Solving a Cauchy problem for the heat equation using cubic smoothing splines

The DASH (Doppler asymmetric spatial heterodyne) is used to detect the upper atmospheric wind speed by its imaging Fizeau interference fringes. There are two wind measurement methods: Fourier series method (FSM) and popular Fourier transform method (FTM). However, the wind speed measurement accuracy of FTM is greatly influenced by window function, and the calculation is relatively complicated. The Four-point algorithm (FPA) for DASH’s wind speed measurement is proposed in this paper. The contents of wind speed measurement principle, forward modeling, noise and inversion by the FSM, FTM and FPA are wholly compared and studied. The three wind speed measurement methods are all derived from the phase difference transformation of DASH Fizeau interference fringes. The Fizeau interference fringes with wind speed of 0–100 m/s at the interval of 10 m/s are simulated, and the forward wind speeds are obtained by FSM, FTM and FPA, and the corresponding wind measurement errors are 2.93%, 4.67% and 3.00%, respectively. After artificially adding Gaussian noise with a mean value of 0 and a standard deviation of 0.1, FSM, FTM and FPA are used to forward the Fizeau interference fringes after flat field, and the corresponding relative errors are 2.30%, 11.66% and 2.27%, respectively. After artificially adding Gaussian noise, the Fizeau interference fringes of wind speeds of 31–39 m/s with 1 m/s interval and 30.1–30.9 m/s with 0.1 m/s interval are simulated, and the forward wind speeds are obtained by FSM and FPA. In both cases, the wind speed measurement errors of FSM are 3.55% and 4.15% higher than those of FPA. The O(<sup>1</sup>S) 557.7 nm airglow at peak altitude of 98 km in Xi’an was photographed by using our GBAII (ground based airglow imaging interferometer)-DASH, and the imaging interferograms with zenith angles of 0° and 45° were obtained. Then by the methods of Fourier series, Fourier transform and FPA are used to obtain the inversion wind speed of 32.21 m/s, 43.55 m/s and 32.17 m/s, respectively. From the forward and inversion results of DASH, we can see that the FPA has a better result for detecting the upper atmospheric wind due to its simple calculation and smaller wind measurement error.

In this article, the partially linear single-index models are discussed based on smoothing spline and average derivative estimation method. This proposed technique consists of two stages: one is to estimate the vector parameter in the linear part using the smoothing cubic spline method, simultaneously, obtaining the estimator of unknown single-index function; the other is to estimate the single-index coefficients in the single-index part by the using average derivative estimator procedure. Some simulated and real examples are presented to illustrate the performance of this method.

Pathway analysis is one way to determine whether there is a causal relationship between extrinsic and intrinsic factors. The linearity assumption is something that can change the model. The shape of the model is subject to linearity assumptions. Path analysis is parametric when the linearity assumption is true, whereas non-parametric path analysis is used when the non-linear shape is unknown and there is no knowledge of the data pattern. Non-linear path analysis is used when the non-linear shape and data pattern are unknown. This work aimed to combine the smoothing spline method and the Fourier series method to compute non-parametric path function and it is believed that they would be able to produce more flexible function estimations for data patterns since both have the benefit of being accurate or close to the real data pattern. As a result, we found that Fourier series and smoothing splines can be used in non-parametric path analysis only if the linearity assumption is violated. Non-parametric regression-based path analysis estimators were then obtained using the ordinary least squares (OLS) approach. It uses a non-parametric approach and therefore gives non-unique estimation results.

To ascertain whether there is a causal connection between exogenous and endogenous factors, one method is to perform path analysis. The linearity assumption is the one that has the power to alter the model. The model's shape is impacted by the linearity assumption. The path analysis is parametric if the linearity assumption is true, but non-parametric path analysis is used if the non-linear form is unknown and there is no knowledge of the data pattern. If the non-linear form is unknown and there is no knowledge of the data pattern, non-linear path analysis is used. This study's goal was to calculate the nonparametric route function using a combination of truncated spline and Fourier series methods. The findings demonstrated that nonparametric path analysis only in cases where the linearity presumption is violated can one employ the Fourier series and truncated spline. Then, using the Ordinary Least Square (OLS) approach, the estimator of Nonparametric Regression-Based Path Analysis was obtained, delivering an estimation result that is not unique because it makes use of a nonparametric approach. The contribution of this paper can be used as reference material, especially analysis in statistics. With this paper, it is hoped that it can be applied in various fields. Suggestions for further research can develop this research with other models.

The railway catenary is the typical tensioned beam structure arranged along the railway lines, supplying power to the moving high-speed trains. The accurate static form-finding and dynamic behaviour prediction of the structure are the fundamental problems. An improved full Fourier series (IFFS) method is proposed to describe the mechanical behaviour of the stitched catenary. Different from the conventional Fourier series method, both the sin and cos series are adopted, and the proportional and constant terms are added. The initialization process and the static characteristics of the stitched catenary are given and compared to the finite element method (FEM). The effects of the expanding orders are discussed, considering the calculation efficiency and accuracy. The differences caused by the stitch wire are analysed, consisting of the static state and the dynamic performance in pantograph–catenary interaction, supporting the structural design of the high-speed railway catenary.

This research, deals with the linear elastic behavior of curved thin beams resting on Winkler foundation with both compressional and tangential resistances. Thin beam theory is extended to include the effect of curvature and externally distributed moments under static conditions. The computer program (CBFFD) coded in fortran_77 is developed to analyze curved thin beams on Winkler foundation by Fourier series and finite difference methods. The results from these methods are plotted with other solutions to compare and check the accuracy of the used methods. INTRODUCION The object of this research is to analyze curved thin beam using finite difference and Fourier series methods. The beam is resting on elastic foundation with Winkler frictional and compresional resistances, and loaded generally (both transverse distributed load and distributed moment). The linear elastic behavior of curved thin beams on elastic foundations is considered. The governing differential equation of curved thin beams (in terms of w only) is developed and converted into finite differences. A computer program in (Fortran language) is developed. This program assembles the finite difference equations to obtain a system of simultaneous algebraic equations and then the solution is obtained by using Gauss elimination method. The deflections and rotations for each node are obtained. The shear and moment are obtained by simple substitutions of the deflections into the finite difference equations of moment and shear. Also, this program used Fourier series method to solve the governing differential equation for simply supported beam and obtain the deflection, moment and shear. The obtained solutions compared with available results to check the accuracy of the used methods. Curved beams are one-dimensional structural elements that can sustain transverse loads by the development of bending, twisting and shearing resistances in the transverse sections of the beam. It's extensively used in engineering and other fields since such beams have many practical applications. The curved beam elements on elastic foundation would be helpful for the analysis of ring foundation of structures such as antennas, water towers structures, transmission towers and various other possible structures and superstructures. These are review of early studies on curved beam.

Assessing the equivalence of nonparametric regression tests based on spline and local polynomial smoothers

In this paper frequency analysis of annual extreme daily rainfall of 14 gauging stations located in an arid zone of Iran were performed using parametric and nonparametric approaches. The parametric methods include normal, two- and three-parameter log-normal, two-parameter gamma, Pearson and log-Pearson type III, extreme value type I (Gumbel), generalized extreme value and generalized logistic distributions. The nonparametric approach is Fourier series method. The data were fitted to all of above mentioned models and the results showed that the goodness of fit of the data to Fourier series is much better than other parametric methods. Thus the Fourier series can be used as an alternative approach for frequency analysis of extreme daily rainfall in an arid zone. In addition, the quantiles can be calculated by the Fourier series.

In the presence of covariate measurement error, estimating a regression function nonparametrically is extremely difficult, the problem being related to deconvolution. Various frequentist approaches exist for this problem, but to date there has been no Bayesian treatment. In this article we describe Bayesian approaches to modeling a flexible regression function when the predictor variable is measured with error. The regression function is modeled with smoothing splines and regression P-splines. Two methods are described for exploration of the posterior. The first, called the iterative conditional modes (ICM), is only partially Bayesian. ICM uses a componentwise maximization routine to find the mode of the posterior. It also serves to create starting values for the second method, which is fully Bayesian and uses Markov chain Monte Carlo (MCMC) techniques to generate observations from the joint posterior distribution. Use of the MCMC approach has the advantage that interval estimates that directly model and adjust for the measurement error are easily calculated. We provide simulations with several nonlinear regression functions and provide an illustrative example. Our simulations indicate that the frequentist mean squared error properties of the fully Bayesian method are better than those of ICM and also of previously proposed frequentist methods, at least in the examples that we have studied.

Receiver operating characteristic (ROC) curve has been employed in classification problems along with the area under the curve (AUC) as the performance indicator of classifiers. Both parametric and non-parametric methods have been widely used to estimate the ROC curve as well as the AUC. In this study, a smoothing spline is proposed in order to provide an alternative of the ROC curve and AUC estimate. A logistic regression is selected as a base classifier for simulation cases of Gaussian and mixture of Gaussian data. The smoothing spline, bi-normal model and empirical method are compared in terms of root mean square error (RMSE) from the true ROC curve and the bias from the true AUC. The results indicate that the ROC curve and its AUC obtained from smoothing spline can provide a trade-off between the parametric bi-normal model and non-parametric empirical method, with 1.4% of bias and 7.75 of RMSE, on average for a dichotomous classification.

A Fourier series technique to enhance signal-to-noise ratio and to estimate the azimuth of signals embedded in a noise background is developed in this paper. The method uses beam measurements made by horizontal line array as inputs. Currently, the average value of incoming energy across a given beam is utilized directly or empirically to estimate the arrival intensity and azimuth. Dividing the beam energy by the beamwidth produces an estimate that is the average of signal times array beam pattern function. The average value of energy from a single source is considerably lower than the corresponding peak value. Also, the azimuth of an incoming signal of interest must be estimated empirically from average intensity levels at adjacent beam centers. The primary objective of this analysis is to provide an estimate of peak values (rather than average values) of the acoustic energy at all azimuths by applying the Fourier series method to utilize the full beam pattern of the array. In the method described in this paper, the acoustic field is expanded in a Fourier series and Fourier coefficients are determined from beam intensity measurements. The method provides a best estimation of the signal plus noise field from beam measurements of a line array system, since the Fourier series is a trigonometric polynomial with the smallest mean square difference from the actual horizontal noise field. The major conclusion of this analysis is that signal-to-noise ratio and estimation of the azimuth of distant signals measured by line array systems are enhanced when the Fourier series method is applied to beam measurements. Enhancement results in the improved prediction of (1) signal arrival azimuth and (2) peak signal level. These conclusions are based on application of the Fourier series method to a limited number of mathematical models of acoustic fields; further evaluation of the Fourier series method for a greater variety of mathematical model acoustic fields is necessary to further quantify the results.

Previous work has demonstrated the advantages of Shannon entropy (H) analysis for the image-based detection of defects in both plexiglas and graphite/epoxy composites [1][2][3]. Application to experimental data shows that the analysis is fast and robust in the presence of noise. However, it suffers from the shortcoming that when signal averaging is employed, H converges to a constant, independent of the underlying waveform characteristics (log2(Ns), where Ns is the number of gated time domain sample points). By considering a generalization of the Shannon entropy to the continuous waveform case, H c, we eliminate this problem and obtain a stable numerical scheme for evaluation of Hc based on the use of Fourier series. As described previously, however, this approach requires a network of 20 workstations over 20 hours to complete analysis of one 41 by 201 pixel image. We describe a new approach for calculating continuous waveform entropy Hc, based on the use of a Green's function. We show that the new approach produces the same or higher image contrast in a time that is roughly three orders of magnitude smaller than that required by the Fourier series method. This improvement arises from two sources. The Green's function approach has greater inherent immunity to noise, and requires fewer calculations than the Fourier series approach. The resulting algorithm makes it feasible to perform Hc analysis on a personal computer

A long time has past since the introduction of the harmonic method for the reconstruction of the ODF from polefigure measurements, and it has been replaced by discrete methods of inversion, because of its incapability with respect to ghost effects. The harmonic method is still not in its best possible state: it disregards the high order harmonics; it disregards measurement errors and therefore gives suboptimal results; it does not provide standard errors, neither for the C-coefficients nor for the ODF; and there are the ghost effects. However, the harmonic method is a well established inversion method and it can improved at these points. Statistical considerations based on geostatistics and a model of the unknown ODF as a random function in a Baysian approach yields an inversion method, which can be characterized as a smoothing spline method. This new method is statistically optimal among all linear methods and resembles favorable features of the harmonic method in an improved way. It provides an optimal linear reconstruction of the even part of the ODF. It does not truncate the harmonic series expansion at a fixed level, but accounts for every even harmonic space in an optimal way with respect to its signal to noise ratio in the polefigure measurements. The method applies for irregularly sampled and incomplete pole figures. The method accomplishes standard errors for the ODF and the C-coefficients. Discrete inversion methods, explicitly or not, reconstruct the odd harmonic part of the function based on the principle of maximum entropy. Based on the theory of exponential families a continuous odd part (and the truncated even part) can be computed based on the entropy principle and the C-coefficients estimated by the spline method.

Bivariate splines with various degrees are considered in this paper. A matrix form of the extended smoothness conditions for these splines is presented. Upon this form, the multivariate spline method for numerical solution of partial differential equations (PDEs) proposed by Awanou, Lai, and Wenston in [The multivariate spline method for scattered data fitting and numerical solutions of partial differential equations, in Wavelets and Splines, G. Chen and M. J. Lai, eds., Nashboro Press, Brentwood, TN, 2006, pp. 24-76] is generalized to obtain a new spline method. It is observed that, combined with prelocal refinement of triangulation and automatic degree raising over triangles of interest, the new spline method of bivariate splines of various degrees is able to solve linear PDEs very effectively and efficiently.

Plates are applied to a wide array of structural applications of varying complexity. Each application requires rigorous analysis to determine the viability of the proposed model. One such application involves modeling a larger structure as a collection of smaller flat plates connected at the plate boundaries. Previous research into these types of structures has led to varying levels of accuracy. It has been dependent on the applications and assumptions involved. To improve the accuracy of these types of structures in a more general context, we propose expanding on current models of coupled plates by modeling the plates using Mindlin plate theory. We analyze the vibration of the improved model with general elastic boundary conditions, point supports and coupling conditions using the Fourier series method and finite element software. When the Fourier series method is applied directly, continuity issues arise at the plate coupling boundaries. To resolve these issues, the Fourier series solution of the vibration displacements is amended to include auxiliary functions. This improved coupled plate model is analyzed and numerically simulated for a variety of elastic boundary conditions and coupling conditions. The numerical results are produced using the Fourier series method and a finite element solution to demonstrate the validity of the improved coupled plate model.