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The need for highly accurate atmospheric wind observations is a high priority in the science community, and in particular numerical weather prediction (NWP). To address this requirement, this study leverages Aeolus wind LIDAR Level-2B data provided by the European Space Agency (ESA) to better characterize atmospheric motion vector (AMV) bias and uncertainty, with the eventual goal of potentially improving AMV algorithms. AMV products from geostationary (GEO) and low-Earth polar orbiting (LEO) satellites are compared with reprocessed Aeolus horizontal line-of-sight (HLOS) global winds observed in August and September 2019. Winds from two of the four Aeolus observing modes are utilized for comparison with AMVs: Rayleigh-clear (derived from the molecular scattering signal) and Mie-cloudy (derived from particle scattering). For the most direct comparison, quality controlled (QC’d) Aeolus winds are collocated with quality controlled AMVs in space and time, and the AMVs are projected onto the Aeolus HLOS direction. Mean collocation differences (MCD) and standard deviation (SD) of those differences (SDCD) are determined from comparisons based on a number of conditions, and their relation to known AMV bias and uncertainty estimates is discussed. GOES-16 and LEO AMV characterizations based on Aeolus winds are described in more detail. Overall, QC’d AMVs correspond well with QC’d Aeolus HLOS wind velocities (HLOSV) for both Rayleigh-clear and Mie-cloudy observing modes, despite remaining biases in Aeolus winds after reprocessing. Comparisons with Aeolus HLOSV are consistent with known AMV bias and uncertainty in the tropics, NH extratropics, and in the Arctic, and at mid- to upper-levels in both clear and cloudy scenes. SH comparisons generally exhibit larger than expected SDCD, which could be attributed to height assignment errors in regions of high winds and enhanced vertical wind shear. GOES-16 water vapor clear-sky AMVs perform best relative to Rayleigh-clear winds, with small MCD (-0.6 m s-1 to 0.1 m s-1) and SDCD (5.4–5.6 m s-1) in the NH and tropics that fall within the accepted range of AMV error values relative to radiosonde winds. Compared to Mie-cloudy winds, AMVs exhibit similar MCD and smaller SDCD (~4.4–4.8 m s-1) throughout the troposphere. In polar regions, Mie-cloudy comparisons have smaller SDCD (5.2 m s-1 in the Arctic, 6.7 m s-1 in the Antarctic) relative to Rayleigh-clear comparisons, which are larger by 1–2 m s-1. The level of agreement between AMVs and Aeolus winds varies per combination of conditions including the Aeolus observing mode coupled with AMV derivation method, geographic region, and height of the collocated winds. It is advised that these stratifications be considered in future comparison studies and impact assessments involving 3D winds. Additional bias corrections to the Aeolus dataset are anticipated to further refine the results.

A method of space ensemble (SE) and time ensemble (TEs) correlation function technique applied on ground penetration radar (GPR) B-scan raw data proves to result in detailed information for tracking buried objects. These computations help in site allocations by time ensemble correlation functions (TESCs) and time stamps of objects scattering by space ensemble correlation functions (SECFs) which are precisely consistent with the simulation results. While these results are applied for given data stack, randomness in physical parameters of tested ground always occurs. An introduction of randomness to the physical parameters of tested models by creating stochastic collocation (SC) ensembles incorporating randomness to B-scan via successive A-scans, SC-SE, and SC-TE ensembles enables sensitivity analysis (SA) study of GPR raw data which gives a tool to change the physical properties of the ground in search for better ground matching. This method can be added to GPR machines or simulation software to enhance raw data analysis. A field experiment of small non-metallic objects' allocation was carried out as a platform for assessing the realistic performance of the method.

An efficient and accurate reliability-based design of the geosynthetic-reinforced slopes (GRS) using the pseudo-static and modified pseudo-dynamic framework is proposed in the present study. Deterministic formulation used in the present study is made robust with the help of nonlinear constrained optimization. The collocation based stochastic response surface method (CSRSM) is used to probabilistically analyze the GRS. The critical modes of failure pertaining to the internal and external stability of the GRS are considered in the formation of the performance functions. The horizontal seismic acceleration coefficient (kh), internal friction angle of soil (φ), soil unit weight (γ), shear wave velocity (Vs), and friction angle at the interface between soil and reinforcement (φb) are chosen as the random variables, owing to their high influence on the stability of the GRS. The influence of correlation on the stability of the reinforced slope is illustrated considering the internal and external stability. System reliability analysis considering the internal and external modes of failure is also performed. An illustrative example is presented showing the steps to design a GRS using the proposed formulation. The results confirm the necessity of performing the system reliability analysis to estimate an accurate value of probability of failure of GRS.

Air pollution is a severe problem growing over time. A dense air-quality monitoring network is needed to update the people regarding the air pollution status in cities. A low-cost sensor device (LCSD) based dense air-quality monitoring network is more viable than continuous ambient air quality monitoring stations (CAAQMS). An in-field calibration approach is needed to improve agreements of the LCSDs to CAAQMS. The present work aims to propose a calibration method for PM2.5 using domain adaptation technique to reduce the collocation duration of LCSDs and CAAQMS. A novel calibration approach is proposed in this work for the measured PM2.5 levels of LCSDs. The dataset used for the experimentation consists of PM2.5 values and other parameters (PM10, temperature, and humidity) at hourly duration over a period of three months data. We propose new features, by combining PM2.5, PM10, temperature, and humidity, that significantly improved the performance of calibration. Further, the calibration model is adapted to the target location for a new LCSD with a collocation time of two days. The proposed model shows high correlation coefficient values (R2) and significantly low mean absolute percentage error (MAPE) than that of other baseline models. Thus, the proposed model helps in reducing the collocation time while maintaining high calibration performance.

Sentinel-3 is an Earth observation satellite constellation launched by the European Space Agency. Each satellite carries two optical multispectral instruments: the Ocean and Land Colour Instrument (OLCI) and the Sea and Land Surface Temperature Radiometer (SLSTR). OLCI and SLSTR sensors produce images covering the visible and infrared spectrum that can be collocated in order to generate synergistic products. In Earth observation, a particular weakness of optical sensors is their high sensitivity to clouds and their shadows. An incorrect cloud and cloud shadow detection leads to mistakes in both land and ocean retrievals of biophysical parameters. In order to exploit both OLCI and SLSTR capabilities, image co-registration at ground level is needed. However, applying such collocation of the images results in cloud location mismatches due to the different viewing angles of OLCI and SLSTR, which complicates the synergistic cloud detection. This study seeks to provide a solution to correctly obtain the projected clouds based on the estimation of cloud top heights in order to better collocate clouds between sensors and detect their shadows. The study presents a forward and backward method to estimate the real nadir position of a cloud on the satellite image starting from an existing cloud mask, as well as the corresponding cloud projections on the surface depending on the solar and sensor viewing angles. The estimation of cloud top heights is based on differences in the cloud projections from SLSTR nadir and oblique views. Experimental results show that the stereo cloud matching based on maximum cross-correlation between SLSTR nadir and oblique spectra was the most robust method to match SLSTR clouds for both nadir and oblique views as compared to spectral distance and spectral angle minimization. We test the method over several images around the world, leading to higher overall accuracy (OA) as compared to Sentinel-3 official products, both in detecting SLSTR clouds and OLCI cloud shadows (SLSTR nadir OA = 93.6%, SLSTR oblique OA = 88.7%, OLCI cloud shadow OA = 93.9% for the stereo matcher, against 82.2%, 81.3% and 90.5%, respectively, for the official Sentinel-3 products). This study also provides a starting point in the development of a cloud screening approach for the upcoming Fluorescence Explorer (FLEX) satellite mission, expected to fly in tandem with Sentinel-3.

The purpose of this study is to formulate a truly and effective boundary-type meshfree method for analysis of 2D transient heat conduction problems including moving laser heat sources. Being based on the method of fundamental solutions (MFS), the present work formulates new time-dependent formulations to solve the problem directly in the time-domain. The motion path and the intensity of the laser heat source over a moving area can be arbitrary functions of time and space. In the proposed formulations, the particular solution associated to the heat source function is evaluated using a simple approach without using any internal points or cells. By solving two example problems, the efficiency and accuracy of the proposed method are demonstrated. The analyses show that the presented boundary-type meshfree method is relatively more efficient than other domain-type techniques. From the analyses, it is also observed that very accurate results can be obtained by the proposed method employing large time-step size and small number of collocation points.

The problem of minimum-time, low-thrust, Earth-to-Mars interplanetary orbital trajectory optimization is considered. The minimum-time orbital transfer problem is modeled as a four-phase optimal control problem where the four phases correspond to planetary alignment, Earth escape, heliocentric transfer, and Mars capture. The four-phase optimal control problem is then solved using a direct collocation adaptive Gaussian quadrature collocation method. The following three models are used in the study: (1) circular planetary motion; (2) elliptic planetary motion; and (3) elliptic planetary motion with gravity perturbations, where the transfer begins in a geostationary orbit and terminates in a Mars-stationary orbit. Results for all three cases are provided, and one particular case is studied in detail to show the key features of the optimal solutions. Using the particular value thrust specific force of $0.00098\times 10^{-4}~\textrm{m}\cdot\textrm{s}^{-2}$, it was found that the minimum times for cases (1), (2), and (3) are, respectively, 215 d, 196 d, and 198 d with departure dates, respectively, of 1 July 2020, 30 June 2020, and 28 June 2020. Finally, the problem formulation developed in this study is compared against prior work on an Earth-to-Mars interplanetary orbit transfer where it is found that the results of this research show significant improvement in transfer time relative to the prior work., Comment: 41 pages, 8 figures, 7 tables

The article deals with stochastic-deterministic thermal dosimetry in lower portion of GHz frequency range for human exposure to radiation from 5G communication systems. Deterministic assessment of a temperature distribution in multilayered head model is based on the finite element solution of Pennes equation while the stochastic analysis uses the stochastic collocation approach. Thermal conductivity and volumetric blood perfusion rate of scalp, skull, and brain tissues are modeled as uniformly distributed random parameters, while values of other thermal input parameters are fixed to their respective average values. The stochastic mean and variance of the heating factor at 3.5 GHz are computed and confidence interval is estimated for some observation points of interest. By using “one-at-a-time” and analysis of variance principles, sensitivity analysis of input parameter set is carried out. Some illustrative numerical results stemming from deterministic and stochastic model, respectively, are presented. When observing the impact of input parameters on heating factor values in the direction of incidence, blood perfusion of scalp is dominant with exception in the interior head parts, where the impact of the brain blood perfusion prevails. For the exposure scenario of interest, the temperature elevation is within the limits given by basic restrictions.

In this letter, a time-domain stochastic radial point interpolation method (SRPIM) is developed for uncertainty quantification of electromagnetic systems. Derivatives of field quantities in Maxwell’s equations are obtained using radial basis function, and stochasticity in the dielectric constant of a part of the model space is incorporated into this formulation. This is validated using the numerical example of a parallel-plate waveguide with dielectrics inside, which is implemented using uniaxial perfectly matched boundary layers. The stochastic variations in the computed field in the time domain and the transmission coefficient in the frequency domain are evaluated. Accuracy of these simulation results is validated using Kolmogorov Smirnov test, with Monte Carlo (MC) simulation as the reference. The computation time of the proposed method is found to be significantly better than MC and superior to stochastic collocation. The proposed method performs better even for large variations.

In this study, a strategy was proposed to determine the interval of the variable shape parameter for the ghost point method using radial basis functions. The determination of a suitable interval for the variable shape parameter remains a challenge. The modified Franke formula was used as an initial predictor of the center of the interval of the variable shape parameter in this study. After extensive tests, a numerical procedure was found for the determination of a suitable interval. The improvement from the imposition of the partial differential equation on the boundary points using the ghost point method was also investigated. To demonstrate the effectiveness of the proposed approach, four numerical examples are presented, including second and fourth order partial differential equations in 2D and 3D.

Design of new materials is quite a difficult task owing to various time and length scales and affiliated uncertainties. The major challenge is to include all these in a conventional model. Hyperparameter models in machine learning can be used to overcome these issues. In this paper, an artificial neural network (ANN) model is developed to estimate the effective elastic parameters of unidirectional fiber reinforced composites using representative volume elements (RVE) considering uncertainty in the fiber diameter. The diameter probability distribution is constructed from the acquired gray images by employing image processing operations. The generalized Polynomial Chaos (gPC) expansion is then used to represent the distribution as a random input parameter for finite element analysis, from where the effective parameters are realized. Similarly, the outputs of the FE model, i.e., elastic parameters, are approximated by gPC expansions having unknown deterministic coefficients and random orthogonal Hermite polynomials. A set of collocation points are generated from roots of the random polynomials, from there, the unknown coefficients are estimated. The realization samples are utilized to train an ANN algorithm based on supervised deep learning. The developed ANN model is later tested and validated for a new sample set of data. It is shown that the ANN model with few hidden layers and neurons has a high accuracy for estimation of the elastic parameters directly from the information on the distribution of fiber diameters.

Due to the imprecise structure originated from the limitation of knowledge level or lack of information, the inverse problems for identifying the unknown parameter of system will face the challenges in the solving mechanism and the computational cost. Compared with the traditional inverse problems, this kind of inverse problems not only need to identify the unknown parameter, but also need to quantify the uncertainty caused by the imprecise structure. In this paper, a basic solving framework based on evidence theory for this inverse uncertainty quantification is established to inversely estimate the effect of the uncertainties in imprecise structure on the unknown parameters. In order to overcome the computing bottleneck problem caused by the coupling of evidence-based uncertainty propagation and the inverse computation, an efficient similar system analysis method is further proposed. Firstly, the marginal interval analysis method is developed to transform the evidence-based uncertain inverse problem into several deterministic inverse problems at the marginal collocation points. Afterwards, the similar system analysis method is proposed by adequately utilizing the similarity of systems at any two adjacent marginal collocation points to further improve the computational efficiency of several deterministic inverse problems. Because of the coupling effect of the above two strategies, the proposed uncertain inverse method decouples the time-consuming multi-layer nested inverse procedure, and then greatly reduces the computational cost of inverse uncertainty quantification based on evidence theory. Two numerical examples and one engineering application are provided to verify the high efficiency of the proposed uncertain inverse method.

In this paper, we developed a new three-step method for numerical solution of third order ordinary differential equations. Interpolation and collocation methods were used by choosing interpolation points at steps points using power series, while collocation points at step points, using a combination of powers series and perturbation terms gotten from the Legendre polynomials, giving rise to a polynomial of degree and equations. All the analysis on the method derived shows that it is zero-stable, convergent and the region of stability is absolutely stable. Numerical examples were provided to test the performance of the method. Results obtained when compared with existing methods in the literature, shows that the method is accurate and efficient

Modeling uncertainties in the input parameters of computer simulations is an established way to account for inevitably limited knowledge. To overcome long run-times and high demand for computational resources, a surrogate model can replace the original simulation. We use spatially adaptive sparse grids for the creation of this surrogate model. Sparse grids are a discretization scheme designed to mitigate the curse of dimensionality, and spatial adaptivity further decreases the necessary number of expensive simulations. We combine this with B-spline basis functions which provide gradients and are exactly integrable. We demonstrate the capability of this uncertainty quantification approach for a simulation of the Hokkaido Nansei–Oki Tsunami with anuga. We develop a better understanding of the tsunami behavior by calculating key quantities such as mean, percentiles and maximum run-up. We compare our approach to the popular Dakota toolbox and reach slightly better results for all quantities of interest. References B. M. Adams, M. S. Ebeida, et al. Dakota. Sandia Technical Report, SAND2014-4633, Version 6.11 User’s Manual, July 2014. 2019. https://dakota.sandia.gov/content/manuals. J. H. S. de Baar and S. G. Roberts. Multifidelity sparse-grid-based uncertainty quantification for the Hokkaido Nansei–Oki tsunami. Pure Appl. Geophys. 174 (2017), pp. 3107–3121. doi: 10.1007/s00024-017-1606-y. H.-J. Bungartz and M. Griebel. Sparse grids. Acta Numer. 13 (2004), pp. 147–269. doi: 10.1017/S0962492904000182. M. Eldred and J. Burkardt. Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification. 47th AIAA. 2009. doi: 10.2514/6.2009-976. K. Höllig and J. Hörner. Approximation and modeling with B-splines. Philadelphia: SIAM, 2013. doi: 10.1137/1.9781611972955. M. Matsuyama and H. Tanaka. An experimental study of the highest run-up height in the 1993 Hokkaido Nansei–Oki earthquake tsunami. National Tsunami Hazard Mitigation Program Review and International Tsunami Symposium (ITS). 2001. O. Nielsen, S. Roberts, D. Gray, A. McPherson, and A. Hitchman. Hydrodymamic modelling of coastal inundation. MODSIM 2005. 2005, pp. 518–523. https://www.mssanz.org.au/modsim05/papers/nielsen.pdf. J. Nocedal and S. J. Wright. Numerical optimization. Springer, 2006. doi: 10.1007/978-0-387-40065-5. D. Pflüger. Spatially Adaptive Sparse Grids for High-Dimensional Problems. Dr. rer. nat., Technische Universität München, Aug. 2010. https://www5.in.tum.de/pub/pflueger10spatially.pdf. M. F. Rehme, F. Franzelin, and D. Pflüger. B-splines on sparse grids for surrogates in uncertainty quantification. Reliab. Eng. Sys. Saf. 209 (2021), p. 107430. doi: 10.1016/j.ress.2021.107430. M. F. Rehme and D. Pflüger. Stochastic collocation with hierarchical extended B-splines on Sparse Grids. Approximation Theory XVI, AT 2019. Springer Proc. Math. Stats. Vol. 336. Springer, 2020. doi: 10.1007/978-3-030-57464-2_12. S Roberts, O. Nielsen, D. Gray, J. Sexton, and G. Davies. ANUGA. Geoscience Australia. 2015. doi: 10.13140/RG.2.2.12401.99686. I. J. Schoenberg and A. Whitney. On Pólya frequence functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves. Trans. Am. Math. Soc. 74.2 (1953), pp. 246–259. doi: 10.2307/1990881. W. Sickel and T. Ullrich. Spline interpolation on sparse grids. Appl. Anal. 90.3–4 (2011), pp. 337–383. doi: 10.1080/00036811.2010.495336. C. E. Synolakis, E. N. Bernard, V. V. Titov, U. Kânoğlu, and F. I. González. Standards, criteria, and procedures for NOAA evaluation of tsunami numerical models. NOAA/Pacific Marine Environmental Laboratory. 2007. https://nctr.pmel.noaa.gov/benchmark/. J. Valentin and D. Pflüger. Hierarchical gradient-based optimization with B-splines on sparse grids. Sparse Grids and Applications—Stuttgart 2014. Lecture Notes in Computational Science and Engineering. Vol. 109. Springer, 2016, pp. 315–336. doi: 10.1007/978-3-319-28262-6_13. D. Xiu and G. E. Karniadakis. The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24.2 (2002), pp. 619–644. doi: 10.1137/S1064827501387826.

This paper investigates a kind of inverse problem for assessing the uncertainties of identified parameters with uncertainties in structural parameters and limited experimental data. The uncertainty is described by the interval model in which only the bounds of uncertain parameters are required. Directly solving this kind of inverse problem involves a double-loop problem where the outer-loop is interval analysis and the inner-loop is deterministic optimization, which requires a large number of calculations. To efficiently evaluate the effect of interval parameters on the identified parameters, a novel method based on the dimension-reduction method and adaptive collocation strategy is proposed. First, the interval inverse problem is transformed into an inverse-propagation problem, and the dimension-reduction interval method is adopted to transform the interval inverse-propagation problem into several one-dimensional interval inverse-propagation problems. Then, an adaptive collocation strategy is proposed to efficiently estimate the lower and upper bounds of identified parameters. Therefore, the double-loop problem can be transformed into several deterministic inverse problems, and the efficiency of solving the uncertain inverse problem is dramatically improved. Two numerical examples and an engineering application are applied to demonstrate the feasibility and efficiency of the proposed method.

In this contribution, two gravimetric geoid models for Egypt; EGY-LGM2019 and EGY-FGM2019 are developed using the least-squares collocation and the Fast-Fourier Transformation methods, applying Rem...

We evaluate temperature and humidity profiles retrieved from an INSAT-3DR sounder against radiosonde measurements for one year during 2017–2018. This evaluation is carried out in terms of slope, intercept, bias, Root Mean Square Error (RMSE), and correlation coefficient in temperature and relative humidity profiles. This comparison provided first-hand information about the performance of INSAT-3DR sounder retrievals. This validation exercise is unique in terms of comparing with the high resolution (1 Hz) radiosonde measurements performed in the afternoon time when there is maximum convection driven by solar radiation. 276 pairs of co-located temperatures have been compared. INSAT-3DR temperature retrievals are strongly linearly correlated to radiosonde measurements, with the slope as 1.001 and correlation coefficient of 0.99, while the mean temperature bias between INSAT-3DR and radiosonde is -0.23°C with RMSE of 1.9°C. Comparison of Relative humidity retrieved from INSAT-3DR to the radiosonde data results in RMSE of 9.8% with a slope of 0.99 with a correlation coefficient of 0.77. The collocation of data points for validation is done matching in space [Latitude, Longitude, Pressure level (height)] and time (±30 minutes). The effect of relaxing the space and time window on the validation statistics is also studied for the first time.

The pollution propagation within the underground water reservoirs is a challenging and important phenomenon. In the current work, the numerical simulation of pollution transport in an underground channel is performed using the meshless method. To account the anomalous dispersion in a general case, the variable order fractional mass transfer equation is utilized for a rectangular channel. The clean fluid stream enters the channel and due to several phenomenon including the leakage of pollution from the channel walls, the internal pollution source, and the occurrence of the chemical reactions, the pollution content is affected. The non-dimensional form of the governing equation is derived to introduce the dominant dimensionless group numbers. The numerical solution of the obtained equation is established based on the meshless local Petrov–Galerkin method using the moving Kriging interpolation. The Dirac delta function is used as a test function over the local sub-domains. To discretize the present formulation in space variables, we apply the moving Kriging shape functions. Also, to estimate the fractional-order versus the time, finite difference relation is utilized. Using Kronecker’s delta property of moving Kriging interpolation shape functions the boundary conditions in the final system are imposed automatically. The main aim of this technique is to investigate a global estimation for the model, which consequently decrease such problems to those of solving a system of algebraic equations. To determine the accuracy and efficiency of the present method on regular and irregular domains, an example is given in various domains and with regular and irregular distributed points. Also, the effect of major parameters including the fractional order exponent, leakage velocity, chemical reaction rate constant, diffusion coefficient in addition to the stationary/moving pollution source is also examined. It will be shown that, by enhancement of the diffusivity from 0.1 to 20, the outlet concentration reduces by 25.1%, while diffusivity increase from 20 to 50 affects the exiting pollution by merely 7.0%.

The complexity of the pore structure, spatial development, fractures, and pore distribution of fractured-vuggy carbonate reservoirs influences the water invasion dynamics of gas reservoirs, which is crucial in the dynamic research of strongly heterogeneous reservoirs. In this study, the collocation relationship of pore-vuggy fractures is described by the quantitative characterization of their attribute parameters. The discrete fracture network model is used to match and construct the fractures in different modes. The distribution classification method is used to model three-dimensional geological reservoirs in terms of their geometric and attribute characteristics. Bottom-water and edge-water gas reservoirs are constructed separately using numerical simulation, and the dynamic characteristics of water invasion are described. The results show that the proposed method is suitable for the geological modeling of fractured-vuggy gas reservoirs with strong heterogeneity and complexity. The modeling accuracy is improved because the gas reservoir heterogeneity and water invasion’s dynamic characteristics can be described accurately. Six stages of water invasion are identified from the numerical simulation of water invasion. This method provides theoretical guidance for the study of heterogeneous gas reservoirs with water. Cited as : Han, X., Tan, X., Li, X., Pang, Y., Zhang, L. Water invasion performance of complex fracture-vuggy gas reservoirs based on classification modeling. Advances in Geo-Energy Research, 2021, 5(2): 222-232, doi: 10.46690/ager.2021.02.11

Brazilian cerrado biodiversity plays an important role in different areas, such as food and medicine due to the variety of plant species that can be found. However, this has been poorly studied in order to explore aspects related to food and analysis of crude extracts or isolated compounds. In this context, this contribution aims to determine kinetic and phenomenological models to evaluate the mass and energy transport during the drying operation of Guapeva (Pouteria gardneriana Radlk) by formulating and solving two inverse problems considering the Differential Evolution algorithm. The mathematical model that represents the Guapeva drying process considers a one-dimensional differential equation system in spherical coordinates. To solve this system, the Normal Collocation is used as a numerical strategy to integrate the mass and energy balances. The proposed inverse problem considers experimental data obtained in an experiment carried out in a tunnel type dryer to obtain kinetics and thermo-physical parameters. The obtained results demonstrated both the feasibility and effectiveness of the proposed methodology to solve inverse problems.

This paper presents a two-step hybrid numerical scheme with one off-grid point for the numerical solution of general second-order initial value problems without reducing to two systems of the first order. The scheme is developed using the collocation and interpolation technique invoked on Bernstein polynomial. The proposed scheme is consistent, zero stable, and is of order four($4$). The developed scheme can estimate the approximate solutions at both steps and off-step points simultaneously using variable step size. Numerical results obtained in this paper show the efficiency of the proposed scheme over some existing methods of the same and higher orders.

In this paper, we develop a machine learning method with the Least Squares Support Vector Regression (LS-SVR) for the numerical solution of Fredholm integral equations. Two different approaches are proposed for training the network by using the shifted Legendre kernel, the collocation and Galerkin LS-SVR approaches. As with the standard LS-SVR for known dataset regression, the formulation of the method gives rise to an optimization problem. An equivalent system of algebraic equations is then derived and in linear cases discussed in terms of the sparsity of the matrices and computational efficiency. Finally, the method is carried out on some numerical examples, including nonlinear and multidimensional cases to show the accuracy and efficiency of the method.

In this paper, we provide further illustrations of prolate interpolation and pseudospectral differentiation based on the barycentric perspectives. The convergence rates of the barycentric prolate interpolation and pseudospectral differentiation are derived. Furthermore, we propose the new preconditioner, which leads to the well-conditioned prolate collocation scheme. Numerical examples are included to show the high accuracy of the new method. We apply this approach to solve the second-order boundary value problem and Helmholtz problem.

The progressive and iterative approximation method for least square fitting (LSPIA) (Deng and Lin in Comput Aided Des 47:32–44, 2014) is an efficient method for fitting a large number of data points. By introducing the global and local relaxation parameters, we develop an extended LSPIA (ELSPIA) method which includes the LSPIA method as its special case. The ELSPIA method constructs the sequence of curves and surfaces by adjusting the control points with the outer and inner iteration. It is proved that the sequence of curves and surfaces converges to the least square fitting curve and surface, respectively, even when the collocation matrix is not of full column rank. The ELSPIA method is flexible to allow the local adjustment of the control points. Moreover, the convergence rate of the ELSPIA method can be faster than that of the LSPIA method under the same assumption. Numerical results verify this phenomenon.

The ill-posedness of the forward-backward heat problem arises in boundary layer problems, fluid dynamics, plasma physics, astrophysics and the study of propagation of an electron beam through the solar corona. In this study, a new truly meshless method is developed for the numerical solution of the two dimensional forward-backward heat equation. We propose a novel method based on the domain decomposition scheme and RBF method with an adaptive nodes technique. Specifically, the physical domain is divided into two subdomains each defining a forward or a backward subproblem. The resulting subproblems are dealt with by a radial basis function meshfree method for spatial dimension and a finite difference scheme for the time derivative followed by an adaptive algorithm to achieve a desired accuracy. In addition, we prove that the time discrete scheme is stable and convergent. Some numerical experiments will be presented to show the performance of our collocation scheme.

Haiyang-2B (HY-2B) is the second marine dynamic environment satellite of China. Sea surface temperature (SST) products from the scanning microwave radiometer (SMR) onboard HY-2B satellite are evaluated against in situ measurements. Approximately, ten months of data are used for the initial evaluation, from January 15, 2019 to November 15, 2019. The temporal and spatial windows for collocation are 30 min and 25 km, respectively, which produce 450 416 matchup pairs between HY-2B/SMR and in situ SSTs. The statistical comparison of the entire data set shows that the mean bias is −0.13 °C (SMR minus buoy), and the corresponding root-mean-square error (RMSE) is 1.06 °C. Time series of collocations for the SST difference shows that a good agreement is found between HY-2B/SMR and in situ SSTs after June 15, revealing a mean bias and an RMSE of only 0.09 °C and 0.72 °C, respectively. A three-way error analysis is conducted between the SMR, Global Precipitation Measurement Microwave Imager (GMI), and in situ SSTs. Individual standard deviations are found to be 0.41 °C for the GMI SST, 0.15 °C for the in situ SST, and 1.03 °C for the SMR SST. The results indicate that the HY2B/SMR SST products need to be improved during the period from January 15, 2019 to June 15, 2019.

In many modern applications, a central task is to model a dynamical behavior with ordinary differential equations. A common way to identify parameters within such a model is to fit its output against given measurements. Since it can be difficult to understand the connection between measurements and the parameter identification result, it is desirable to develop methods for analyzing this aspect. In this paper, we show how information from parametric sensitivity analysis can be used to gain a better insight into the impact of certain measurement regions on the identified model parameters. We parameterize the measurements by a B-spline regression and formulate the task of parameter identification using a collocation approach. In the resulting nonlinear optimization problem, we consider the B-spline coefficients as perturbation parameters. Next, we identify the model parameters and compute their parametric sensitivities with respect to these perturbations. In a final step, we evaluate a newly developed measure to characterize the desired influence. The corresponding optimization problems are solved with the nonlinear programming solver WORHP in combination with its integrated module WORHP Zen, which computes the required parametric sensitivities efficiently. We demonstrate the proposed approach by applying it to the example of parameter identification of a driving car.

This paper presents a method to predict the pilot workload in helicopter landing after one engine failure. The landing procedure is simulated numerically via applying nonlinear optimal control method in the form of performance index, path constraints and boundary conditions based on an augmented six-degree-of-freedom rigid-body flight dynamics model, solved by collocation and numerical optimization method. UH-60A helicopter is taken as the sample for the demonstration of landing after one engine failure. The numerical simulation was conducted to find the trajectory of helicopter and the controls from pilot for landing after one engine failure with different performance index considering the factor of pilot workload. The reasonable performance index and corresponding landing trajectory and controls are obtained by making a comparison with those from the flight test data. Furthermore, the pilot workload is evaluated based on wavelet transform analysis of the pilot control activities. The workloads of pilot control activities for collective control, longitudinal and lateral cyclic controls and pedal control during the helicopter landing after one engine failure are examined and compared with those of flight test. The results show that when the performance index considers the factor of pilot workload properly, the characteristics of amplitudes and constituent frequencies of pilot control inputs in the optimal solution are consistent with those of the pilot control inputs in the flight test. Therefore, the proposed method provides a tool of predicting the pilot workload in helicopter landing after one engine failure.

This study presents a comparison of tropospheric parameters, namely zenith tropospheric delay and total refractivity calculated using meteorological measurements from a Meteodrone and Global Navigation Satellite Systems (GNSS) stations. A Meteodrone is a unique Unmanned Aerial Vehicle (UAV) originally designed for automatic measurements of meteorological parameters in vertical profiles (up to 3 km above ground) for the assimilation into numerical weather prediction models. One profile measured with both ascending and descending drone takes approximately 20 min, and there are multiple flights possible during one night. The measurements for this campaign have been conducted in March 2019, in Marbach, Switzerland. We have collected pressure, temperature and humidity profiles from nine flights. From the measured meteorological parameters, we have calculated the refractivity and zenith tropospheric delays (ZTDs). The parameters are compared with models calculated from 32 neighboring GNSS stations interpolated using the in-house developed least-squares collocation software COMEDIE (COllocation of MEteorological Data for Interpretation and Estimation of tropospheric path delays). The comparisons between the Meteodrones and the GNSS-derived models show a good agreement with an average absolute bias of 2.4 mm with 1.5 mm standard deviation for ZTD and 2.7 ppm with 1.7 ppm standard deviation for the comparison of the refractivity. The bias is a result of local changes of meteorological conditions captured by the drone, but not by the interpolation model.

Characterization of forests is an important aspect for the management and conservation of biodiversity and satellite remote sensing provides opportunity to rapidly assess biodiversity patterns across large geographical areas. The present study aimed to characterize vegetation in a landscape with heterogenous topography using medium resolution Sentinel-2 satellite data and random forest (RF) machine learning algorithm. The study area falls in the Nandhaur landscape of western Himalayan foothills and forms an important part of the Terai Arc Landscape. Sentinel-2 satellite data of January, April and May months was used along with SRTM Digital Elevation Model. Primary field data was collected for dominant vegetation community characteristics and used for training and validation of the model in RF. A total of eleven forest vegetation communities was characterized with an accuracy of 70–87% for individual imageries as well as in collocation of images. Satellite image of April was found most useful for discriminating forest communities with high spectral separability among communities. Variable important analysis revealed Vegetation Red Edge, Narrow NIR and NIR bands were useful for discriminating communities with maximum information along with SWIR and Red bands. Study found that collocation of the multi-date images was useful for classifying communities with higher accuracies. This attempt at characterizing heterogeneous vegetation communities with advanced machine learning tools in Himalayan foothills showed encouraging outcomes which can be used in future studies for monitoring and management of key habitats in this area.

In this paper, we concern with the data interpolation by using extended cubic uniform B-splines with shape parameters. Two iterative formats, namely the progressive iterative approximation (PIA) and the weighted progressive iterative approximation (WPIA), are proposed to interpolate given data points. We study the optimal shape parameter and the optimal weight for the proposed methods by solving the eigenvalues of the collocation matrix. The optimal shape parameter can make the iterative methods not only have the fastest convergence speed but also have smallest initial interpolation error. Numerical experiments are given to illustrate the effectiveness of the proposed methods.

In this paper, a numerical approach is proposed based on least squares support vector regression for solving Volterra integral equations of the first and second kind. The proposed method is based on using a hybrid of support vector regression with an orthogonal kernel and Galerkin and collocation spectral methods. An optimization problem is derived and transformed to solving a system of algebraic equations. The resulting system is discussed in terms of the structure of the involving matrices and the error propagation. Numerical results are presented to show the sparsity of resulting system as well as the efficiency of the method.

In this paper, on the basis of the reproducing kernel functions, a novel meshless algorithm is explored for fractional advection–diffusion-reaction equations (ADREs) with Caputo time variable order. Firstly, the Gaussian kernel function and the Mittag–Leffler kernel function are introduced and combined to construct a new binary reproducing kernel function. Secondly, base on the constructed reproducing kernel function, by employing the space–time collocation technique, a novel meshless method is proposed for fractional ADREs. Numerical experiments are implemented and the numerical results show the potential of our new approach for fractional ADREs.

In this paper, formulation of an efficient numerical schemes for the approximation first-order initial value problems (IVPs) of ordinary differential equations (ODE) is presented. The method is a block scheme for some k-step linear multi-step methods (and) using the Hermite Polynomials a basis function. The continuous and discrete linear multi-step methods (LMM) are formulated through the technique of collocation and interpolation. Numerical examples of ODE have been examined and results obtained show that the proposed scheme can be efficient in solving initial value problems of first order ODE.

We propose kernel-based collocation methods for numerical solutions to Heath–Jarrow–Morton models with Musiela parametrization. The methods can be seen as the Euler–Maruyama approximation of some f...

The Orbiting Carbon Observatory 2 (OCO-2) carries a hyperspectral A-band sensor that can obtain information about cloud geometric thickness (H). The OCO2CLD-LIDAR-AUX product retrieved H with the aid of collocated CALIPSO (Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation) lidar data to identify suitable clouds and provide a priori cloud top pressure (Ptop). This collocation is no longer possible, since CALIPSO's coordination flying with OCO-2 has ended, so here we introduce a new cloud flagging and a priori assignment using only OCO-2 data, restricted to ocean footprints where solar zenith angle <45∘. Firstly, a multi-layer perceptron network was trained to identify liquid clouds over the ocean with sufficient optical depth (τ>1) for a valid retrieval, and agreement with MODIS–CALIPSO (Moderate Resolution Imaging Spectroradiometer) is 90.0 %. Secondly, we developed a lookup table to simultaneously retrieve cloud τ, effective radius (re) and Ptop from A-band and CO2 band radiances, with the intention that these will act as the a priori state estimate in a future retrieval. Median Ptop difference vs. CALIPSO is 12 hPa with an inter-decile range of [-11,87]hPa, substantially better than the MODIS–CALIPSO range of [-83,81]hPa. The MODIS–OCO-2 τ difference is 0.8[-3.8,6.9], and re is -0.3[-2.8,2.1]µm. The τ difference is due to optically thick and horizontally heterogeneous cloud scenes. As well as an improved passive Ptop retrieval, this a priori information will allow for a purely OCO-2-based Bayesian retrieval of cloud droplet number concentration (Nd). Finally, our cloud flagging procedure may also be useful for future partial-column above-cloud CO2 abundance retrievals.

A steady-state diffusion model comprising of diffusion and reaction in a catalyst pellet was used to study and understand the effect of the approximation of the catalyst effectiveness factor in the modeling of trickle bed reactors. The hydrogenation of alpha-methylstyrene was chosen as the case study reaction from literature for liquid-limited reaction conditions. The effectiveness factor values in this study were sensitive to reactor scale equations during the prediction of reactor performance. The approximation of accounting for the overall catalyst effectiveness factor in the reactor scale model equations from different models and solving the reactor scale model equations by different modeling approaches was assessed. It was observed that evaluating the overall effectiveness factor from pellet scale model equations and integrating the parameter in the reactor scale axial dispersion model at every local axial collocation point to simulate the reactor performance showed better agreement to the experimental data. This approach evaluates the effectiveness factor locally with the variation of the reactant concentration at different axial points across the reactor rather than using one value for the entire reactor. The approach of using a single effectiveness factor for the whole bed and using the effectiveness factor approximated as a fitted polynomial to the reactant concentrations did not properly predict the reactor performance.

To quantify waveguide dispersion uncertainty, we propose a stochastic analysis technique based on the polynomial chaos method, where Smolyak sparse grids are used as sample spaces for the recently developed nonintrusive stochastic testing (ST) technique. This strategy allows for a smaller set of candidate collocation nodes in the ST algorithm compared with the computationally expensive tensor grid approach. The method is tailored to the analysis of waveguide structures with a statistically spatially varying dielectric-property profile, which is then captured by a discrete set of uncorrelated random variables by means of Karhunen–Loeve transforms. In this way, the propagation properties of realistic stochastic waveguides can be predicted, which is of major importance in the electronic design process. The presented method, which is constructed around a second-order full-wave deterministic finite element solver, is illustrated through the uncertainty quantification of two waveguide dispersion problems and is validated using commercial electromagnetic field solvers and the Monte Carlo method.

In this paper, a numerical algorithm combining the meshless collocation technique based on the globally supported Multiquadric Radial Basis Function (MQRBF) approximation method and the Asymptotic Numerical Method (ANM) is developed to investigate the nonlinear and linear static behaviors of Single-Walled Carbone Nanotubes (SWCNTs) based on a Nonlocal Continuum Beam Model (NCBM). The NCBM which accounts for the effects of small scale, shear deformation and geometric nonlinearity of von-Karman type is constructed by implementing the Nonlocal Elasticity Theory of Eringen (NET) into the First-Order Shear Deformation Elastic Beam Theory (FOSDEBT) and used to model the SWCNTs as a Continuous Elastic Nanobeam (CEN). Based on the Total Lagrangian Formulation (TLF) the nonlocal nonlinear governing equations of SWCNT are elaborated in a quadratic matrix strong form. These equations are then solved numerically by applying the proposed algorithm. This algorithm permits to transform the quadratic nonlocal nonlinear governing equations into a sequence of linear ones shared the same tangent stiffness matrix which are then solved numerically by the MQRBF approximation method, to get the whole branch of solution a continuation procedure is required. In order to highlight the reliability and efficiency of the present numerical algorithm, comparative studies are conducted.

The ensemble Kalman filter (EnKF) has received substantial attention in hydrologic data assimilation due to its ease of implementation. In EnKF, a large enough ensemble size is often required to ensure accuracy, which may result in considerable computational overhead, especially for large-scale problems. Motivated by recent developments in multi-fidelity simulation, we develop a novel data assimilation method that provides an alternative to EnKF, namely adaptive multi-fidelity probabilistic collocation-based Kalman filter (AMF-PCKF). The appealing feature is to approximate the system response with polynomial chaos expansion (PCE) using the adaptive multi-fidelity probabilistic collocation method, which improves the computational efficiency without sacrificing accuracy. This constitutes the forecast step of AMF-PCKF, while the analysis step is established by sequentially updating the PCE coefficients. As demonstrated by a synthetic numerical case of heat transport in unsaturated flow and a real-world two-phase flow experiment, AMF-PCKF can provide more accurate estimations than EnKF under the same amount of computation, even when the number of unknown parameters is as high as 100.

Triple collocation (TC) is a popular technique for determining the data quality of three products that estimate the same geophysical variable using mutually independent methods. When TC is applied to a triplet of one point-scale in situ and two coarse-scale datasets that have the similar spatial resolution, the TC-derived performance metric for the point-scale dataset can be used to assess its spatial representativeness. In this study, the spatial representativeness of in situ snow depth measurements from the meteorological stations in northeast China was assessed using an unbiased correlation metric estimated with TC. Stations are considered representative if ; that is, in situ measurements explain no less than 50% of the variations in the “ground truth” of the snow depth averaged at the coarse scale (0.25°). The results confirmed that TC can be used to reliably exploit existing sparse snow depth networks. The main findings are as follows. 1) Among all the 98 stations in the study region, 86 stations have valid values, of which 57 stations are representative for the entire snow season (October–December, January–April). 2) Seasonal variations in are large: 63 stations are representative during the snow accumulation period (December–February), whereas only 25 stations are representative during the snow ablation period (October–November, March–April). 3) The is positively correlated with mean snow depth, which largely determines the global decreasing trend in from north to south. After removing this trend, residuals in can be explained by heterogeneity features concerning elevation and conditional probability of snow presence near the stations.

Achieving the long-endurance of a solar-powered unmanned aerial vehicle is difficult under a limited amount of energy and specific meteorological environments. Therefore, it is essential to manage the energy used by a solar-powered UAV during a mission. In this paper, a framework for energy-based path planning for a solar-powered UAV is presented. In order to generate the optimal path in terms of energy in a specific meteorological environment, the energy model of a solar-powered UAV is considered with regard to wind field. Additionally, the optimal energy path is examined through the optimal control technique. The optimal path is then applied to some case studies regarding solar-powered UAVs, and evaluated in order to verify the feasibility of the proposed approach. The optimization results show that the resulting path is indeed the optimal path, which increases the endurance of a flight and takes weather conditions into accounts. The proposed approach is also expected to find optimal paths in accordance with real weather environments and/or various weather situations.

The spaceborne active sounders have been contributing invaluable vertically resolved information of atmospheric optical properties since the launch of CALIPSO (Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation) in 2006. To ensure the continuity of climate studies and monitoring the global changes, one has to understand the differences between lidars operating at different wavelengths, flying at different orbits, and utilizing different observation geometries, receiving paths, and detectors. In this article, we show the results of an intercomparison study of ALADIN (Atmospheric Laser Doppler INstrument) and CALIOP (Cloud-Aerosol Lidar with Orthogonal Polarization) lidars using their scattering ratio (SR) products for the period of 28/06/2019−31/12/2019. We suggest an optimal set of collocation criteria (Δdist&thinsp

Low-cost gas and particulate matter sensor packages offer a compact, lightweight, and easily transportable solution to address global gaps in air quality (AQ) observations. However, regions that would benefit most from widespread deployment of low-cost AQ monitors often lack the reference-grade equipment required to reliably calibrate and validate them. In this study, we explore approaches to calibrating and validating three integrated sensor packages before a 1-year deployment to rural Malawi using colocation data collected at a regulatory site in North Carolina, USA. We compare the performance of five computational modeling approaches to calibrate the electrochemical gas sensors: k-nearest neighbors (kNN) hybrid, random forest (RF) hybrid, high-dimensional model representation (HDMR), multilinear regression (MLR), and quadratic regression (QR). For the CO, Ox, NO, and NO2 sensors, we found that kNN hybrid models returned the highest coefficients of determination and lowest error metrics when validated. Hybrid models were also the most transferable approach when applied to deployment data collected in Malawi. We compared kNN hybrid calibrated CO observations from two regions in Malawi to remote sensing data and found qualitative agreement in spatial and annual trends. However, ARISense monthly mean surface observations were 2 to 4 times higher than the remote sensing data, partly due to proximity to residential biomass combustion activity not resolved by satellite imaging. We also compared the performance of the integrated Alphasense OPC-N2 optical particle counter to a filter-corrected nephelometer using colocation data collected at one of our deployment sites in Malawi. We found the performance of the OPC-N2 varied widely with environmental conditions, with the worst performance associated with high relative humidity (RH >70 %) conditions and influence from emissions from nearby residential biomass combustion. We did not find obvious evidence of systematic sensor performance decay after the 1-year deployment to Malawi. Data recovery (30 %–80 %) varied by sensor and season and was limited by insufficient power and access to resources at the remote deployment sites. Future low-cost sensor deployments to rural, low-income settings would benefit from adaptable power systems, standardized sensor calibration methodologies, and increased regional regulatory-grade monitoring infrastructure.

The Pilot-Mission Exploitation Platform (Pi-MEP) for salinity is an ESA initiative originally meant to support and widen the uptake of Soil Moisture and Ocean Salinity (SMOS) mission data over the ocean. Starting in 2017, the project aims at setting up a computational web-based platform focusing on satellite sea surface salinity data, supporting studies on enhanced validation and scientific process over the ocean. It has been designed in close collaboration with a dedicated science advisory group in order to achieve three main objectives: gathering all the data required to exploit satellite sea surface salinity data, systematically producing a wide range of metrics for comparing and monitoring sea surface salinity products’ quality, and providing user-friendly tools to explore, visualize and exploit both the collected products and the results of the automated analyses. The Salinity Pi-MEP is becoming a reference hub for the validation of satellite sea surface salinity missions by providing valuable information on satellite products (SMOS, Aquarius, SMAP), an extensive in situ database (e.g., Argo, thermosalinographs, moorings, drifters) and additional thematic datasets (precipitation, evaporation, currents, sea level anomalies, sea surface temperature, etc.). Co-localized databases between satellite products and in situ datasets are systematically generated together with validation analysis reports for 30 predefined regions. The data and reports are made fully accessible through the web interface of the platform. The datasets, validation metrics and tools (automatic, user-driven) of the platform are described in detail in this paper. Several dedicated scienctific case studies involving satellite SSS data are also systematically monitored by the platform, including major river plumes, mesoscale signatures in boundary currents, high latitudes, semi-enclosed seas, and the high-precipitation region of the eastern tropical Pacific. Since 2019, a partnership in the Salinity Pi-MEP project has been agreed between ESA and NASA to enlarge focus to encompass the entire set of satellite salinity sensors. The two agencies are now working together to widen the platform features on several technical aspects, such as triple-collocation software implementation, additional match-up collocation criteria and sustained exploitation of data from the SPURS campaigns.

Buoys provide key observations of wind speed over the ocean and are routinely used as a source of validation data for satellite wind products. However, the movement of buoys in high seas and the airflow over waves might cause inaccurate readings, raising concern when buoys are used as a source of wind speed comparison data. The relative accuracy of buoy winds is quantified through a triple collocation (TC) exercise comparing buoy winds to winds from ASCAT and ERA5. Differences between calibrated buoy winds and ASCAT are analyzed through separating the residuals by anemometer height and testing under high wind-wave and swell conditions. First, we converted buoy winds measured near 3, 4, and 5 m to stress-equivalent winds at 10 m (U10S). Buoy U10S from anemometers near 3 m compared notably lower than buoy U10S from anemometers near 4 and 5 m, illustrating the importance of buoy choice in comparisons with remote sensing data. Using TC calibration of buoy U10S to ASCAT in pure wind-wave conditions, we found that there was a small, but statistically significant difference between height adjusted buoy winds from buoys with 4 and 5 m anemometers compared to the same ASCAT wind speed ranges in high seas. However, this result does not follow conventional arguments for wave sheltering of buoy winds, whereby the lower anemometer height winds are distorted more than the higher anemometer height winds in high winds and high seas. We concluded that wave sheltering is not significantly affecting the winds from buoys between 4 and 5 m with high confidence for winds under 18 ms−1. Further differences between buoy U10S and ASCAT winds are observed in high swell conditions, motivating the need to consider the possible effects of sea state on ASCAT winds.

In this paper, a kind of localized radial basis function-based collocation method with reproducibility has been designed for nonlocal diffusion models. The basic idea of the method is to localize the RBF shape function by a corrected kernel with compact support, and meanwhile make the interpolation function to meet the reproducing conditions by modifying the coefficient contained in such kernel. Three types of nonlocal diffusion problems including constant and singular kernels are solved by our method in numerical experiments, which indicates that our method shows almost the same convergent behavior compared with RBF collocation methods, but it is much better conditioning and more time-efficient. It also overcomes the shortcoming that the RK-enhanced RBF method is not convergent for nonlocal diffusion models.