Parabolic Spline Research Articles

A model instability issue in the National Centers for Environmental Prediction Global Forecast System version 16 and potential solutions

Abstract. The National Centers for Environmental Prediction (NCEP) Global Forecast System version 16 (GFSv16) encountered a few model instability failures during the pre-operational real-time parallel runs. The model forecasts failed when an extremely small thickness depth appeared at the model's lowest layer during the landfall of strong tropical cyclones. A quick solution was to increase the value of minimum thickness depth, an arbitrary parameter introduced to prevent numerical instability. This modification solved the model's numerical instability with a small impact on forecast skills. It was adopted in GFSv16 to implement this version of the operational system as planned. Upon further investigation, it was determined that the extremely thin depth was a result of the advection of geopotential heights at the interfaces of model layers. In the Finite-Volume Cubed-Sphere (FV3) dynamical core, the horizontal winds at interfaces for advection are calculated from the layer-mean values by solving a tridiagonal system of equations in the entire vertical column based on the parabolic spline method (PSM) with high-order boundary conditions (BCs). We replaced the high-order BCs with zero-gradient BCs for the interface-wind reconstruction. The impact of the zero-gradient BCs was investigated by performing sensitivity experiments with GFSv16, idealized mountain ridge tests and the Rapid Refresh Forecast System (RRFS). The results showed that zero-gradient BCs can fundamentally solve the instability and have little impact on the forecast performances and the numerical solutions of idealized mountain tests. This option has been added to FV3 and will be utilized in the GFS (GFSv17/Global Ensemble Forecast System version 13 – GEFSv13) and RRFS for operations.

Feature curve-based registration for airborne LiDAR bathymetry point clouds

The airborne LiDAR bathymetry (ALB) system is widely used in the fields of sea–land measurement, including seafloor topographic feature description, 3D seafloor model construction, coral reef monitoring, and underwater archaeology. A manned ALB data registration method based on feature curves was proposed considering problems such as sparse ALB data features, low point cloud density, and difficult corresponding feature extraction. First, triangulation network interpolation was used to extract the isoline points of the seafloor point cloud. The curves describing the seafloor topographic trend were generated through cubic parabolic spline function interpolation. Second, the curve deformation energy function was constructed based on the curve features, and a similarity measurement was carried out on the registration curve by integrating the energy function with the longest common subsequence (LCSS) algorithm. The RANSAC and ICP algorithms were used for rough and fine registration, respectively. The effectiveness and robustness of the proposed method were evaluated using two sets of experimental data. The experimental results showed that the average distance between corresponding points in the coarse registration stage reached 0.128 m and 0.136 m. After fine registration, the average point distance between the point cloud coordinates and ground truth reached 0.073 m and 0.267 m, providing an effective and reliable solution for ALB data registration.

A Fast Cross-Correlation Combined with Interpolation Algorithms for the LiDAR Working in the High Background Noise

Processing speed and accuracy of measurements are important factors reflecting the performance quality of light detection and ranging (LiDAR) systems. This study proposed a fast cross-correlation (fCC) algorithm to improve the computation loading in the LiDAR system operating in high background noise environments. To reduce the calculation time, we accumulated cycles of the receiver waveform to increase the signal-to-noise ratio. In this way, the stop pulse can be easily distinguished from the background noise by applying the cross-correlation (CC) on the accumulated receiver waveform with the first start pulse. In addition, the proposed fCC combined with variant interpolation techniques: the parabolic (fCCP), gaussian (fCCG), cosine (fCCC), and cubic spline (fCCS) to increase the measurement accuracy were also investigated and compared. The experiments were performed on the real-time LiDAR system under high background light intensity. The tested results showed that the proposed method fCCP achieved 879 ns per measurement, 38 times faster than the original CC method combined with the same parabolic interpolation algorithm (CCP) 33.5 μs. Meanwhile, the fCCS method resulted in the highest accuracy/precision, reaching 5.193 cm/8.588 cm, respectively. These results demonstrated that our proposed method significantly improves the measurements speed in the LiDAR system operating under strong background light.

SIMULATION OF THE INFLUENCE OF BALLISTIC FACTORS ON SHELL DISSOLUTION

The influence of initial random variations of ballistic factors, such as mass, initial projectile velocity and throwing angle, on the scattering of projectiles by the Monte Carlo method is studied. The study used a computer external ballistic model of the projectile flight of a 30-mm gun, based on a system of differential equations. A parabolic spline is used to interpolate the air drag law of 1943. For each of the selected ballistic indicators (projectile mass, initial velocity of the projectile, and throwing angle), 1000 normally distributed random values were generated, and hit points were calculated, from which the standard deviation was calculated.
 Several series of computational experiments were performed to study the influence of each factor separately, the pairwise influence of factors and the combined influence of three factors. In the case of the combined effect on the scattering of all three studied factors, the greatest influence is the magnitude of the variation of the initial velocity of the projectile. Software and techniques can be used by munitions developers to improve their combat capabilities and performance ammunition. When firing a firearm, the phenomenon of scattering occurs due to the variety of factors that change randomly from shot to shot. Scattering characteristics are determined by statistical processing of firing data in the range. Projectile scattering is closely linked to the effectiveness of hitting targets, and therefore the study of scattering is given much attention by researchers.
 For practical application, scattering estimates obtained by statistical analysis of experimentally obtained data are usually used.

Approximation of non - interpolatory complex parabolic spline on the unit circle

In this paper we have constructed a non-interpolatory spline on the unit circle. The rate of convergence and the error in approximation corresponding to the complex valued function has been considered.

Astroinformatics: Statistically Optimal Approximations of Near-Extremal Parts with Application to Variable Stars

The software MAVKA is described, which was elaborated for statistically optimal determination of the characteristics of the extrema of 1000+ variable stars of different types, mainly eclipsing and pulsating. The approximations are phenomenological, but not physical. As often, the discovery of a new variable star is made on time series of a single-filter (single-channel) data, and there is no possibility to determine parameters needed for physical modelling (e.g. temperature, radial velocities, mass ratio of binaries). Besides classical polynomial approximation "AP" (we limited the degree of the polynomial from 2 to 9), there are realized symmetrical approximations (symmetrical polynomials "SP", "wall-supported" horizontal line "WSL" and parabola "WSP", restricted polynomials of non-integer order based on approximations of the functions proposed by Andronov (2012) and Mikulasek (2015) and generally asymmetric functions (asymptotic parabola "AP", parabolic spline "PS", generalized hyperbolic secant function "SECH" and "log-normal-like" "BSK"). This software is a successor of the "Observation Obscurer" with some features for the variable star research, including a block for "running parabola" "RP" scalegram and approximation. Whereas the RP is oriented on approximation of the complete data set. MAVKA is pointed to parts of the light curve close to extrema (including total eclipses and transits of stars and exoplanets). The functions for wider intervals, covering the eclipse totally, were discussed in Andronov (2017). Global and local approximations are reviewed in Andronov (2020). The software is available at http://uavso.org.ua/mavka and https://katerynaandrych.wixsite.com/mavka. We have analyzed the data from own observations, as well as from monitoring obtained at ground-based and space (currently, mainly, TESS) observatories. It may be used for signals of any nature.

MAVKA: Program of statistically optimal determination of phenomenological parameters of extrema. Parabolic spline algorithm and analysis of variability of the semi-regular star Z UMa

Advanced MAVKA software for the approximation of extrema observations is used to analyze the variability of the brightness of pulsating and eclipsing stars, but may be useful in analyzing signals of any nature. A new algorithm using a parabolic (quadratic) spline is proposed. In contrast to the traditional definition of a spline as a piecewise-defined function at fixed intervals, a spline is proposed to be divided into three intervals, but the positions of the boundaries between the intervals are additional parameters. The spline defect is 1, that is, the function and its first derivative are continuous and the second derivative can be discontinuous at the boundaries. Such a function is an enhancement of the "asymptotic parabola" (Marsakova and Andronov 1996). The dependence of the fixed signal approximation accuracy on the location of the boundaries of the interval is considered. The parameter accuracy estimates using the least squares method and bootstrap are compared. The variability of the semi-regular pulsating star Z UMa is analyzed. The presence of multicomponent variability of an object, including, four periodic oscillations and significant variability of the amplitudes and phases of individual oscillations is shown.

Применение параболического сплайна для анализа распределения населения по доходам

The aim of the study is an empirical test of the hypothesis about the possibility of using the lognormal function as an adequate model of the distribution of the population by income. The method of spline functions is used as the main research tool. The values of the free parameters of the parabolic spline were determined from the solution of the problem of minimizing the integral smoothness of the distribution function under the condition of its monotonic increase. As the information base of the study, statistical data on the distribution of wages of employees of organizations of the Russian Federation in 2019 were used. This work was supported by the Russian Foundation for Basic Research, grant No. 19-010-000921.

Использование параболических сплайнов в САПР линейных сооружений

The article continues studying the problem of the calculus of variations that occurs in line structures routing, in particular, roads. The task is to find a line that satisfies all technical constraints and gives a minimum of a given functional, for example, construction costs. The unknown extremal is a parabolic spline, that is, a plane curve, the elements of which are parabolas of the second order conjugated by line segments. The principal feature of the problem is that the number of spline elements is unknown. The spline parameters must satisfy the constraints on the first derivative and curvature. Besides, also the ordinates of the individual points may be restricted. In addition, the lengths of the spline elements must be at least the given values. The problem is solved in two stages. First, the number of elements is determined, and then their parameters are optimized. Algorithms of nonlinear and dynamic programming are used. The structural features of the constraint system are taken into account, and an algorithm for constructing a basis in the null space of the matrix of active constraints is given. As an alternative, an algorithm is implemented that uses penalty functions for violation of constraints on ordinates of given points. The successful implementation of algorithms is reported.

Parabolic spline interpolation for functions with large gradient in the boundary layer

We consider the problem of Subbotin’s parabolic spline interpolation for functions with large gradient domains. In the case of the common piecewise uniform Shishkin’s mesh we obtain two-sided accuracy estimates for the class of functions with exponential boundary layer. The spline interpolation accuracy estimates are not uniform in a small parameter, while the error itself can grow unboundedly as the small parameter vanishes and the number N of nodes remains fixed. We include the results of some simulations.

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.

On the parabolic spline-interpolation of functions with a boundary layer

A subject of the article is a parabolic spline-interpolation of functions having high gradient domains. Uniform grids’ inefficiency was shown. The piecewise-uniform grids, concentrated in the boundary layer were scrutinized. There were announced asymptotically exact estimates on a class of functions with an exponential boundary layer in consideration with the traditional parabolic spline interpolation. There were obtained results showing non-uniform in small parameter estimates. A probable divergence of interpolation processes was extrapolated. A modified parabolic spline with obtained uniform in small parameter interpolation error estimations was introduced. The paper contains results of numerical experiments, which strengthen theoretical estimates.

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

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

Using parabolic splines for interpolation of the air drag functions

The work is dedicated to the interpolation function of air drag by using parabolic spline. Used spline does not require any additional conditions and is stablе. The efficiency of the application of parabolic spline for different drag functions is shown. The spline approximation demonstates good accuracy, saves the behavior of drag functions and can be used in models of external ballistics.

실시간 비선형 최적화 알고리즘을 이용한 족형 로봇의 Swing 궤적 최적화 방법

An effective swing trajectory of legged robots is different from the swing trajectories of humans or animals because of different dynamic characteristics. Therefore, it is important to find optimal parameters through experiments. This paper proposes a real-time nonlinear programming (RTNLP) method for optimization of the swing trajectory of the legged robot. For parameterization of the trajectory, the swing trajectory is approximated to parabolic and cubic spline curves. The robotic leg is position-controlled by a high-gain controller, and a cost function is selected such that the sum of the motor inputs and tracking errors at each joint is minimized. A simplified dynamic model is used to simulate the dynamics of a robotic leg. The purpose of the simulation is to find the feasibility of the optimization problem before an actual experiment occurs. Finally, an experiment is carried out on a real robotic leg with two degrees of freedom. For both the simulation and the experiment, the design variables converge to a feasible point, reducing the cost value.

The analytical solution of the 3D model with Robin's boundary conditions for 2 peat layers

<p>In this paper we consider averaging methods for solving the 3-D boundary value problem in domain containing 2 layers of the peat block. We consider the metal concentration in the peat blocks. Using experimental data the mathematical model for calculation of concentration of metal in different points in every peat layer is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations of second order with piece-wise diffusion coefficients in every direction and peat layers.</p><p>The special parabolic and exponential spline, which interpolation middle integral values of piece-wise smooth function, are considered. With the help of this splines is reduce the problems of mathematical physics in 3-D with piece-wise coefficients to respect one coordinate to problems for system of equations in 2-D. This procedure allows reduce the 3-D problem to a problem of 2-D and 1-D problems and the solution of the approximated problem is obtained analytically.</p><p>The solution of corresponding averaged 2-D initial-boundary value problem is obtained also numerically, using for approach differential equations the discretization in space applying the central differences. The approximation of the 2-D non-stationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.</p>

Dedicated three dimensional numerical models for the inverse determination of the heat flux and heat transfer coefficient distributions over the metal plate surface cooled by water

The inverse method has been developed to determine three dimensional heat flux and heat transfer coefficient distributions in space and time. The numerical tests conducted for simulated temperature sensor indications have shown that the dedicated heat conduction model has to be employed to achieve correct solutions for limited number of temperature sensors. The dedicated three dimensional finite element method based on nonlinear shape functions has been developed to effectively solve the heat conduction problem. The accuracy of 5 finite element models has been compared to analytical solution and to a reference finite element solution. The reduced nonlinear finite element model with 384 degrees of freedom has given in direct simulation of the temperature field errors at a level of 2°C only. Heat transfer boundary condition over the cooled surface has been approximated by serendipity family elements with cubic shape functions. Heat transfer coefficients at surface element nodes have been extended in time of cooling with the parabolic spline functions. Inverse solutions based on the developed three dimensional heat condition and boundary condition models have been obtained without additional regularization. Solutions have been achieved for measured temperatures as well. Temperature of EN 1.4724 steel plate heated to 900°C and then cooled has been measured by thermocouples located 2mm below the cooled surface. The plate has been cooled by 1 and 2 water jets. Equations for heat transfer coefficient as functions of dimensionless plate surface temperature have been developed and verified in direct simulations of EN 1.4724 steel cooling.

A novel gridding algorithm to create regional trace gas maps from satellite observations

Abstract. The recent increase in spatial resolution for satellite instruments has made it feasible to study distributions of trace gas column densities on a regional scale. For this application a new gridding algorithm was developed to map measurements from the instrument's frame of reference (level 2) onto a longitude–latitude grid (level 3). The algorithm is designed for the Ozone Monitoring Instrument (OMI) and can easily be employed for similar instruments – for example, the upcoming TROPOspheric Monitoring Instrument (TROPOMI). Trace gas distributions are reconstructed by a continuous parabolic spline surface. The algorithm explicitly considers the spatially varying sensitivity of the sensor resulting from the instrument function. At the swath edge, the inverse problem of computing the spline coefficients is very sensitive to measurement errors and is regularised by a second-order difference matrix. Since this regularisation corresponds to the penalty term for smoothing splines, it similarly attenuates the effect of measurement noise over the entire swath width. Monte Carlo simulations are conducted to study the performance of the algorithm for different distributions of trace gas column densities. The optimal weight of the penalty term is found to be proportional to the measurement uncertainty and the width of the instrument function. A comparison with an established gridding algorithm shows improved performance for small to moderate measurement errors due to better parametrisation of the distribution. The resulting maps are smoother and extreme values are more accurately reconstructed. The performance improvement is further illustrated with high-resolution distributions obtained from a regional chemistry model. The new algorithm is applied to tropospheric NO2 column densities measured by OMI. Examples of regional NO2 maps are shown for densely populated areas in China, Europe and the United States of America. This work demonstrates that the newly developed gridding algorithm improves regional trace gas maps; its application could be very helpful for the study of satellite-derived trace gas distributions.

Using spline approximation to differentiate a function defined with errors

The article considers the problem of differentiating a function defined by tabular values with errors. Parabolic spline approximation is shown to produce an efficient result. On the one hand, the construction of a parabolic spline does not require a large volume of computations, while on the other hand the minimum norm derivative makes it possible to construct an approximation that remains stable on a larger interval than with a difference derivative.

A two-dimensional minimum-derivative spline

We consider the construction of a C (1,1) interpolation parabolic spline function of two variables on a uniform rectangular grid, i.e., a function continuous in a given region together with its first partial derivatives which on every partial grid rectangle is a polynomial of second degree in x and second degree in y. The spline function is constructed as a minimum-derivative one-dimensional quadratic spline in one of the variables, and the spline coefficients themselves are minimum-derivative quadratic spline functions of the other variable.