Shack–Hartmann wavefront sensor and a deformable mirror can successfully correct a distorted wavefront by the conjugation principle. However, if a wave propagates over such a path that scintillation is not negligible, the appearance of branch points makes least-squares reconstruction fail to estimate the wavefront effectively. An adaptive optics technique based on the stochastic parallel gradient descent (SPGD) control algorithm is an alternative approach which does not need wavefront information but optimizes the performance metric directly. Performance was evaluated by simulating a SPGD control system and conventional adaptive correction with least-squares reconstruction in the context of a laser beam projection system. We also examined the relative performance of coping with branch points by the SPGD technique through an example. All studies were carried out under the conditions of assuming the systems have noise-free measurements and infinite time control bandwidth. Results indicate that the SPGD adaptive system always performs better than the system based on the least-squares wavefront reconstruction technique in the presence of relatively serious intensity scintillations. The reason is that the SPGD adaptive system has the ability of compensating a discontinuous phase, although the phase is not detected and reconstructed.     

基于小波变换的最小二乘相位解缠算法

最小二乘法是求解二维相位解缠问题最稳健的方法之一,其本质是在最小二乘意义下使缠绕相位的离散偏导数与解缠相位的偏导数整体偏差最小,并等效为可求解一大型的稀疏线性方程系统。由于系统矩阵结构的稀疏性,在采用迭代法求解时收敛速度非常慢。为了改善收敛特性,提出一种基于多分辨率表示的离散小波变换相位解缠算法。利用小波变换将原线性系统转化成具有较好收敛条件的等价新系统。仿真实验表明,该方法能够很好的恢复真实相位,其解缠效果优于Gauss-Seidel松弛迭代和多重网格法。     

Analysis and design of nonlinear fiber Bragg gratings and their application for optical compression of reflected pulses

We demonstrate a novel split-step solution for analyzing nonlinear fiber Bragg gratings. The solution is used for designing nonlinear fiber Bragg gratings with a low reflectivity. The structure of the grating is designed according to the profiles of the incident and reflected pulses. We demonstrate our method for nonlinear compression of a pulse reflected from a fiber Bragg grating. The method allows us to obtain compressed pulses with a very low wing intensity.     

TXRF intensity dependence on position of dried residue on sample carrier and TXRF determination of halogen in liquid samples

Total reflection X‐ray fluorescence (TXRF) spectroscopy is an effective technique for simultaneous multi‐elemental trace analysis of a small volume of a sample placed on a flat substrate. An internal standard method is usually applied for quantitative TXRF analysis of liquid samples such as drinking water and environmental samples. However, it was difficult to determine Cl and Br because they were lost as volatile hydrogen halide compounds by adding an acid internal standard solution. Thus, we attempted to apply the traditional calibration curve method for the determination of halogens without internal standard. If internal standard method is not applied, the TXRF intensity drastically changes depending on the relative position of the dried residue to the detector. Therefore, we investigated the relationship between TXRF intensity and the position of dried residue relative to the detector. As a result, it was confirmed that TXRF intensity critically depended on the position of the dried residue on the sample carrier. The position of the droplet of the sample solution was carefully controlled by using an air blower in order to place the dried residue at the most effective position on the sample carrier. We could successfully make a calibration curve for Cl with a good relationship without internal standard. Finally, Cl in the NaCl solutions (0 –5 ppm, 10 µl) was successfully determined by the calibration curve method using a table‐top TXRF instrument. The limit of detection of Cl was 63 ppb (ng/ml). Copyright © 2016 John Wiley & Sons, Ltd.     

On the structure of galactic cosmic ray intensity during its long-term variations

The structure of GCR intensity and its changes during the solar magnetic cycle are analyzed by numerically solving its transport equation. We propose a research method that allows us to use the intermediate results from numerical calculations, to resolve the intensity of galactic cosmic rays into components caused by sunspot and magnetic solar cycles, and to monitor the changes in energy in different parts of the heliosphere and different phases of the cycle. A method for dividing intensity into its components associated with the main physical processes determining the modulation of galactic cosmic rays is also proposed.     

Allene nu9 and nu10: Low-Temperature Measurements of Line Intensity

We report laboratory intensity measurements for the weak nu9 (998.8 cm-1) and intense nu10 (841.1 cm-1) bands of allene. Allene is predicted to be a constituent of Titan's atmosphere, and measurements of its abundance would yield important information about the atmospheric chemistry of that body. Spectra were obtained at a temperature of 200 K (approximating Titan conditions) using the high-resolution FTS instrument at Kitt-Peak National Observatory's McMath-Pierce observatory. A total of 505 nu9 and 687 nu10 line intensities were fit using a least-squares method to accurately determine two sets of transition dipole moments. Integrated band intensities computed utilizing the fitted parameters were found to be 36 +/- 4% cm-2 atm-1 and 301 +/- 4% cm-2 atm-1 for nu9 and nu10, respectively, at 200 K. Copyright 1999 Academic Press.