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We consider the extremal problem of interpolation of convex scattered data in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$ by smooth edge convex curve networks with minimal Lp$$ {L}_p $$‐norm of the second derivative for 1

This paper presents an approach to scattered data interpolation using q-Bézier triangular patches via an efficient algorithm. While existing studies have formed q-Bézier triangular patches through convex combination, their application to scattered data interpolation has not been previously explored. Therefore, this study aims to extend the use of q-Bézier triangular patches to scattered data interpolation by achieving C 1 continuity throughout the data points. We test the proposed scheme using both established data points and real-life engineering problems. We compared the performance of the proposed interpolation scheme with a well-known existing scheme by varying the q parameter. The comparison was based on visualization and error analysis. Numerical and graphical results were generated using MATLAB. The findings indicate that the proposed scheme outperforms the existing scheme, demonstrating a higher coefficient of determination ( R 2 ), smaller root mean square error (RMSE), and faster central processing unit (CPU) time. These results highlight the potential of the proposed q-Bézier triangular patches scheme for more accurate and reliable scattered data interpolation via the proposed algorithm. [ABSTRACT FROM AUTHOR]

Scattered data interpolation is important in sciences, engineering, and medical-based problems. The degree smoothness attained is either C1 or C2. However, based on our simulations, we found that, Hussain and Hussain (2011) scheme is not producing C1 everywhere. Motivated by this, in this study we will propose an improve sufficient condition for the C1 on each adjacent triangle. Our scheme is guaranteed to produce C1 everywhere. To support our claim, we have implemented both schemes and perform some numerical comparison including error measurement. From the results, our proposed scheme is producing C1 interpolating surface while the interpolating surface produced by Hussain and Hussain (2011) is not C1 everywhere. Furthermore, our proposed scheme give smaller error compared with Hussain and Hussain (2011) scheme. All numerical and graphical results are presented by using MATLAB programming.

Image quality degradation is one of the main problems in Digital Holography. A novel method for noise reduction based on the Hilbert-Huang transform is introduced that reduces the normalized contrast and partially removes the DC-term. It is successfully applied in the hologram plane and requires a single hologram to function. The method causes no reduction in spatial resolution. Moreover, it can be combined with existing noise-filtering methods to improve image quality even further. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new nonconforming finite element as a polynomial enrichment of the standard triangular linear element. Based on this new element, we propose an improvement of the triangular Shepard operator. We prove that the order of this new approximation operator is at least cubic. Numerical experiments demonstrate the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we use the knowledge of positive definite tensors to develop a concept of positive definite multi-kernels to construct the kernel-based interpolants of scattered data. By the techniques of reproducing kernel Banach spaces, the optimal recoveries and error analysis of the kernel-based interpolants are shown for a special class of strictly positive definite multi-kernels. [ABSTRACT FROM AUTHOR]

Scattered data interpolation is frequently encountered problems for reconstructing an unknown function from given scattered data, and radial basis function (RBF) methods have proved to be highly efficient for such problems. Here we extend quantum algorithms to scattered data interpolation problems using Gaussian RBFs. Our algorithm builds upon coherent states to form the dense interpolation matrix, a nonsparse matrix exponentiation to perform matrix inversion, and a swap test to evaluate on a new given data. Compared with the theoretical performance of a standard classical method—the conjugate gradient method, our quantum algorithm achieves a quadratic improvement in the number of data. [ABSTRACT FROM AUTHOR]

Scattered data interpolation and approximation techniques allow for the reconstruction of a scalar field based upon a finite number of scattered samples of the field. In general, the fidelity of the reconstruction with respect to the original scalar field tends to deteriorate as the number of samples decreases. For the situation of very sparse sampling, the results may not be acceptable at all. However, if it is known that the scalar field of interest is correlated with a known flow field - as is the case when the scalar field represents the value of an oceanographic tracer that propagates under the influence of the ocean's flow - then this knowledge can be exploited to enhance the scattered data reconstruction method. One way to exploit flow field information is to use it to construct a modified notion of distance between points. Replacing the standard Euclidean distance metric with a flow-field-aware notion of distance provides a method for extending standard scattered data interpolation methods into flow-based methods that produce superior results for very sparse data. The resulting reconstructions typically have lower root-mean-square errors than reconstructions that do not use the flow information, and qualitatively they often appear physically more realistic.

Scattered data interpolation and approximation techniques allow for the reconstruction of a scalar field based upon a finite number of scattered samples of the field. In general, the fidelity of the reconstruction with respect to the original scalar field tends to deteriorate as the number of samples decreases. For the situation of very sparse sampling, the results may not be acceptable at all. However, if it is known that the scalar field of interest is correlated with a known flow field - as is the case when the scalar field represents the value of an oceanographic tracer that propagates under the influence of the ocean's flow - then this knowledge can be exploited to enhance the scattered data reconstruction method. One way to exploit flow field information is to use it to construct a modified notion of distance between points. Replacing the standard Euclidean distance metric with a flow-field-aware notion of distance provides a method for extending standard scattered data interpolation methods into flow-based methods that produce superior results for very sparse data. The resulting reconstructions typically have lower root-mean-square errors than reconstructions that do not use the flow information, and qualitatively they often appear physically more realistic.

In this paper, we consider the problem of learning prediction models for spatiotemporal physical processes driven by unknown partial differential equations (PDEs). We propose a deep learning framework that learns the underlying dynamics and predicts its evolution using sparsely distributed data sites. Deep learning has shown promising results in modeling physical dynamics in recent years. However, most of the existing deep learning methods for modeling physical dynamics either focus on solving known PDEs or require data in a dense grid when the governing PDEs are unknown. In contrast, our method focuses on learning prediction models for unknown PDE-driven dynamics only from sparsely observed data. The proposed method is spatial dimension-independent and geometrically flexible. We demonstrate our method in the forecasting task for the two-dimensional wave equation and the Burgers-Fisher equation in multiple geometries with different boundary conditions, and the ten-dimensional heat equation.

Abstract This paper discusses the functional scattered data interpolation to interpolate the general scattered data. Compared with the previous works, we construct a new cubic Bézier-like triangular basis function controlled by three shape parameters. This is an advantage compared with the existing schemes since it gives more flexibility for the shape design in geometric modeling. By choosing some suitable value of the parameters, this new triangular basis is reduced to the cubic Ball and cubic Bézier triangular patches, respectively. In order to apply the proposed bases to general scattered data, firstly the data is triangulated using Delaunay triangulation. Then the sufficient condition for C 1 $C^{1}$ continuity using cubic precision method on each adjacent triangle is implemented. Finally, the interpolation scheme is constructed based on a convex combination between three local schemes of the cubic Bézier-like triangular patches. The detail comparison in terms of maximum error and coefficient of determination r 2 $r^{2}$ with some existing meshfree methods i.e. radial basis function (RBF) such as linear, thin plate spline (TPS), Gaussian, and multiquadric are presented. From graphical results, the proposed scheme gives more visually pleasing interpolating surfaces compared with all RBF methods. Based on error analysis, for all four functions, the proposed scheme is better than RBFs except for data from the third function. Overall, the proposed scheme gives r 2 $r^{2}$ value between 0.99920443 and 0.99999994. This is very good for surface fitting for a large scattered data set.

This paper discusses scattered data interpolation using cubic trigonometric Bézier triangular patches with C¹ continuity everywhere. We derive the C¹ condition on each adjacent triangle. On each triangular patch, we employ convex combination method between three local schemes. The final interpolant with the rational corrected scheme is suitable for regular and irregular scattered data sets. We tested the proposed scheme with 36,65, and 100 data points for some well-known test functions. The scheme is also applied to interpolate the data for the electric potential. We compared the performance between our proposed method and existing scattered data interpolation schemes such as Powell-Sabin (PS) and Clough-Tocher (CT) by measuring the maximumerror, root mean square error (RMSE) and coefficient of determination (R²). From the results obtained, our proposed method is competent with cubic Bézier, cubic Ball, PS and CT triangles splitting schemes to interpolate scattered data surface. This is very significant since PS and CT requires that each triangle be splitting into several micro triangles. [ABSTRACT FROM AUTHOR]

In this article we present a new adaptive algorithm for solving 2D interpolation problems of large scattered data sets through the radial basis function partition of unity method. Unlike other time-consuming schemes this adaptive method is able to efficiently deal with scattered data points with highly varying density in the domain. This target is obtained by decomposing the underlying domain in subdomains of variable size so as to guarantee a suitable number of points within each of them. The localization of such points is done by means of an efficient search procedure that depends on a partition of the domain in square cells. For each subdomain the adaptive process identifies a predefined neighborhood consisting of one or more levels of neighboring cells, which allows us to quickly find all the subdomain points. The algorithm is further devised for an optimal selection of the local shape parameters associated with radial basis function interpolants via leave-one-out cross validation and maximum likelihood estimation techniques. Numerical experiments show good performance of this adaptive algorithm on some test examples with different data distributions. The efficacy of our interpolation scheme is also pointed out by solving real world applications. [ABSTRACT FROM AUTHOR]

We compare a recently proposed multivariate spline based on mixed partial derivatives with two other standard splines for the scattered data smoothing problem. The splines are defined as the minimiser of a penalised least squares functional. The penalties are based on partial differential operators, and are integrated using the finite element method. We compare three methods to two problems: to remove the mixture of Gaussian and impulsive noise from an image, and to recover a continuous function from a set of noisy observations. [ABSTRACT FROM AUTHOR]

Scattered data interpolation can be used to approximate a multivariate function by a linear combination of positive definite radial basis functions (RBFs). In practice, the approximation error stagnates (due to numerical instability) even if the function is smooth and the number of data centers is increased. A smaller approximation error can be obtained using multilevel interpolation on a sequence of nested subsets of the initial set of centers. For the construction of these nested subsets, we compare two thinning algorithms from the literature, a greedy algorithm based on nearest neighbor computations and a Poisson point process. The main novelty of our approach lies in the use of H -matrices both for the solution of linear systems and for the evaluation of residual errors at each level. For the solution of linear systems, we use GMRes combined with a domain decomposition preconditioner. Using H -matrices allows us to solve larger problems more efficiently compared with multilevel interpolation based on dense matrices. Numerical experiments with up to 50,000 scattered centers in two and three spatial dimensions demonstrate that the computational time required for the construction of the multilevel interpolant using H -matrices is of almost linear complexity with respect to the number of centers. [ABSTRACT FROM AUTHOR]

In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernel-based basis functions. The linear spaces spanned by these discontinuous kernels lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon in reconstructing functions with jumps. For the new basis we provide error bounds and numerical results that support our claims. The method is also effectively tested for approximating satellite images. [ABSTRACT FROM AUTHOR]

This paper discusses the functional scattered data interpolation to interpolate the general scattered data. Compared with the previous works, we construct a new cubic Bézier-like triangular basis function controlled by three shape parameters. This is an advantage compared with the existing schemes since it gives more flexibility for the shape design in geometric modeling. By choosing some suitable value of the parameters, this new triangular basis is reduced to the cubic Ball and cubic Bézier triangular patches, respectively. In order to apply the proposed bases to general scattered data, firstly the data is triangulated using Delaunay triangulation. Then the sufficient condition for C 1 continuity using cubic precision method on each adjacent triangle is implemented. Finally, the interpolation scheme is constructed based on a convex combination between three local schemes of the cubic Bézier-like triangular patches. The detail comparison in terms of maximum error and coefficient of determination r 2 with some existing meshfree methods i.e. radial basis function (RBF) such as linear, thin plate spline (TPS), Gaussian, and multiquadric are presented. From graphical results, the proposed scheme gives more visually pleasing interpolating surfaces compared with all RBF methods. Based on error analysis, for all four functions, the proposed scheme is better than RBFs except for data from the third function. Overall, the proposed scheme gives r 2 value between 0.99920443 and 0.99999994. This is very good for surface fitting for a large scattered data set. [ABSTRACT FROM AUTHOR]

The problem of Lagrange interpolation of functions of two variables by quadratic polynomials based on nodes which are vertices of a triangulation has been recently studied and local six-tuples of vertices which assure the uniqueness and the optimal-order of the interpolation polynomial are known. Following the idea of Little and the theoretical results on the approximation order and accuracy of the triangular Shepard method, we introduce an hexagonal Shepard operator with quadratic precision and cubic approximation order for the classical problem of scattered data approximation without least square fit. [ABSTRACT FROM AUTHOR]

In this paper we present a trivariate algorithm for fast computation of tetrahedral Shepard interpolants. Though the tetrahedral Shepard method achieves an approximation order better than classical Shepard formulas, it requires to detect suitable configurations of tetrahedra whose vertices are given by the set of data points. In doing that, we propose the use of a fast searching procedure based on the partitioning of domain and nodes in cubic blocks. This allows us to find the nearest neighbor points associated with each ball that need to be used in the 3D interpolation scheme. Numerical experiments show good performance of our interpolation algorithm. [ABSTRACT FROM AUTHOR]

Natural-neighbor interpolation methods, such as Sibson's method, are well-known schemes for multivariate data fitting and reconstruction. Despite its many desirable properties, Sibson's method is computationally expensive and difficult to implement, especially when applied to higher-dimensional data. The main reason for both problems is the method's implementation based on a Voronoi diagram of all data points. We describe a discrete approach to evaluating Sibson's interpolant on a regular grid, based solely on finding nearest neighbors and rendering and blending d-dimensional spheres. Our approach does not require us to construct an explicit Voronoi diagram, is easily implemented using commodity three-dimensional graphics hardware, leads to a significant speed increase compared to traditional approaches, and generalizes easily to higher dimensions. For large scattered data sets, we achieve two-dimensional (2D) interpolation at interactive rates and 3D interpolation (3D) with computation times of a few seconds.

Visualization of scattered data over a volumetric spatial domain is often done by reconstructing a trivariate function on some grid using scattered data interpolation methods and visualizing the function using standard visualization techniques. Scattered data reconstruction algorithms are often computationally expensive and difficult to implement. In order to visualize streaming scattered data, where visualization needs to take place in real time while new data is constantly streaming in, efficient approaches to scattered data reconstruction are required. We present a general framework for scattered data interpolation operating on discrete domains. Since common visualization methods require an underlying grid, it suffices to compute the scattered data reconstruction over the same grid. The key idea for speeding up the reconstruction is a re-factorization of the algorithm. The re-factorized version is designed such that it easily maps to graphics hardware architectures exploiting their performance and parallelism. Moreover, it naturally extends to applications for streaming data. As a proof of concept, we have implemented inverse-distance-weighted interpolation, natural neighbor interpolation, and radial Hermite interpolation using our general framework. We apply the framework to two kinds of streaming data: progressive scattered data and real-time sensor data with moving sensors delivering asynchronous measurements. To account for the scattered spatial and temporal distribution of streaming sensor data, we use a four-dimensional extension of our framework, which elegantly handles representation of time-varying data and leads to reconstructions that are smooth in both space and time.

Abstract In this paper, we present an efficient algorithm for the computation of triangular Shepard interpolation method. More precisely, it is well known that the triangular Shepard method reaches an approximation order better than the Shepard one (DellAccio et al., 2016), but it needs to identify useful general triangulation of the node set. Here we propose a searching technique used to detect and select the nearest neighbor points in the interpolation scheme (Cavoretto et al., 2016, 2017). It consists in determining the closest points belonging to the different neighborhoods and subsequently applies to the triangulation-based approach. Numerical experiments and some geological applications show efficiency and accuracy of the interpolation procedure. [ABSTRACT FROM AUTHOR]

Scattered data interpolation using conditionally positive definite radial basis functions typically leads to large, dense, and indefinite systems of saddle-point type. Due to ill-conditioning, the iterative solution of these systems requires an effective preconditioner. Using the technique of H-matrices, we propose, analyze, and compare two preconditioning approaches: transformation of the indefinite into a positive definite system using either Lagrangian augmentation or the nullspace method combined with subsequent H-Cholesky preconditioning. Numerical tests support the theoretical condition number estimates and illustrate the performance of the proposed preconditioners which are suitable for problems with up to N≈40000 centers in two or three spatial dimensions. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • Alignment residuals provide local estimates of the sample motion at the 3D fiducial positions. • Scattered data interpolation techniques can be applied to model the sample motion for the entire 3D space. • The motion model is then integrated in the tomographic reconstruction to yield motion-compensated tomograms. Abstract Recently, it has been shown that the resolution in cryo-tomography could be improved by considering the sample motion in tilt-series alignment and reconstruction, where a set of quadratic polynomials were used to model this motion. One requirement of this polynomial method is the optimization of a large number of parameters, which may limit its practical applicability. In this work, we propose an alternative method for modeling the sample motion. Starting from the standard fiducial-based tilt-series alignment, the method uses the alignment residual as local estimates of the sample motion at the 3D fiducial positions. Then, a scattered data interpolation technique characterized by its smoothness and a closed-form solution is applied to model the sample motion. The motion model is then integrated in the tomographic reconstruction. The new method improves the tomogram quality similar to the polynomial one, with the important advantage that the determination of the motion model is greatly simplified, thereby overcoming one of the major limitations of the polynomial model. Therefore, the new method is expected to make the beam-induced motion correction methodology more accessible to the cryoET community. [ABSTRACT FROM AUTHOR]

Abstract Scattered data interpolation using conditionally positive definite radial basis functions (RBFs) requires the solution of a symmetric saddle-point system. Based on an approximation of the system matrix as a hierarchical matrix, we solve the system iteratively using the GMRes algorithm and a domain decomposition preconditioner. The novelty of our work lies in the proposed solution of the subdomain problems using the nullspace method with an orthogonal basis represented as a sequence of Householder reflectors. The resulting positive definite subdomain systems are solved either directly or using an inner GMRes iteration with H -Cholesky preconditioning. Numerical tests demonstrate the effectiveness of this solution process for up to N = 160000 centers in two and three dimensions. [ABSTRACT FROM AUTHOR]

In this report, three methods of image-to-image registration using control points are evaluated. We assume that ephemeris sensor and platform data are unavailable. These techniques are the polynomial method, the piecewise linear transformation and the multiquadric method. The motivation for this research is the need for more accurate geometric correction of digital remote sensing data. This is especially important for airborne scanned imagery which is characterized by greater distortions than satellite data.The polynomial and piecewise linear methods were developed for use with satellite imagery and have remained popular due to their relative simplicity in theory and implementation. With respect to airborne data however, both of these methods have serious shortcomings. The polynomial method, a global model, is generally applied as a least-squares approximation to the control points. Mathematically it is unconstrained between points leading to undesirable excursions in the warp. The piecewise linear method (or finite element method), a local procedure, produces a faceted irregular warp when the distortions between the control points are highly nonlinear.The multiquadric method is a radial basis function. Two radial basis functions show promise for image warping: the multiquadric and thin plate spline. The multiquadric method is a global technique which captures local variations and interpolates, passing through the control points. It includes a tension-like parameter which can be used to adjust its behavior relative to local distortions. The principal shortcoming of the multiquadric method is that it is quite computationally intensive. Both the multiquadric method and thin plate splines have been evaluated extensively for scattered data interpolation.In a test application using badly warped aircraft imagery, the multiquadric method produced better results both visually, e.g. crooked lines were straightened, and quantitatively with lower residual errors. The results for the multiquadric method are encouraging for improved environmental remote sensing and geographic information systems integration. The technique may be applied to satellite data as well as to airborne scanner data. The multiquadric method may be used for warping polygons and applied to mosaicking as well. Its present functional form is flexible and may be modified quite easily to further adapt to local distortions, a task not performed for this report. Advances in the rapid evaluation of radial basis functions will make both the multiquadric and thin plate spline techniques even more attractive in the future.

The paper presents a new interactive technique for correcting CAD/CAM data containing errors. An algorithm is described that can be used to correct surface data containing undesirable discontinuities (''gaps''/ ''holes'', and ''overlaps'') and intersections among surface patches. Such surface problems commonly arise in the aircraft, automobile, and ship industry and make later processing of the data difficult, if not impossible. The new method provides a tool to correct wrong data requiring minimal user interaction. The imput for the scheme can be a set of parametrically defined surfaces (e.i., Bezier, B-spline, or NURBS surfaces) or a set of triangles (or quadrilaterals) discretizing the original geometry. The output is a set of G0 or G1 (tangent plane)continuous, bicubic B-splines surfaces approximating the given data. Each of these B-spline surfaces is constructed using just four user-specified boundary curves.

Some families of Haar spaces in ℝd,d≥1, whose basis functions are d-variate piecewise polynomials, are highlighted. The starting point is a sequence of univariate piecewise polynomials, called Lobachevsky splines, arised in probability theory and asymptotically related to the normal density function. Then, it is shown that d-variate Lobachevsky splines can be expressed as products of Lobachevsky splines. All these splines have simple analytic expressions and subsets of them are suitable for scattered data interpolation, allowing efficient computation and plain error analysis. [ABSTRACT FROM AUTHOR]

In this paper we discuss an improvement of the triangular Shepard operator proposed by Little to extend the Shepard method. In particular, we use triangle based basis functions in combination with a modified version of the linear local interpolant on the vertices of the triangle. We deeply study the resulting operator, which uses functional and derivative data, has cubic approximation order and a good accuracy of approximation. Suggestions on how to avoid the use of derivative data, without losing both order and accuracy of approximation, are given. [ABSTRACT FROM AUTHOR]

In this paper, firstly, based on new recursive algorithms of non-tensor-product-typed bivariate divided differences, scattered data interpolation schemes are constructed in the cases of odd and even interpolating nodes, respectively. Moreover, the corresponding error estimation is worked out, and equivalent formulae are obtained between bivariate high-order non-tensor-product-typed divided differences and high-order partial derivatives. Furthermore, the operation count for the addition/subtractions, multiplication, and divisions approximates O ( n 2 ) in the computation of the interpolating polynomials presented, while the operation count approximates O ( n 3 ) in the case of radial basis functions for sufficiently large n . Finally, several numerical examples show that it is valid for the recursive interpolating polynomial schemes, and these interpolating polynomials change as the order of the interpolating nodes, although the node collection is the same. [ABSTRACT FROM AUTHOR]

This paper discusses scattered data interpolation by using cubic Timmer triangular patches. In order to achieve C1 continuity everywhere, we impose a rational corrected scheme that results from convex combination between three local schemes. The final interpolant has the form quintic numerator and quadratic denominator. We test the scheme by considering the established dataset as well as visualizing the rainfall data and digital elevation in Malaysia. We compare the performance between the proposed scheme and some well-known schemes. Numerical and graphical results are presented by using Mathematica and MATLAB. From all numerical results, the proposed scheme is better in terms of smaller root mean square error (RMSE) and higher coefficient of determination (R2). The higher R2 value indicates that the proposed scheme can reconstruct the surface with excellent fit that is in line with the standard set by Renka and Brown’s validation.

We present a parallel algorithm for multivariate Radial Basis Function Partition of Unity Method (RBF-PUM) interpolation. The concurrent nature of the RBF-PUM enables designing parallel algorithms for dealing with a large number of scattered data-points in high space dimensions. To efficiently exploit this concurrency, our algorithm makes use of shared-memory parallel processors through the OpenCL standard. This efficiency is achieved by a parallel space partitioning strategy with linear computational time complexity with respect to the input and evaluation points. The speed of our approach allows for computationally more intensive construction of the interpolant. In fact, the RBF-PUM can be coupled with a cross-validation technique that searches for optimal values of the shape parameters associated with each local RBF interpolant, thus reducing the global interpolation error. The numerical experiments support our claims by illustrating the interpolation errors and the running times of our algorithm. [ABSTRACT FROM AUTHOR]

The partition of unity (PU) method, performed with local radial basis function (RBF) approximants, has been proved to be an effective tool for solving large scattered data interpolation problems. However, in order to achieve a good accuracy, the question about how many points we have to consider on each local subdomain, i.e. how large can be the local data sets, needs to be answered. Moreover, it is well-known that also the shape parameter affects the accuracy of the local RBF approximants and, as a consequence, of the PU interpolant. Thus here, both the shape parameter used to fit the local problems and the size of the associated linear systems are supposed to vary among the subdomains. They are selected by minimizing an a priori error estimate. As evident from extensive numerical experiments and applications provided in the paper, the proposed method turns out to be extremely accurate also when data with non-homogeneous density are considered. [ABSTRACT FROM AUTHOR]

Scattered data interpolation by radial kernel functions leads to linear equation systems with large, fully populated, ill-conditioned interpolation matrices. A successful iterative solution of such a system requires an efficient matrix-vector multiplication as well as an efficient preconditioner. While multipole approaches provide a fast matrix-vector multiplication, they avoid the explicit setup of the system matrix which hinders the construction of preconditioners, such as approximate inverses or factorizations which typically require the explicit system matrix for their construction. In this paper, we propose an approach that allows both an efficient matrix-vector multiplication as well as an explicit matrix representation which can then be used to construct a preconditioner. In particular, the interpolation matrix will be represented in hierarchical matrix format, and several approaches for the blockwise low-rank approximation are proposed and compared, of both analytical nature (separable expansions) and algebraic nature (adaptive cross approximation). The validity of using an approximate system matrix in the iterative solution of the interpolation equations is demonstrated through a range of numerical experiments. [ABSTRACT FROM AUTHOR]

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This paper introduces techniques for the estimation of solutions to fractional order differential equations (FODEs) and the approximation of a function's Caputo fractional derivative. These techniques are based on scattered data interpolation via reproducing kernel Hilbert spaces (RKHSs). Specifically, an RKHS is generated for the purpose of estimating fractional derivatives from the Mittag-Leffler function. The RKHS, called the Mittag-Leffler RKHS, as well as others are utilized to estimate Caputo fractional derivatives and to introduce a modified Adams-Bashforth- Moulton method for the estimation of the solution to FODEs. [ABSTRACT FROM AUTHOR]

Meshfree methods are often tested in literature for the rectangular domain with uniform aquifer properties (e.g. transmissivity or hydraulic conductivity) and constant groundwater head boundary conditions. In this paper, a multiquadric meshfree (Mfree) groundwater model is developed for a real field irregular domain aquifer system which is capable of incorporating variability like heterogeneity, temporally and spatially varied groundwater head and flow boundary conditions. The developed model is free from the inadequacy of meshing which eventually saves the time on computing resources by calculating the groundwater head-derivatives at different nodes sprinkled over the entire flow domain. Initially, two synthetic problems with pre- existing analytical and grid based results are tested against the presented model. Sensitivity analysis of different parameters like time step size, nodal density and support size of basis function are also investigated. Subsequently Mfree solutions are obtained for a large Mahi Right Bank Canal (MRBC) unconfined aquifer located in Gujarat India. The Mfree solution performance on two synthetic and one real field large aquifer problem revealed that the projected method is advantageous over other grid based simulator in terms of computational time consideration. Such a simulation is preferable in groundwater management models where flow models are coupled with the optimization problem. [ABSTRACT FROM AUTHOR]

In this paper we propose a new stable and accurate approximation technique which is extremely effective for interpolating large scattered data sets. The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, the approach consists in computing, for each PU subdomain, a stable basis. Such technique, taking advantage of the local scheme, leads to a significant benefit in terms of stability, especially for flat kernels. Furthermore, an optimized searching procedure is applied to build the local stable bases, thus rendering the method more efficient. [ABSTRACT FROM AUTHOR]

With the recent development of CubeSats, several ultracompact, low cost, and rapidly deployable satellites have been developed for earth observation missions. Because of the geometry of the acquisition process, measurements are irregularly sampled, whereas in meteorological applications, data are preferred on a regular grid. This problem is further complicated by the fact that, due to CubeSats’ compact sizes and constraints, such as limited power, errors occur in geolocation calibration, resulting in positional errors. In this letter, we analyze how the commonly used triangulation-based linear data interpolation scheme behaves under probabilistic models for the positional errors. The derived distribution of interpolation error caused by positional error is intractable even under a Gaussian distribution for positional errors. To address this problem, we developed an analytical closed-form solution to the first two moments of the interpolation error. Using models for positional errors motivated by our prior work, experimental results show that, compared with the first-order linear model, the second-order one provides a better approximation in terms of the mean and variance, which is very close to that is obtained using more computationally intensive Monte Carlo simulations. This model also allows for the closed-form calculation of mean squared interpolation error, which can be of use in the context of system design where the impact of positional errors on remote sensing products must be considered. [ABSTRACT FROM PUBLISHER]