Classical Interpolation Research Articles

Carbon Footprint Optimization with Data Resolution Conversion via Kalman Filter for Smart Energy Hub

The severe climate changes due to global warming pushed the world to strive to reduce carbon emissions. And transitioning to renewable energy sources is essential for achieving sustainability and combating climate change. This study aims to minimize the carbon emissions for a proposed smart energy hub. This paper prioritizes the minimization of carbon emissions, considering the minimization of operational running costs, and the maximization of profit. In this paper, two optimization scenarios were studied to compare the results. In the first scenario, the minimization of carbon emissions was achieved. In the second scenario, the minimization of running costs and the maximization of profit from the hub assets were studied. The proposed model was designed in MATLAB. Then, the results were verified by CPLEX and validated by RTDS. The multi-objective model was presented to obtain the optimal operation. The mitigation of data uncertainty was achieved by applying the Kalman filter. In this work, a novel method was proposed for the estimation of the quarter-hour resolution data from the hourly ones via the Kalman filter rather than by applying the classic polynomial interpolation methods.

Machine learning-augmented modeling on the formation of Si-dominated Non-β″ early-stage precipitates in Al-Si-Mg alloys with Si supersaturation induced by non-equilibrium solidification

Recent experiments have shown that Al-Si-Mg alloys solidified under high cooling rates may lead to the nucleation of Si-enriched clusters that are remarkably different from the conventional Mg-Si co-clusters (e.g. β″ particles), and yet the responsible mechanism remains to be elucidated. Here we tackle the problem using a multiscale modeling framework that integrates atomistic modeling, energy landscape sampling, and lattice-based kinetic Monte Carlo (kMC) simulation. The migration energy barriers for vacancy-mediated diffusion amid complex local chemical environments are predicted on-the-fly using a surrogate machine learning model. We discover that the actual vacancy-Si migration barriers are much lower than those assumed in the classic linear interpolation approximation. Such a strong deviation from conventional wisdom, in conjunction with differing Si solute composition, can lead to a great variety in the nucleated early-stage precipitates. More specifically, a high-level supersaturation of Si solute (i.e.xSi/(xSi+xMg)>0.75) would lead to an unexpectedly high enrichment of Si in the nucleated clusters with the Si:Mg ratio up to 5∼6; while at a lower-level supply of Si solute the Mg-Si co-clusters (i.e. Si:Mg ratio around 1∼2) are nucleated instead. These findings provide a viable explanation for the diverse types of early-stage precipitates observed in various experiments, from Si-enriched precipitates in high-pressure die cast Al alloys to β″ particles in conventional casting and/or heat-treated alloys. The implications of our findings are also discussed.

Geostatistical Simulation of Daily Rainfall Fields—Performance Assessment for Extremes in West Africa

Abstract The spatial description of high-resolution extreme daily rainfall fields is challenging because of the high spatial and temporal variability of rainfall, particularly in tropical regions due to the stochastic nature of convective rainfall. Geostatistical simulations offer a solution to this problem. In this study, a stochastic geostatistical simulation technique based on the spectral turning bands method is presented for modeling daily rainfall extremes in the data-scarce tropical Ouémé River basin (Benin). This technique uses meta-Gaussian frameworks built on Gaussian random fields, which are transformed into realistic rainfall fields using statistical transfer functions. The simulation framework can be conditioned on point observations and is computationally efficient in generating multiple ensembles of extreme rainfall fields. The results of tests and evaluations for multiple extremes demonstrate the effectiveness of the simulation framework in modeling more realistic rainfall fields and capturing their variability. It successfully reproduces the empirical cumulative distribution function of the observation samples and outperforms classical interpolation techniques like ordinary kriging in terms of spatial continuity and rainfall variability. The study also addresses the challenge of dealing with uncertainty in data-poor areas and proposes a novel approach for determining the spatial correlation structure even with low station density, resulting in a performance boost of 9.5% compared to traditional techniques. Additionally, we present a low-skill reference simulation method to facilitate a comprehensive comparison of the geostatistical simulation approaches. The simulations generated have the potential to provide valuable inputs for hydrological modeling.

On the Univariate Vector-Valued Rational Interpolation and Recovery Problems

In this paper, we consider a novel vector-valued rational interpolation algorithm and its application. Compared to the classic vector-valued rational interpolation algorithm, the proposed algorithm relaxes the constraint that the denominators of components of the interpolation function must be identical. Furthermore, this algorithm can be applied to construct the vector-valued interpolation function component-wise, with the help of the common divisors among the denominators of components. Through experimental comparisons with the classic vector-valued rational interpolation algorithm, it is found that the proposed algorithm exhibits low construction cost, low degree of the interpolation function, and high approximation accuracy.

SR4KVQA: Video quality assessment database and metric for 4K super-resolution

The quality assessment for 4K super-resolution (SR) videos can be conducive to the optimization of video SR algorithms. To improve the subjective and objective consistency of the SR quality assessment, a 4K video database and a blind metric are proposed in this paper. In the database SR4KVQA, there are 30 4K pristine videos, from which 600 SR 4K distorted videos with mean opinion score (MOS) labels are generated by three classic interpolation methods, six SR algorithms based on the deep neural network (DNN), and two SR algorithms based on the generative adversarial network (GAN). The benchmark experiment of the proposed database indicates that video quality assessment (VQA) of the 4K SR videos is challenging for the existing metrics. Among those metrics, the Video-Swin-Transformer backbone demonstrates tremendous potential in the VQA task. Accordingly, a blind VQA metric based on the Video-Swin-Transformer backbone is established, where the normalized loss function and optimized spatio-temporal sampling strategy are applied. The experiment result manifests that the Pearson linear correlation coefficient (PLCC) and Spearman rank-order correlation coefficient (SROCC) of the proposed metric reach 0.8011 and 0.8275 respectively on the SR4KVQA database, which outperforms or competes with the state-of-the-art VQA metrics. The database and the code proposed in this paper are available in the GitHub repository, https://github.com/AlexReadyNico/SR4KVQA.

Binary metaheuristic algorithms for 0–1 knapsack problems: Performance analysis, hybrid variants, and real-world application

This paper examines the performance of three binary metaheuristic algorithms when applied to two distinct knapsack problems (0–1 knapsack problems (KP01) and multidimensional knapsack problems (MKP)). These binary algorithms are based on the classical mantis search algorithm (MSA), the classical quadratic interpolation optimization (QIO) method, and the well-known differential evolution (DE). Because these algorithms were designed for continuous optimization problems, they could not be used directly to solve binary knapsack problems. As a result, the V-shaped and S-shaped transfer functions are used to propose binary variants of these algorithms, such as binary differential evolution (BDE), binary quadratic interpolation optimization (BQIO), and binary mantis search algorithm (BMSA). These binary variants are evaluated using various high-dimensional KP01 examples and compared to several classical metaheuristic techniques to determine their efficacy. To enhance the performance of those binary algorithms, they are combined with repair operator 2 (RO2) to offer better hybrid variants, namely HMSA, HQIO, and HDE. Those hybrid algorithms are evaluated using several medium- and large-scale KP01 and MKP instances, as well as compared to other hybrid algorithms, to demonstrate their effectiveness. This comparison is conducted using three performance metrics: average fitness value, Friedman mean rank, and computational cost. The experimental findings demonstrate that HQIO is a strong alternative for solving KP01 and MKP. In addition, the proposed algorithms are applied to the Merkle-Hellman Knapsack Cryptosystem and the resource allocation problem in adaptive multimedia systems (AMS) to illustrate their effectiveness when applied to optimize those real applications. The experimental findings illustrate that the proposed HQIO is a strong alternative for handling various knapsack-based applications.

Meshless track assimilation (MTA) of 3D PTV data

We propose a data assimilation algorithm to improve the resolution of 3D particle tracking velocimetry (PTV) data and, as a byproduct, it estimates the pressure field. The proposed method, called meshless tracking assimilation (MTA), does not rely on a mesh to compute derivatives. Instead, it exploits the analytical formulation of the obtained field to extract the gradients of the involved quantities in arbitrary points. MTA actively constraints PTV data points to satisfy the Navier–Stokes equations in both time and space at user-defined spatial locations. Three test cases are investigated. The first one is a synthetic dataset reconstructed from a direct numerical simulation and it serves to ensure that the method is properly implemented and working. The second one is the 2020 data assimilation challenge which gives a qualitative comparison with other data assimilation methods. Finally, MTA is used on a Tomographic PTV experiment of a Jet Flow with ReD≃3000 to ensure it can be implemented successfully on experimental data. The first and third test cases show a significant performance boost compared to the classic linear interpolation. This improvement allows for a higher level of structural resolution without sacrificing the temporal and spatial smoothness of the solution. The second test case generally aligns well with other state-of-the-art data assimilation techniques in terms of performance, but there are some noticeable differences.

Classical multivariate Hermite coordinate interpolation on [formula omitted]-dimensional grids

In this work, we study the Hermite interpolation on n-dimensional non-equally spaced, rectilinear grids over a field k of characteristic zero, given the values of the function at each point of the grid and the partial derivatives up to a maximum degree. Initially, we prove the uniqueness of the interpolating polynomial, followed by deriving a concise closed expression that employs a solitary summation, independent of dimensionality, which is algebraically simpler than the only alternative closed form for the n-dimensional classical Hermite interpolation (Gasca and Sauer, 2000). We also provide the remainder of the interpolation in integral form; we derive the ideal of the interpolation and express the interpolation remainder using only polynomial divisions, in the case of interpolating a polynomial function. Moreover, we prove the continuity of Hermite polynomials defined on adjacent n-dimensional grids, thus establishing spline behavior. Finally, we perform illustrative numerical examples to showcase the applicability and high accuracy of the proposed interpolant, in the simple case of few points, as well as hundreds of points on 3D-grids using a spline-like interpolation, which compares favorably to state-of-the-art spline interpolation methods.

Towards a Unified Approach for Continuously-Variable Impedance Control of Powered Prosthetic Legs over Walking Speeds and Inclines.

Research in powered prosthesis control has explored the use of impedance-based control algorithms due to their biomimetic capabilities and intuitive structure. Modern impedance controllers feature parameters that smoothly vary over gait phase and task according to a data-driven model. However, these recent efforts only use continuous impedance control during stance and instead utilize discrete transition logic to switch to kinematic control during swing, necessitating two separate models for the different parts of the stride. In contrast, this paper presents a controller that uses smooth impedance parameter trajectories throughout the gait, unifying the stance and swing periods under a single, continuous model. Furthermore, this paper proposes a basis model to represent intertask relationships in the impedance parameters-a strategy that has previously been shown to improve model accuracy over classic linear interpolation methods. In the proposed controller, a weighted sum of Fourier series is used to model the impedance parameters of each joint as continuous functions of gait cycle progression and task. Fourier series coefficients are determined via convex optimization such that the controller best reproduces the joint torques and kinematics in a reference able-bodied dataset. Experiments with a powered knee-ankle prosthesis show that this simpler, unified model produces competitive results when compared to a more complex hybrid impedance-kinematic model over varying walking speeds and inclines.

Fractal Analysis of Polarizability in Graphite Deposits: Methodological Integration for Geological Prediction and Exploration Efficiency

Most geophysical and geochemical data are commonly acknowledged to exhibit fractal and multifractal properties, but the fractal characteristics of polarizability have received limited attention from the literature. The present study demonstrates that the polarizability data of the graphite deposits have fractal characteristics and introduces the fractal method for its quantitative analysis to indicate and predict the properties of graphite deposits. The results show that the concentration-area (C-A) method is superior to classical interpolation in anomaly extraction but inferior to the spectrum-area (S-A) method in the coverage region. Because the type of graphite ore is sedimentary-metamorphic in this area, the graphite ore-bodies can be regarded as a special stratum, which is different from most metal deposits, and the anomaly of graphite ore are shown in the background mode of the S-A method. The high values of the background mode effectively indicate the potential areas where the graphite-bearing strata occur, while observing a decrease in the power-law exponent (β) of the background mode as the width of ore-bodies increases. The validity of this conclusion was confirmed based on the vertical profiles of the predicted area, and the uncharted ore vein was thereby identified. Furthermore, it was found that the anomaly mode can serve as a grade indicator of graphite ore rather than delineating the fault. By integrating the background and anomaly modes of the S-A method, we can quantitatively predict and effectively identify high-grade targets from sedimentary deposits containing minerals in future exploration.

Local physical gradient-based coupling information interpolate method and application on double-wall turbine blade multidisciplinary analysis

A novel generalization interpolate modification method based on local gradient is presented. The main idea was to remove the sample points located in the gradient direction away from the unknown points compared to the sample points located in the perpendicular direction. This method first uses the Taylor expansion to construct a set of equations and solves the local physical gradient. Subsequently, a scale vector is derived to modify the interpolate space. Three classical interpolation methods were modified: Inverse Distance Weighted (IDW), Radial Basis Function (RBF), and Kriging method. The results of the test functions prove that the proposed method can effectively decrease the interpolate error regardless of low or high dimensional interpolate problem and irrespective of the sample point size. Surface and volume interpolate validations demonstrate that the local gradient generates a large interpolate error, and the modified method can essentially diminish this error with a maximum reduction of 9.94 %. Subsequently, this method is applied to implement a multidisciplinary analysis of a practical double-wall cooling blade with a large physical gradient. The results showed that the pressure and temperature interpolate errors could be decreased by 5.18 % and 14.5 %, respectively, with only an additional 115 s, at the fillet and film hole area where the local gradient was high. When the strength analysis was implemented, the maximum stress error reduction was 25.04 MPa at the stress concentration region.

Applied the Cokriging interpolation method to survey Air Quality Index (AQI) for dust TSP in Da Nang city

Mapping to forecast the air pollution concentration in Da Nang city is an urgent issue for management agencies and researchers of environmental pollution. Although the simulation of spatial location has become popular, it uses the classical interpolation methods with low reliability. Based on the distribution of air quality monitoring stations located in industrial parks, residential areas, transport axes... and sources of air pollution, the application of geostatistical theories, this study presents the results of the Cokriging's interpolation selection which provides forecast results of air pollution distribution in Da Nang city with high reliability. In this article, we use the recorded TSP concentrations (one of major air pollution causes at large metropolis) at several observational stations in Da Nang city, employ the Cokriging interpolation method to find suitable models, then predict TSP dust concentrations at some unmeasured stations in the city. Our key contribution is finding good statistical models by several criteria, then fitting those models with high precision.

Integer multiple quantum image scaling based on NEQR and bicubic interpolation

As a branch of quantum image processing, quantum image scaling has been widely studied. However, most of the existing quantum image scaling algorithms are based on nearest-neighbor interpolation and bilinear interpolation, the quantum version of bicubic interpolation has not yet been studied. In this work, we present the first quantum image scaling scheme for bicubic interpolation based on the novel enhanced quantum representation (NEQR). Our scheme can realize synchronous enlargement and reduction of the image with the size of 2 n × 2 n by integral multiple. Firstly, the image is represented by NEQR and the original image coordinates are obtained through multiple CNOT modules. Then, 16 neighborhood pixels are obtained by quantum operation circuits, and the corresponding weights of these pixels are calculated by quantum arithmetic modules. Finally, a quantum matrix operation, instead of a classical convolution operation, is used to realize the sum of convolution of these pixels. Through simulation experiments and complexity analysis, we demonstrate that our scheme achieves exponential speedup over the classical bicubic interpolation algorithm, and has better effect than the quantum version of bilinear interpolation.

Costate estimation for a multiple-interval pseudospectral method using collocation at the flipped legendre-Gauss-Radau points

A multiple-interval pseudospectral method is presented for direct trajectory optimization and costate estimation of optimal control problems using collocation at the flipped Legendre-Gauss-Radau points. The recently developed global flipped Radau pseudospectral method is one of the special case. The necessary optimality conditions of the multiple-interval differential scaled optimal control problem are rigorously derived for the first time by applying the calculus of variations. Then, the costate and constraint multiplier estimates for the proposed method are derived in detail by comparing the discretized optimality conditions with the Karush-Kuhn-Tucker conditions of the nonlinear programming problem resulting from collocation. Furthermore, in order to improve the numerical stability and alleviate the computational error when calculating the state pseudospectral differentiation matrix, the barycentric Lagrange interpolation is adopted in the state approximation instead of the classic Lagrange interpolation without losing the computational efficiency. The method presented in this paper is applied to two benchmark optimal control problems and is compared with the de-facto standard pseudospectral methods. Extensive numerical results indicate that it has the ability to obtain highly accurate solutions to a wide variety of complex constrained optimal control problems.

Screening and optimization of interpolation methods for mapping soil-borne polychlorinated biphenyls

There is yet no scientific consensus, and for now, on how to choose the optimal interpolation method and its parameters for mapping soil-borne organic pollutants. Take the polychlorinated biphenyls (PCBs) for instance, we present the comparison of some classic interpolation methods using a high-resolution soil monitoring database. The results showed that empirical Bayesian kriging (EBK) has the highest accuracy for predicting the total PCB concentration, while root mean squared error (RMSE) in inverse distance weighting (IDW) is among the highest in these interpolation methods. The logarithmic transformation of non-normally distributed data contributed to enhance considerably the semivariogram for modeling in kriging interpolation. The increasing of search neighborhood reduced IDW's RMSE, but slightly affected in ordinary kriging (OK), while both of them resulted in over smooth of prediction map. The existence of outliers made the difference between two points increase sharply, and thereby weakening spatial autocorrelation and decreasing the accuracy. As predicted error increased continuously, the prediction accuracy of different interpolation methods reached unanimity gradually. The attempt of the assisted interpolation algorithm did not significantly improve the prediction accuracy of the IDW method. This study constructed a standardized workflow for interpolation, which could reduce human error to reach higher interpolation accuracy for mapping soil-borne PCBs.

Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes.

Lasso trigonometric polynomial approximation for periodic function recovery in equidistant points

This paper introduces a fully discrete soft thresholding trigonometric polynomial approximation on [−π,π], named Lasso trigonometric interpolation. This approximation is an ℓ1-regularized discrete least squares approximation under the same conditions of classical trigonometric interpolation on an equidistant grid. Lasso trigonometric interpolation is a sparse scheme which is efficient in dealing with noisy data. We theoretically analyze Lasso trigonometric interpolation quality for continuous periodic function. The L2 error bound of Lasso trigonometric interpolation is less than that of classical trigonometric interpolation, which improved the robustness of trigonometric interpolation. The performance of Lasso trigonometric interpolation for several testing functions (sin wave, triangular wave, sawtooth wave, square wave), is illustrated with numerical examples, with or without the presence of data errors.

On solving some Cauchy singular integral equations by de la Vallée Poussin filtered approximation

The paper deals with the numerical solution of Cauchy Singular Integral Equations based on some non standard polynomial quasi–projection of de la Vallée Poussin type. Such kind of approximation presents several advantages over classical Lagrange interpolation such as the uniform boundedness of the Lebesgue constants, the near–best order of uniform convergence to any continuous function, and a strong reduction of Gibbs phenomenon. These features will be inherited by the proposed numerical method which is stable and convergent, and provides a near-best polynomial approximation of the sought solution by solving a well conditioned linear system. The numerical tests confirm the theoretical error estimates and, in case of functions subject to Gibbs phenomenon, they show a better local approximation compared with analogous Lagrange projection methods.

A Novel Approach for Bathymetry Estimation through Bayesian Gravity Inversion

The bathymetry is the most superficial layer of the Earth’s crust on which it is possible to perform direct measurements. However, it is also well known that water covers more than 70% of the Earth’s surface, so an enormous expenditure of acquisition campaigns should be performed to produce a high-resolution map of this layer. Currently exploiting mainly commercial navigation routes, the sea floor coverage with shipborne sounding is only at 11%, and the remaining part is currently modeled by classical interpolation techniques or satellite-based gravity inversion methods. In the present work, a new method to refine bathymetry modeling based on the exploitation of global gravity field models is presented. In the proposed solution, once modeled and removed from the observed gravity field, the gravitational signals related to the deepest structures, a 3D Bayesian inversion algorithm is used to improve the actual knowledge of bathymetry. The proposed inversion method also enables limiting the solution to shipborne sounding measurements in such a way as to improve the seafloor grid where no local, high-quality information is available. Two test cases are discussed in the Mediterranean Sea region. Promising results are presented, opening the possibility of applying an analogous method to refine the bathymetry modeling at larger scales up to the global one.

Improving Rainfall Fields in Data-Scarce Basins: Influence of the Kernel Bandwidth Value of Merging on Hydrometeorological Modeling

Accurate rainfall fields are important for several hydrological and meteorological applications. In poorly instrumented basins, the lack of rain gauges heavily affects the spatial rainfall estimation, yet neither remote sensed nor climate model estimates are good enough for management applications. To tackle this problem, we investigate the impacts of combining both sources of information by varying the kernel bandwidth value of the double smoothing merging algorithm and analyzing the error of the rainfall fields. We explored the correlation between rain gauge density and bandwidth and compared the results against classical geostatistical interpolation methods based solely on in-situ measurements. Propagation of rainfall error into hydrological modelling is usual, and therefore we evaluated the influence of the bandwidth in streamflow simulations implementing two hydrological models. The hydrological evaluation considered the analysis of hydrological signatures rather than just performance metrics. We found that there is a clear correlation between kernel bandwidth and monitoring network density and that the bandwidth also affects hydrological performance. Simple bilinear downscaling did not produce a significant difference in meteorological or hydrological errors, and rain gauge network configuration also impacts the error of the field. We conclude that merging outperforms the results of classical interpolation methods, in some cases by 20% or 50%, suggesting the suitability of the method for being applied in data-scarce domains.