Global Interpolation Research Articles

Incorporating uncertainty of groundwater modeling in sea-level rise assessment: a case study in South Florida

Researchers can assist in effective decision making by reducing uncertainty in marine and groundwater inundation due to Sea-level Rise (SLR). A majority of studies consider marine inundation, but only recently has groundwater inundation been incorporated into the SLR mapping. However, the effect of including uncertainty in groundwater modeling is still not well understood. In this study, we evaluate the effect of considering groundwater modeling uncertainty in assessing land area inundated vulnerable to marine and groundwater inundation in South Florida. Six Water Table Elevation Model (WTEM) techniques (Multiple Linear Regression (MLR), Geographic Weighted Regression (GWR), Global Polynomial Interpolation (GPI), Inverse Distance Weighted (IDW), Ordinary Kriging (OK), and Empirical Bayesian Kriging (EBK)) are tested to identify the best approach. Simple inundation models excluding uncertainty with and without WTEM are examined. Refined inundation models using Monte Carlo simulation that include uncertainty in future SLR estimates, LiDAR elevation, vertical datums and the transformations made between datums with and without WTEM uncertainty are evaluated. GPI and EBK are recognized as the best for producing WTEMs in two primary physiographic regions (the Southern Slope and Atlantic Coastal Ridge). Excluding uncertainty without WTEM underestimates total land area by 14 %, while including uncertainty without WTEM overestimates total vulnerable land area by 16 % at the 95 % probability threshold. It is significant to include WTEM uncertainty in SLR vulnerability analysis for more effective adaptation decisions.

Subdomain Chebyshev Spectral Method for 2D and 3D Numerical Differentiations in a Curved Coordinate System

A new numerical approach, called the “subdomain Chebyshev spectral method” is presented for calculation of the spatial derivatives in a curved coordinate system, which may be employed for numerical solutions of partial differential equations defined in a 2D or 3D geological model. The new approach refers to a “strong version” against the “weak version” of the subspace spectral method based on the variational principle or Galerkin’s weighting scheme. We incorporate local nonlinear transformations and global spline interpolations in a curved coordinate system and make the discrete grid exactly matches geometry of the model so that it is achieved to convert the global domain into subdomains and apply Chebyshev points to locally sampling physical quantities and globally computing the spatial derivatives. This new approach not only remains exponential convergence of the standard spectral method in subdomains, but also yields a sparse assembled matrix when applied for the global domain simulations. We conducted 2D and 3D synthetic experiments and compared accuracies of the numerical differentiations with traditional finite difference approaches. The results show that as the points of differentiation vector are larger than five, the subdomain Chebyshev spectral method significantly improve the accuracies of the finite difference approaches.

A Trivariate Interpolation Algorithm Using a Cube-Partition Searching Procedure

In this paper we propose a fast algorithm for trivariate interpolation, which is based on the partition of unity method for constructing a global interpolant by blending local radial basis function interpolants and using locally supported weight functions. The partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cube-partition searching procedure. More precisely, we construct a cube structure, which partitions the domain and strictly depends on the size of its subdomains, so that the new searching procedure and, accordingly, the resulting algorithm enable us to efficiently deal with a large number of nodes. Complexity analysis and numerical experiments show high efficiency and accuracy of the proposed interpolation algorithm.

Surface Pressure Tide Climatologies Deduced from a Quality-Controlled Network of Barometric Observations*

Abstract Global “ground truth” knowledge of solar diurnal S1 and semidiurnal S2 surface pressure tides as furnished by barometric in situ observations represents a valuable standard for wide-ranging geophysical and meteorological applications. This study attempts to aid validations of the air pressure tide signature in current climate or atmospheric analysis models by developing a new global assembly of nearly 6900 mean annual S1 and S2 estimates on the basis of station and marine barometric reports from the International Surface Pressure Databank, version 2 (ISPDv2), for a principal time span of 1990–2010. Previously published tidal compilations have been limited by inadequate spatial coverage or by internal inconsistencies and outliers from suspect tidal analyses; here, these problems are mostly overcome through 1) automated data filtering under ISPDv2’s quality-control framework and 2) a meticulously conducted visual inspection of station harmonic decompositions. The quality of the resulting compilation is sufficient to support global interpolation onto a reasonably fine mesh of 1° horizontal spacing. A multiquadric interpolation algorithm, with parameters fine-tuned by frequency and for land or ocean regions, is employed. Global charts of the gridded surface pressure climatologies are presented, and these are mapped to a wavenumber versus latitude spectrum for comparison with long-term means of S1 and S2 from four present-day atmospheric analysis systems. This cross verification, shown to be feasible even for the minor stationary modes of the tides, reveals a small but probably significant overestimation of up to 18% for peak semidiurnal amplitudes as predicted by global analysis models.

Global–local nonlinear model reduction for flows in heterogeneous porous media

Abstract In this paper, we combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equations, we employ the GMsFEM to represent the solution on a coarse grid with multiscale basis functions and apply proper orthogonal decomposition on a coarse grid. Computing the GMsFEM solution involves calculating the residual and the Jacobian on a fine grid. As such, we use local and global empirical interpolation concepts to circumvent performing these computations on the fine grid. The resulting reduced-order approach significantly reduces the flow problem size while accurately capturing the behavior of fully-resolved solutions. We consider several numerical examples of nonlinear multiscale partial differential equations that are numerically integrated using fully-implicit time marching schemes to demonstrate the capability of the proposed model reduction approach to speed up simulations of nonlinear flows in high-contrast porous media.

Pseudo-spectral methods in one-dimensional magnetostriction

In this paper a pseudo-spectral method is proposed to solve a one-dimensional model of a saturated hard ferromagnetic thin-film structure within the Euler–Bernoulli kinematics. The model accounts for the non-local nature of the magneto-elastic coupling and interaction is in the form of a logarithmic potential. The proposed solution method adopts global polynomial interpolation at a main grid, given by the Gauss–Lobatto points, and it employs a secondary grid, consisting of the Gauss points, to perform the Gaussian quadrature. The two grids are non-overlapping to avoid the singularity. Interpolation relates the unknowns, evaluated at the secondary grid, to their values at the collocation grid. Furthermore, the integration interval is parted about the singularity point. The procedure is assessed through the relative equilibrium residual for different values of the approximating polynomial degree and of the quadrature order. Maximum, average and standard deviation of the error are presented. An asymptotic analysis yields the Boundary Solution to the problem and results are compared when the latter is introduced in the numerical scheme. It is shown that its contribution is important in reducing the overall error. The equilibrium residual is plotted and its behavior discussed. It is further shown that numerical precision significantly affects the results at midspan, owing to the self-equilibrium of the system, thereby a limit exists to the best accuracy which may be gained through a more accurate interpolation.

Determining suitable model for zoning drinking water distribution network based on corrosion potential in Sanandaj City, Iran

Corrosion in general is a complex interaction between water and metal surfaces and materials in which the water is stored or transported. Water quality monitoring in terms of corrosion and scaling is crucial, and a key element of preventive maintenance, given the economic and health hurts caused by corrosion and scaling in water utilities. The aim of this study is to determine the best model for zoning and interpolation corrosive potential of water distribution networks. For this purpose, 61 points of Sanandaj, Iran, distribution network were sampled and using Langelier indices, we investigated corrosivity potential of drinking water. Then, we used geostatistical methods such as ordinary kriging (OK), global polynomial interpolation, local polynomial interpolation, radius-based function, and inverse distance weighted for interpolation, zoning and quality mapping. Variogram analysis of variables was performed to select appropriate models. The results of the calculation of the Langelier index represented scaling potential of drinking water. Suitable model for fitness on exponential variogram was selected based on less (residual sums of squares) and high (R 2 ) value. Moreover, the best method for interpolation was selected using the

Evaluation of Geographic Information Systems-Based Spatial Interpolation Methods Using Ohio Indoor Radon Data

This paper evaluates the performance of six different Geographic Information System based interpolation methods: inverse distance weighting (IDW), radial basis function (RBF), global polynomial interpolation, local polyno- mial interpolation, kriging, and cokriging, using the Ohio homes database developed between 1987 and 2011. The best performing interpolation method to be used in the prediction of radon gas concentrations in the unmeasured areas of Ohio, USA was determined by validating the model predictions with operational performance measures. Additionally, this study performed a zip code level-based analysis that provided a complete picture of the radon gas concentration distribution in Ohio. The RBF method was identified to be the best performing method. While the RBF method performed significantly better than the IDW, it was statistically similar to the other interpolation methods. The RBF predicted radon gas concentration results indicated a significant increase in the number of zip codes that exceeded the United States Environmental Protec- tion Agency and the World Health Organization action limits, thereby, indicating the need to mitigate the Ohio radon gas concentrations to safe levels in order to reduce the health effects. The approach demonstrated in this paper can be applied to other radon-affected areas around the world.

Preparing the anatomical model for ablation of unresectable liver tumor.

IntroductionNowadays the best treatment of the primary and secondary hepatic tumor is surgical resection, but only 5–15% of all patient with hepatocellular carcinoma and 20–25% of all patients with liver metastases are indicated for resection. In these cases some kind of ablation and other technique could be used.AimTo present the methodology of preparing the anatomical model for ablation of unresectable liver tumor.Material and methodsThe presented method is based on abdomen computed tomography (CT) dynamic examination. Three methods of segmentation are used: rolling vector for liver volume, modified Frangi filter for liver vessels, and fuzzy expert system with initial region-of-interest anisotropic filtration for liver metastases. Segmentation results are the input data for creating 3D anatomical models in the form of B-spline curves and surfaces performing the surface global interpolation algorithm. A graphical user interface for presentation and evaluation of models, presented in color against DICOM images in grayscale, is designed and implemented.ResultsThe proposed approach was tested on 20 abdominal CT obtained from the Department of Clinical Radiology of Silesian Medical University. The lack of a “gold standard” provides for the correction of the results.ConclusionsPreparation of the anatomical model is one of the important early stages of the use of image-based navigation systems. This process could not take place in a fully automatic manner and verification of the results obtained is performed by the radiologist. Using the above anatomical model in surgical workflow is presented.

Creating an advanced backpropagation neural network toolbox within GIS software

An artificial neural network (ANN) toolbox is created within GIS software for spatial interpolation, which will help GIS users to train and test ANNs, perform spatial analysis, and display results as a single process. The performance is compared to that of the open source Fast Artificial Neural Network library and conventional interpolation methods by creating digital elevation models (DEMs) given that nearly exact solutions exist. Simulation results show that the advanced backpropagations such as iRprop speed up the learning, while they can get stuck in a local minimum depending on initial weight sets. Besides, the division of input–output examples into training and test data affects the accuracy, particularly when the distribution of the examples is skewed and peaked, and the number of data is small. ANNs, however, show the similar performance to inversed distance weighted or kriging and outperform polynomial interpolations as a global interpolation method in high-dimensional data. In addition, the neural network residual kriging (NNRK) model, which combines the ANN toolbox and kriging within GIS software, is performed. The NNRK outperforms conventional methods and well captures global trends and local variations. A key outcome of this work is that the ANN toolbox created within the de facto standard GIS software is applicable to various spatial analysis including hazard risk assessment over a large area, in particular when there are multiple potential causes, the relationship between risk factors and hazard events is not clear, and the number of available data is small given its performance for DEM generation.

Comparison of interpolation methods for estimating spatial distribution of precipitation in Ontario, Canada

ABSTRACTIn this study, different interpolation techniques in a geographical information system (GIS) environment are analysed and compared for estimating the spatial distribution of precipitation in the province of Ontario, Canada. A high‐resolution regional climate modelling system [Providing Regional Climates for Impacts Studies (PRECIS)] is used to simulate the present (1961–1990) and future (2071–2100) precipitation events for 12 meteorological stations over Ontario. The results verify that for the present case PRECIS simulates well the precipitation events when compared with observed data. The future precipitation events can be projected after the validation of PRECIS. Six interpolation methods are then used to generate spatial distribution of precipitation based on the projections of future precipitation of 12 meteorological stations; they include inverse distance weighting (IDW), global polynomial interpolation (GPI), local polynomial interpolation (LPI), radial basis functions (RBF), ordinary kriging (OK), and universal kriging (UK). Cross‐validation is applied to evaluate the accuracy of interpolation methods in terms of the root mean square error (RMSE). The results indicate that LPI is the optimal method with the least RMSE for interpolating the PRECIS precipitation. LPI is then used to analyse spatial variations of the average annual precipitation for the period of 2071–2100 over Ontario.

A meshless interpolation algorithm using a cell-based searching procedure

In this paper we propose a fast algorithm for bivariate interpolation of large scattered data sets. It is based on the partition of unity method for constructing a global interpolant by blending radial basis functions as local approximants and using locally supported weight functions. The partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cell-based searching procedure. More precisely, we construct a cell structure, which partitions the domain and strictly depends on the dimension of the subdomains, thus providing a meaningful improvement in the searching process compared to the nearest neighbour searching techniques presented in Allasia et al. (2011) and Cavoretto and De Rossi (2010, 2012). In fact, this efficient algorithm and, in particular, the new searching procedure enable us a fast computation also in several applications, where the amount of data to be interpolated is often very large, up to many thousands or even millions of points. Analysis of computational complexity shows the high efficiency of the proposed interpolation algorithm. This is also supported by numerical experiments.

A Meshfree Quasi-Interpolation Method for Solving Burgers’ Equation

The main aim of this work is to consider a meshfree algorithm for solving Burgers’ equation with the quartic B-spline quasi-interpolation. Quasi-interpolation is very useful in the study of approximation theory and its applications, since it can yield solutions directly without the need to solve any linear system of equations and overcome the ill-conditioning problem resulting from using the B-spline as a global interpolant. The numerical scheme is presented, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. Compared to other numerical methods, the main advantages of our scheme are higher accuracy and lower computational complexity. Meanwhile, the algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.

Optimal Domain Splitting for Interpolation by Chebyshev Polynomials

Polynomial interpolants defined using Chebyshev extreme points as nodes converge uniformly at a geometric rate when sampling a function that is analytic on an interval. However, the convergence rate can be arbitrarily close to unity if the function has a singularity close to the interval when extended to the complex plane. In such cases, splitting the interval and doing piecewise interpolation may be more efficient in the total number of nodes than the global interpolant. Because the convergence rate is determined by Bernstein ellipses obtained through a Joukowski conformal map, relative efficiency of splitting at any point in the interval can be calculated and then optimized over the interval. The optimal splitting may be applied recursively. The Chebfun software project uses a simple rule of thumb without prior singularity information to create a binary search that can be shown to do an excellent job of finding the optimal splitting in most cases. However, the process can use a large number of intermediate function evaluations that are not needed in the final approximant. A Chebyshev--Padé technique generates approximate singularity locations that are good enough to get close to the optimal splitting more directly in test cases. The technique is applied to a singularly perturbed boundary-value problem.

A Review on Attempts towards CAD/CAE Integration Using Macroelements

In this paper we review several aspects of older and contemporary attempts to integrate computer-aided design (CAD: geometric model) and computer-aided engineering (CAE: finite elements, boundary elements, etc.). After a short review on formulas for the description of CAD surfaces, a systematic mechanism for creating several types of corresponding isoparametric macroelements is presented. Gordon-Coons is initially applied in conjunction with piecewise linear, Lagrange polynomials and natural B-splines. Then, it is extended to more basis and blending functions. In addition to the well-known 'Lagrangian'-type elements, equivalent 'Bezierian'-type elements are introduced. Tensor product B-splines and aspects of NURBS isogeometric formulation are given. In addition to quadrilaterals, triangular macroelements based on Barnhill's interpolation are presented for the first time. The review covers applications of CAD-based macroelements in conjunction with the Galerkin-Ritz formulation, the Boundary Element Method, as well as recent Global Collocation procedures. A numerical example on a vibrating membrane elucidates the performance of the CAD-based global interpolation and depicts its superiority over the conventional finite element method.

Surrogate modeling for the fast optimization of energy systems

While software simulations in the building technology field are essential for designing efficient systems, for specific applications, the tools currently available are computationally too expensive. This paper presents a novel approach for the fast computation of accurate simulation results and demonstrates its efficacy for solving a multiobjective optimization problem.A surrogate modeling approach is proposed, which is based on an effective interpolation algorithm and an optimal distribution of sampling nodes. The highest accuracy is shown to be achieved by a Poisson disk node distribution and an interpolation function developed from a cubic radial basis function. Furthermore, since global interpolation algorithms in high-dimensional spaces are fundamentally inefficient, a local adaptation is developed. The fidelity of the surrogate model is shown to generalize to higher dimensions with respect to the average number of nodes per dimension ηs.The application potential is tested on a case study consisting of a multiobjective design optimization problem. The results show that the surrogate and the fine model converge to the same solution. Being 150 times faster than the original model, the surrogate model is thus a valid choice for computationally demanding applications.

Bi-material V-notch stress intensity factors by the fractal-like finite element method

The fractal-like finite element method (FFEM) is developed to provide stress intensity factor (SIF) values for bi-material notches. The displacement fields around a bi-material notch tip are derived and employed as global interpolation functions in the FFEM to transform the large number of nodal displacements in the singular region to a small set of generalised co-ordinates leading to direct computation of the SIFs and the coefficients of the higher order terms. Various numerical examples for bi-material crack and notch cases are presented. New results for bi-material notches are reported.

A piecewiseP2-nonconforming quadrilateral finite element

We introduce a piecewise P 2 -nonconforming quadrilateral finite element. First, we decompose a convex quadrilateral into the union of four triangles divided by its diagonals. Then the finite element space is defined by the set of all piecewise P 2 -polynomials that are quadratic in each triangle and continuously differentiable on the quadrilateral. The degrees of freedom (DOFs) are defined by the eight values at the two Gauss points on each of the four edges plus the value at the intersection of the diagonals. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are eight. Global basis functions are defined in three types: vertex-wise, edge-wise, and element-wise types. The corresponding dimensions are counted for both Dirichlet and Neumann types of elliptic problems. For second-order elliptic problems and the Stokes problem, the local and global interpolation operators are defined. Also error estimates of optimal order are given in both broken energy and L 2 (Ω ) norms. The proposed element is also suitable to solve Stokes equations. The element is applied to approximate each component of velocity fields while the discontinuous P 1 -nonconforming quadrilateral element is adopted to approximate the pressure. An optimal error estimate in energy norm is derived. Numerical results are shown to confirm the optimality of the presented piecewise P 2 -nonconforming element on quadrilaterals.

A Global Interpolator With Low Sample Rate for Multilevel Fast Multipole Algorithm

A new, improved version of a global interpolator utilizing trigonometric polynomials is presented for the high-frequency multilevel fast multipole algorithm. The number of required points to sample the outgoing and incoming field patterns is low, almost half in some levels, compared with the earlier published versions. Compared with local interpolators based on Lagrange interpolating polynomials, the proposed technique performs even more favorably and reduces the number of sample points by a factor of eight. The numerical examples demonstrate that the interpolator allows full numerical accuracy control during the aggregation and disaggregation phases, regardless of the number of the levels in the octree.

Central West Antarctica among the most rapidly warming regions on Earth

There is clear evidence that the West Antarctic Ice Sheet is contributing to sea-level rise. In contrast, West Antarctic temperature changes in recent decades remain uncertain. West Antarctica has probably warmed since the 1950s, but there is disagreement regarding the magnitude, seasonality and spatial extent of this warming. This is primarily because long-term near-surface temperature observations are restricted to Byrd Station in central West Antarctica, a data set with substantial gaps. Here, we present a complete temperature record for Byrd Station, in which observations have been corrected, and gaps have been filled using global reanalysis data and spatial interpolation. The record reveals a linear increase in annual temperature between 1958 and 2010 by 2.4±1.2 °C, establishing central West Antarctica as one of the fastest-warming regions globally. We confirm previous reports of West Antarctic warming, in annual average and in austral spring and winter, but find substantially larger temperature increases. In contrast to previous studies, we report statistically significant warming during austral summer, particularly in December–January, the peak of the melting season. A continued rise in summer temperatures could lead to more frequent and extensive episodes of surface melting of the West Antarctic Ice Sheet. These results argue for a robust long-term meteorological observation network in the region. The West Antarctic Ice Sheet is contributing to sea-level rise, but temperature trends in the region have remained uncertain. A complete temperature record for Byrd Station in central West Antarctica, spanning from 1958 to 2010, establishes West Antarctica as one of the fastest-warming regions globally.