Local Interpolation Method Research Articles

Local Polynomial Interpolation Method Optimization in the Process of Digital Terrain Model Creation Based on Data Collected From a Multibeam Echosounder

Local Polynomial Interpolation Method Optimization in the Process of Digital Terrain Model Creation Based on Data Collected From a Multibeam Echosounder

Information technologies in the construction of the field of radiation pollution of the locality and forecasting

The article examines the improvement of information technologies in the construction of the field of radiation contamination of the area based on the measured values of the impact factors of a nuclear explosion. Forecasting uses summaries of radioactive contamination centers and meteorological information. This process is quite well formalized and its practical implementation does not present great difficulties in automated systems for monitoring the radiation environment.Studies have shown that with a large heterogeneity of the source data by location (the presence of significant zones in which measurements were not carried out), this approach can unjustifiably overestimate or underestimate the calculated values in these zones. The local interpolation method was used to correct this effect. It is based on the assumption of the presence of local linear dependencies in the source data.In automated radiation monitoring systems, interpolation methods are used in the presence of a widespread network of stationary sensors connected to communication channels.The interpretation of radiation reconnaissance data using interpolation methods does not allow using a priori information about the parameters of the cell and the conditions for the formation of radiation contamination of the area. Joint accounting in the processing of these data allows you to combine the stages of forecasting and detection of the actual situation into a single process.

Digital twin method for the stress field of a deep-diving spherical shell based on a simulation database

In this paper, a Digital Twin (DT) method is proposed for quickly predicting the stress fields of deep-diving spherical shells based on a simulation database. The DT method can completely cover the stress field distribution of the entire pressure-resistant shell and provide stress predictions for different sizes of deep-diving protection shells. The DT model has a three-level structure. The Level-1 DT maps the Finite Element Model to the digital model, which is verified by experiments. The difference between the experimental and numerical results is less than 9.4%. Then, the Level-2 DT deduces the digital model and expands the simulation database samples. The uncalculated stress field results in the simulation database could be obtained by the local Lagrangian interpolation method. Furthermore, the Level-3 DT can rapidly predict the stress field distribution in deep-diving spherical shell digital models. By comparison with different optimization algorithms, the Particle Swarm Optimization algorithm demonstrates strong optimization capabilities and fast convergence, and the error compared with the simulation results is less than 1%. Finally, combined with actual sensor measurements and historical experience data, this method can be used to carry out real-time mapping and synchronous deduction of the service state of the dynamic stress digital twin.

Temporal and Spatial Analysis of Rainfall and Evapotranspiration in Erbil Plain and the Peripheral Areas

Since analysis of spatial events is region specific and cannot be generalized, local studies are of vital importance to identify the most accurate interpolation methods. Furthermore, the trend analysis of climatic parameters at different time scales is helpful for making a better climate change adaptation and mitigation plan to overcome water scarcity. Accordingly, the current study was proposed to detect trends in rainfall and evapotranspiration at monthly and annual time scales over the Erbil plain and the surrounding area using parametric and non-parametric tests. Moreover, four deterministic and five geostatistical methods were evaluated for searching the best interpolation method to generate a continuous surface for the indicated climatic variables. The results revealed that the majority of data sets are categorized useful class and serially independent. Furthermore, it was found both rainfall and potential evapotranspiration have a mix of upward and downward trends and most of them are insignificant at 5% level of significance. Further, the interpolation analysis indicated that local polynomial interpolation (LPI) method was proven to be best interpolator for generating continuous surfaces for rainfall and ETo over the area under study followed by the empirical Bayesian Kriging method.

Triquintic interpolation in three dimensions

This paper develops a local triquintic interpolation method for a function defined over a unit cube mesh element. The resulting polynomial interpolant is isotropic and has global C2 continuity. The triquintic polynomial coefficients are related to the function and its derivatives at the eight corners of a mesh element by an invertible square matrix of rank 216. The proposed method is an extension of a previously developed tricubic interpolation scheme with global C1 continuity.

A high-order generalized differential quadrature method with lattice Boltzmann flux solver for simulating incompressible flows

This paper presents a high-order generalized differential quadrature method with lattice Boltzmann flux solver (LBFS-GDQ) for simulating incompressible isothermal flows. In this method, high-order polynomials are adopted to approximate both the solution and fluxes globally across the computational domain. Solution derivatives and flux divergence are conveniently computed by the GDQ method. At the interior solution points, the viscous and inviscid fluxes are evaluated simultaneously via LBFS. Treatments to prevent the global accuracy from being contaminated by the streaming error of LBFS are proposed and studied, including the choice for the local streaming spacing and interpolation methods for the local reconstruction. The present method inherits the advantages of both GDQ and LBFS, i.e., global spectral accuracy, direct evolution of macroscopic variables, and convenient implementation of boundary conditions. Numerical experiments with a wide selection of incompressible flow problems confirm the excellent accuracy, efficiency, and flexibility of the proposed method.

Learning with Partition of Unity-based Kriging Estimators

For supervised regression tasks we propose and study a new tool, namely Kriging Estimator based on the Partition of Unity (KEPU) method. Its background belongs to the framework of local kernel-based interpolation methods. Indeed, even if the latter needs to be accurately tailored for gaussian process regression, the KEPU scheme provides a global estimator which is constructed by gluing together the local Kriging predictors via compactly supported weights. The added value of this investigation is twofold. On the one hand, our theoretical studies about the propagation of the uncertainties from the local predictors to the global one offer the opportunity to define the PU method in a stochastic framework and hence to provide confidence intervals for the PU approximant. On the other hand, as confirmed by extensive numerical experiments, when the number of instances grows, such a method enables us to significantly reduce the usually high complexity cost of fitting via gaussian processes.

4-dimensional local radial basis function interpolation of large, uniformly spaced datasets

Background and ObjectiveLarge, uniformly spaced, complex and time varying datasets derived from high resolution medical image velocimetry can provide a wealth of information regarding small-scale transient physiological flow phenomena and pulsation of anatomical boundaries. However, there remains a need for interpolation techniques to effectively reconstruct a fully 4-dimensional functional relationship from this data. This paper presents a preliminary evaluation of a 4-dimensional local radial basis function (RBF) algorithm as a means of addressing this problem for laminar flows. MethodsA 4D interpolation algorithm is proposed based on a Local Hermitian Interpolation (LHI) using a combination of multi-quadric RBF with a partition of unity scheme. The domain is divided into uniform sub-systems with size restricted to immediately neighbouring points. The validity of the algorithm is first established on a known 4D analytical dataset and a CFD based laminar flow phantom. Application is then demonstrated through characterisation of a large 4D laminar flow dataset obtained from magnetic resonance imaging (MRI) measurements of cerebrospinal fluid velocities in the brain. ResultsPerformance of the algorithm is compared to that of a quad-linear interpolation, demonstrating favourable improvement in accuracy. The technique is shown to be robust, computationally efficient and capable of refined interpolation in Euclidean space and time. Application to MR velocimetry data is shown to produce promising results for the 4D reconstruction of the transient flow field and movement of the fluid boundaries at spatial and temporal locations intermediate to the original data. ConclusionThis study has demonstrated feasibility of an accurate, stable and efficient 4-dimensional local RBF interpolation method for large, transient laminar flow velocimetry datasets. The proposed approach does not suffer from ill-conditioning or high computational cost due to domain decomposition into local stencils where the RBF is only ever applied to a limited number of points. This work offers a potential tool to assist medical diagnoses and drug delivery through better understanding of physiological flow fields such as cerebrospinal fluid. Further work will evaluate the technique on a wider range of flow fields and against CFD simulation.

Stacking machine learning models versus a locally weighted linear model to generate high-resolution monthly precipitation over a topographically complex area

This study applied a stacked generalization ensemble approach to generate high-resolution precipitation estimates and compared its performance with an optimized local weighted linear regression (LWLR) algorithm, a well-known local precipitation-elevation interpolation method incorporating physiographical factors. The stacked generalization ensemble consists of multilayer perceptron neural network (MLP), support vector machine (SVM), and random forest (RF) combined through a meta-learning algorithm with/without rescanning input covariates. Sixteen input covariates, including 2 topographic features, 5 cloud properties, 5 environmental variables, 3 precipitation products (PPs), and inverse distance weighted (IDW) estimates as the precipitation background field were fed into the machine learning models. Hold out approach was adopted for performance evaluation in which 50% of the 174 gauges were used for training, and the rest was used for validation. The results indicated that the overall monthly MAE, RMSE, and rBias of the proposed stacking model for the validation dataset were 3.3%, 6.8%, and 50%, respectively, less than that of the RF model, which is the best individual model. Also, the stacking model outperformed LWLR by decreasing monthly MAE, RMSE, and rBias 10.7%, 19.1%, and 28.6%, respectively. Further analysis implied that (1) the stacked model is more robust than LWLR and less dependent on the density of gauges, thus suitable for areas with scarce gauge coverage; (2) comparing the spatial distribution of mean monthly precipitation maps, generated by stacking and LWLR models, stacking algorithm can successfully screen out the bulls' eyes of background IDW precipitation field, and both patterns are consistent with the topography of the area; and (3) the stacking scheme is found to have a better extrapolation ability than LWLR over high elevations.

ChiliPDF: Chebyshev interpolation for parton distributions

Parton distribution functions (PDFs) are an essential ingredient for theoretical predictions at colliders. Since their exact form is unknown, their handling and delivery for practical applications relies on approximate numerical methods. We discuss the implementation of PDFs based on a global interpolation in terms of Chebyshev polynomials. We demonstrate that this allows for significantly higher numerical accuracy at lower computational cost compared with local interpolation methods such as splines. Whilst the numerical inaccuracy of currently used local methods can become a nontrivial limitation in high-precision applications, in our approach it is negligible for practical purposes. This holds in particular for differentiation and for Mellin convolution with kernels that have end point singularities. We illustrate our approach for these and other important numerical operations, including DGLAP evolution, and find that they are performed accurately and fast. Our results are implemented in the C++ library ChiliPDF.

Comparison of the effectiveness of GIS-based interpolation methods for estimating the spatial distribution of agrochemical soil properties

Various interpolation methods were compared in the ArcGIS (a complex of geoinformation software products of the American company ESRI) geographical information system environment for assessing agrochemical soil properties on the territory of LLC “Neral-Chishmy” agricultural enterprise. The study aims to compare the effectiveness of interpolation methods based on GIS technologies for assessing the spatial distribution of agrochemical properties in the soil. Six interpolation methods were used to generate the spatial distribution of the studied indicator: inverse distance weighting, local polynomial interpolation, radial basis functions method, simple Kriging, ordinary Kriging, Universal Kriging. Cross-check was used to evaluate the interpolation methods accuracy by comparing the values of mean error, mean square error, mean square normalized error. It was found that the model created using the Local Polynomial Interpolation method had the lowest values of the average difference between the measurement and the interpolated value and the mean square error equal to 165.9908, indicating how closely the model predicts the measured values. An artificial neural network was created for spatial and temporal modelling of agrochemical properties in the soil, introducing new technologies into agricultural production. It is established that universal Kriging is the most suitable method for interpolating the spatial distribution of agrochemical properties of soils, namely humus, with the lowest RMSE value of 0.7198. By 2025, current agricultural technologies will gradually reduce humus content in layer 0...40 cm. Implementing the interpolation method for assessing the agrochemical properties spatial distribution in the soil will significantly increase the humus content.

Technology for restoring functional dependencies to determine reliability parameters

The problem of determining the properties of the object by analyzing the numerical and qualitative characteristics of a discrete sample is considered. A method has been developed to determine the probability of trouble-free operation of electronic systems for the case if the interpolation fields are different between several interpolation nodes. A method has been developed to determine the probability of trouble-free operation if the interpolation polynomial is the same for the entire interpolation domain. It is shown that local interpolation methods give more accurate results, in contrast to global interpolation methods. It is shown that in the case of global interpolation it is possible to determine the value of the function outside the given values by extrapolation methods, which makes it possible to predict the probability of failure. It is shown that the use of approximation methods to determine the probability of trouble-free operation reduces the error of the second kind. A method for analyzing the qualitative characteristics of functional dependences has been developed, which allows us to choose the optimal interpolation polynomial. With sufficient statistics, using the criteria of consent, it is possible to build mathematical models for the analysis of failure statistics of electronic equipment. Provided that the volume of statistics is not large, such statistics may not be sufficient and the application of consent criteria will lead to unsatisfactory results. Another approach is to use an approximation method that is applied to statistical material that was collected during testing or controlled operation. In this regard, it is extremely important to develop a method for determining the reliability of electronic systems in case of insufficiency of the collected statistics of failures of electronic equipment.

RBF collocation and hybrid-LHI methods for Stokes systems and its application to controllability problems

The purpose of this article is to introduce radial basis function (RBF) methods for solving both direct Stokes equations and controllability problems for the Stokes system with few internal scalar controls. In both cases, Dirichlet or Navier-slip boundary conditions are considered. We introduce two radial basis function solvers, one global and the other local, to solve Stokes equations. These methods are used to discretize the primal and adjoint systems related to the controllability problems. Both techniques are based on divergence-free global RBFs. A global colocation technique based on Div-free inverse multi-quadrics is formulated and analyzed. A generalization of scalar hybrid kernels to a vector divergence-free hybrid RBFs setting is defined. Based on these kernels, the local Hermite interpolation (LHI) method in vector form is introduced. Due to the properties of the hybrid kernel, we show that due to the properties of the hybrid kernel this local method, can reduce up to double precision, the value of the condition number of the local Gram matrices. Simultaneously, it is proved that the real components of the eigenvalues corresponding to the global LHI matrix are negative and that consequently backward difference formulas are stable for time integration. The conjugate gradient algorithm is adapted to the radial basis function setting to solve the controllability problems. Several benchmarks problems in two dimensions with a non-convex domain (a star shape) are numerically solved by these RBFs methods to display and compare their feasibility. The solutions to these problems are also implemented by finite element techniques to study their relative performance.

A tolerance constrained robot path circular interpolation method for industrial SCARA robots

The linear robot path is tangential and curvature discontinuity, which will lead to vibration and unnecessary hesitation during execution. Local corner transition method and local spline interpolation method are used in state-of-art industrial robot controller to reduce vibration, while local corner transition method cannot interpolate target points and local spline interpolation method cannot constrain chord errors. This research proposes a robot path local interpolation method that eliminates deficiencies of each method. The smoothing method satisfies all of the following requirements: G1continuity, target point interpolation and chord tolerance confined, shape-preserving (free of self-intersection), and unified parameterization. The generated smooth path consists of linear path and circular arc path with G1continuity. A geometric iterative method cooperating with local corner transition method is used to generate local interpolation path. Simulations and actual experiments verify the generated smooth path is G1continuous, tolerance constrained, shape-preserving, and have high computational efficiency.

A decoupled five-axis local smoothing interpolation method to achieve continuous acceleration of tool axis

Five-axis machines are widely used in high-speed and high-precision machining of complex sculptured surfaces for the ability to adjust the tool orientation. But, most of the five-axis machining trajectories generated by computer-aided manufacturing (CAM) software are G01 blocks in the form of a large number of linear segments. The G01 blocks for surface machining show inadequacies as their high-order discontinuities. Although there are a lot of researches to deal with the discontinuities, there are still many problems such as smoothing error control, motion synchronization, kinematic constraints limitation. Besides, the kinematic constraints of the tool orientation motion are always neglected. In this paper, a two-step real-time decoupling local smoothing method is proposed for the problem of the five-axis tool path smoothing. The C2 continuity of the tool path is guaranteed within the error limited. Not only the kinematic constraints of tool tip motion but also the kinematic constraints of the tool orientation motion are considered. The continuous acceleration of each axis motion of the machine tool is realized through feed-rate scheduling by finite impulse response (FIR) filtering. Finally, through numerical simulations and experiments, compared with the existing method and G01 linear interpolation, it is verified that the proposed smoothing interpolation method has a higher computation efficiency and can improve the processing efficiency and surface quality of the tool path while satisfying the specified smoothing error constraints and kinematic constraints.

Two-Stage Point Cloud Super Resolution with Local Interpolation and Readjustment via Outer-Product Neural Network

This paper proposes a two-stage point cloud super resolution framework that combines local interpolation and deep neural network based readjustment. For the first stage, the authors apply a local interpolation method to increase the density and uniformity of the target point cloud. For the second stage, the authors employ an outer-product neural network to readjust the position of points that are inserted at the first stage. Comparison examples are given to demonstrate that the proposed framework achieves a better accuracy than existing state-of-art approaches, such as PU-Net, PointNet and DGCNN (Source code is available at https://github.com/qwerty1319/PC-SR ).

A circumscribed local interpolation methodology for CNC machining along linear tool paths

ABSTRACT The G01 tool path, which consisted of linear segments, is commonly used in NC machining. There is the tangential and curvature discontinuity at the junctions of adjacent linear segments, which can cause the fluctuations of feedrate and deteriorate surface quality. In this paper, a novel circumscribed local interpolation method is proposed to smooth the discrete path and consequently improve the kinematic performance of the machine tool. The polygons are firstly constructed with a constraint of the maximum approximation error. Then, the double cubic Bezier curves are used to smooth the corner between two linear segments and the control points are derived according to the G2 continuity condition and the length constraint. Further, the motion profile along the blended tool path is generated by utilizing the quantic polynomial feedrate scheduling strategy. Finally, the proposed smoothing method is validated by simulations and experiments.

Meshless local radial point interpolation (MLRPI) to two dimensional wave equation with Neumann’s boundary conditions

In this article, the meshless local radial point interpolation (MLRPI) methods are applied to simulate two dimensional wave equation subject to given appropriate initial and Neumann’s boundary conditions. In MLRPI method, all integrations are carried out locally over small quadrature domains of regular shapes such as square or circle. The radial point interpolation method is proposed to construct shape functions for MLRPI. A weak formulation with a Heaviside step function transforms the set of governing equations into local integral equations on local sub domains where Neumann’s boundary condition is imposed naturally. A two-step time discretization method with the help of Crank-Nicolson technique is employed to approximate the time derivatives. Convergence studies in the numerical example show that MLRPI method possesses excellent rates of convergence.

Fractal Characterization of Subviral Particle Motion: On the Influence of Spatio-Temporal Interpolation Methods.

Due to the rapid globalization there is an increasing danger for pandemic outbreaks. The high death toll of fast spreading diseases like the Ebola infection demand the fast development of new medicines. Thus, the automation of pharmaceutical processes is an indispensable but challenging task. In cooperation with the Institute for Virology, Philipps-University, Marburg, Germany, recently, algorithms to detect and evaluate subviral particle tracks in live-cell fluorescence image sequences were developed. In steady interdisciplinary exchange between pharmacists and engineers it turned out that new measures to identify and classify subviral particle motion are required. This article focuses on the evaluation and optimization of a new method to classify subviral particle motion using fractal dimension estimation. The influence of global and local interpolation methods on fractal dimension estimation is investigated. The methods are tested on simulated data and applied to real image sequences. The results prospect a high benefit of using the presented methods for an effective classification of subviral particle behavior.

Peruvian Subduction Surface Model for Seismic Hazard Assessments

Throughout the years seismic hazard calculations in Peru have been developed using area sources models, having to date a great variety of models, however, since they are discretized planar models, they cannot adequately represent the continuity and subduction characteristics of the Nazca Plate. The main objective of this work is the developing of a surface subduction model (SSM), useful for seismic hazard assessments as well as the revision and control of previous models used in this sort of assessments. In this study a spatial interpolation was performed employing the Local Polynomial Interpolation method to capture short-range variation in addition to long-range trends. The data base is based on the compilation of seismic catalogs from Peruvian and international institutions such as the IGP, the USGS, the ISC and others, subsequently, in order to have independent events the elimination of duplicate events, aftershocks and foreshocks was carried out. Then, by interpolation of the focal depths of the independent events, a subduction surface model (SSM) was generated as well as a Standard Error Surface which supports a good correlation of the model. Furthermore, 14 transversal sections of the SSM was employed to compare with the hypocenter’s distributions, evidencing a good correlation with the spatial distribution of the events, in addition to adequately capturing the subduction characteristics of the Nazca Plate. Finally, a comparison was made between 2 Peruvian area models for seismic hazard and SSM developed in the present research, evidencing that seismic source models of the area type have deficiencies mainly in the depths they consider, thus is recommended the use of the present model for future seismic hazard assessments.