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In this paper, a kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator is studied by combining the known multiquadric quasi-interpolation operator with the generalized Taylor polynomial as the expansion in the bivariate Bernoulli polynomials. Some error bounds and convergence rates of the combined operators are studied. A selection of numerical examples is presented to compare the performances of the obtained scheme. Furthermore, our method can be applied to time-dependent differential equations. Its advantage is that the algorithm is very simple and easy to implement. [ABSTRACT FROM AUTHOR]

We propose AAA rational approximation as a method for interpolating or approximating smooth functions from equispaced samples. Although it is always better to approximate from large numbers of samples if they are available, whether equispaced or not, this method often performs impressively even when the sampling grid is coarse. In most cases it gives more accurate approximations than other methods. We support this claim with a review and discussion of nine classes of existing methods in the light of general properties of approximation theory as well as the “impossibility theorem” for equispaced approximation. We make careful use of numerical experiments, which are summarized in a sequence of nine figures. Among our new contributions is the observation, summarized in Fig. 7, that methods such as polynomial least-squares and Fourier extension may be either exponentially accurate and exponentially unstable, or less accurate and stable, depending on implementation. [ABSTRACT FROM AUTHOR]

We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees of freedom. Importantly, this approach is compatible with discrete exterior calculus and expands the framework to cover higher order methods on cubical meshes, complementing the earlier strategy based on simplices. [ABSTRACT FROM AUTHOR]

We describe a general approach for constructing a broad class of operators approximating high-dimensional curves based on geometric Hermite data. The geometric Hermite data consists of point samples and their associated tangent vectors of unit length. Extending the classical Hermite interpolation of functions, this geometric Hermite problem has become popular in recent years and has ignited a series of solutions in the 2D plane and 3D space. Here, we present a method for approximating curves, which is valid in any dimension. A basic building block of our approach is a Hermite average — a notion introduced in this paper. We provide an example of such an average and show, via an illustrative interpolating subdivision scheme, how the limits of the subdivision scheme inherit geometric properties of the average. Finally, we prove the convergence of this subdivision scheme, whose limit interpolates the geometric Hermite data and approximates the sampled curve. We conclude the paper with various numerical examples that elucidate the advantages of our approach. [ABSTRACT FROM AUTHOR]

Hermite interpolation is classically used to reconstruct smooth data when the function and its first order derivatives are available at certain nodes. If first order derivatives are not available, it is easy to set a system of equations imposing some regularity conditions at the data nodes in order to obtain them. This process leads to the construction of a Hermite spline. The problem of the described Hermite splines is that the accuracy is lost if the data contains singularities. The consequence is the appearance of oscillations, if there is a jump discontinuity in the function, that globally affects the accuracy of the spline, or the smearing of singularities, if the discontinuities are in the derivatives of the function. This paper is devoted to the construction and analysis of a new technique that allows for the computation of accurate first order derivatives of a function close to singularities using a Hermite spline. The idea is to correct the system of equations of the spline in order to attain the desired accuracy even close to the singularities. Once we have computed the first order derivatives with enough accuracy, a correction term is added to the Hermite spline in the intervals that contain a singularity. The aim is to reconstruct piecewise smooth functions with O (h 4) accuracy even close to the singularities. The process of adaption will require some knowledge about the position of the singularity and the jumps of the function and some of its derivatives at the singularity. The whole process can be used as a post-processing, where a correction term is added to the classical cubic Hermite spline. Proofs for the accuracy and regularity of the corrected spline and its derivatives are given. We also analyse the mechanism that eliminates the Gibbs phenomenon close to jump discontinuities in the function. The numerical experiments presented confirm the theoretical results obtained. [ABSTRACT FROM AUTHOR]

We present a new image scaling method both for downscaling and upscaling, running with any scale factor or desired size. The resized image is achieved by sampling a bivariate polynomial which globally interpolates the data at the new scale. The method's particularities lay in both the sampling model and the interpolation polynomial we use. Rather than classical uniform grids, we consider an unusual sampling system based on Chebyshev zeros of the first kind. Such optimal distribution of nodes permits to consider near-best interpolation polynomials defined by a filter of de la Vallée-Poussin type. The action ray of this filter provides an additional parameter that can be suitably regulated to improve the approximation. The method has been tested on a significant number of different image datasets. The results are evaluated in qualitative and quantitative terms and compared with other available competitive methods. The perceived quality of the resulting scaled images is such that important details are preserved, and the appearance of artifacts is low. Competitive quality measurement values, good visual quality, limited computational effort, and moderate memory demand make the method suitable for real-world applications. [ABSTRACT FROM AUTHOR]

In this paper, based on the Tikhonov regularized approximation formulae derived by An and Wu, we focus on their error estimates, the selection of the regularization parameter and the numerical stability. First, we quantify certain terms in the previous L 2 error bound and the uniform error bound. From these two error bounds, we know that the selection of the regularization parameter is important. Then, we employ two strategies, the balancing principle and Brezinski–Rodriguez–Seatzu estimators, to select the regularization parameter. Some numerical experiments are given to illustrate that both parameter choice strategies can select suitable regularization parameters. The two parameter choice strategies are also compared by testing some oscillatory functions. Finally, we study the numerical stability of Tikhonov regularized barycentric interpolation formula and Tikhonov regularized modified Lagrange interpolation formula. Both interpolation formulae are shown to be forward stable, and the Tikhonov regularized modified Lagrange interpolation formula is also backward stable. We give numerical examples to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Finite elements with polynomial basis functions on the simplex with a symmetric distribution of nodes should have a unique polynomial representation. Unisolvence not only requires that the number of nodes equals the number of independent polynomials spanning a polynomial space of a given degree, but also that the Vandermonde matrix controlling their mapping to the Lagrange interpolating polynomials can be inverted. Here, a necessary condition for unisolvence is presented for polynomial spaces that have non-decreasing degrees when going from the edges and the various faces to the interior of the simplex. It leads to a proof of a conjecture on a necessary condition for unisolvence, requiring the node pattern to be the same as that of the regular simplex. [ABSTRACT FROM AUTHOR]

We propose an efficient mesh adaptive method for the numerical solution of time-dependent partial differential equations considered in the fixed space-time cylinder Ω × (0 , T) . We employ the space-time discontinuous Galerkin method which enables us to use different meshes at different time levels in a natural way. The mesh adaptive algorithm is based on control of the interpolation error in the L ∞ (0 , T ; L q (Ω)) -norm. The goal is to construct a sequence of conforming triangular meshes in such a way that the interpolation error bound is under a given tolerance and the number of degrees of freedom is minimal. The resulting grids consist of anisotropic mesh elements with varying polynomial approximation degrees with respect to space. We present a theoretical framework of this approach as well as several numerical examples demonstrating the accuracy, efficiency, and applicability of the method. [ABSTRACT FROM AUTHOR]

The main concern of this paper is to develop a new class of A-stable fourth-order numerical scheme for solving initial value problems. The idea is an evolutionary and heuristic approach; using the Grasshopper optimization, along with the Hermite interpolation for stages, we obtain a class of A-stable methods. Four types of weighting rules are introduced for the current formulation. The fundamental weighting rule (FWR) is the most important rule, which emphasizes on the symmetric and central structure of the method. A systematic strategy is proposed to obtain the FWR based on swarm intelligence and regression. The new techniques are called α I -(e + i)P, where e and i are the number of terminal and internal points, respectively. The numerical experiments demonstrate the reasonable behaviour of the algorithms on several test problems from different applications. Finally, we find that the new formulas are well suited for long time behaviour of the time-history analysis in earthquake engineering. [ABSTRACT FROM AUTHOR]

In this study, the numerical solutions of the modified Fornberg–Whitham (mFW) equation, which describes immigration of the solitary wave and peakon waves with discontinuous first derivative at the peak, have been obtained by the collocation finite element method using quintic trigonometric B-spline bases. Although there are solutions of this equation by semi-analytical and analytical methods in the literature, there are very few studies on the solution of the equation by numerical methods. Any linearization technique has not been used while applying the method. The stability analysis of the applied method is examined by the von-Neumann Fourier series method. To show the performance of the method, we have considered three test problems with nonhomogeneous boundary conditions having analytical solutions. The error norms L2 and L∞ are calculated to demonstrate the accuracy and efficiency of the presented numerical scheme. [ABSTRACT FROM AUTHOR]

Numerical analysts and scientists working in applications often observe that once they improve their techniques to get a better accuracy, some instability of the evaluation creeps in through the back door. This paper shows for a large class of numerical methods that such a Trade-off Principle between error and evaluation stability is unavoidable. It is an instance of a no free lunch theorem. Here, evaluation is the mathematical map that takes input data to output data. This is independent from the numerical routine that calculates the output. Therefore, evaluation stability is different from computational stability. The setting is confined to recovery of functions from data, but it includes solving differential equations by writing such methods as a recovery of functions under constraints imposed by differential operators and boundary values. The trade-off principle bounds the product of two terms from below. The first is related to errors, and the second turns out to be related to evaluation instability. Under certain conditions satisfied for splines and kernel-based interpolation, both can be minimized. Then the lower bound is attained, and the error term is the inverse of the instability term. As a byproduct, it is shown that Kansa’s Unsymmetric Collocation Method sacrifices accuracy for improved evaluation stability, when compared to symmetric collocation. [ABSTRACT FROM AUTHOR]

In this digital, internet-based world, it is not new to face cyber attacks from time to time. A number of heavy viruses have been made by hackers, and they have successfully given big losses to our systems. In the family of these viruses, the Stuxnet virus is a well-known name. Stuxnet is a very dangerous virus that probably targets the control systems of our industry. The main source of this virus can be an infected USB drive or flash drive. In this research paper, we study a mathematical model to define the dynamical structure or the effects of the Stuxnet virus on our computer systems. To study the given dynamics, we use a modified version of the Caputo-type fractional derivative, which can be used as an old Caputo derivative by fixing some slight changes, which is an advantage of this study. We demonstrate that the given fractional Caputo-type dynamical model has a unique solution using fixed point theory. We derive the solution of the proposed non-linear non-classical model with the application of a recent version of the Predictor–Corrector scheme. We analyze various graphs at different values of the arrival rate of new computers, damage rate, virus transmission rate, and natural removal rate. In the graphical interpretations, we verify the values of fractional orders and simulate 2-D and 3-D graphics to understand the dynamics clearly. The major novelty of this study is that we formulate the optimal control problem and its important consequences both theoretically and mathematically, which can be further extended graphically. The main contribution of this research work is to provide some novel results on the Stuxnet virus dynamics and explore the uses of fractional derivatives in computer science. The given methodology is effective, fully novel, and very easy to understand.

Two-dimensional biharmonic boundary-value problems are considered by the linear barycentric rational collocation method, and the unknown function is approximated by the barycentric rational polynomial. With the help of matrix form, the linear equations of the discrete biharmonic equation are changed into a matrix equation. From the convergence rate of barycentric rational polynomial, we present the convergence rate of linear barycentric rational collocation method for biharmonic equation. Finally, several numerical examples are provided to validate the theoretical analysis.

In this work, our motivation is to design an impressive new numerical approximation on non-uniform grid points for the Caputo fractional derivative in time 0 C D t α with the order α ∈ (1,2). An adaptive high-order stable implicit difference scheme is developed for the time-fractional diffusion wave equations (TFDWEs) by using estimation of order O (N t α - 5) for the Caputo derivative in the time domain on non-uniform mesh and well-known second-order central difference approximation for estimating the spatial derivative on a uniform mesh. The designed algorithm allows one to build adaptive nature where the scheme is adjusted according to the behaviour of α in order to keep the numerical errors very small and converge to the solution very fast as compared to the previously investigated scheme. We rigorously analyze the local truncation errors, unconditional stability of the proposed method, and its convergence of (5 − α)-th order in time and second-order in space for all values of α ∈ (1,2). A reduced order technique is implemented by using moving mesh refinement and assemble with the derived scheme in order to improve the temporal accuracy at several starting time levels. Furthermore, the numerical stability of the derived adaptive scheme is verified by imposing random external noises. Some numerical tests are given to show that the numerical results are consistent with the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a new Hermite interpolatory subdivision scheme for curve interpolation is introduced. The scheme is constructed from the Rational [3/2] Bernstein Bezier polynomial. We call it the [3/2]-scheme. The limit function of the [3/2]-scheme interpolates both the function values and their derivatives. The proposed scheme has three shape parameters w 0 , w 1 and w 2 . It is shown that if w 1 = w 0 + w 2 2 , then the [3/2]-scheme reproduces linear polynomial and is C 1 provided w 0 and w 2 lie in a region of convergence. The scheme also satisfies the shape preserving properties, i.e., monotonicity and convexity. We also compare the [3/2]-scheme with other existing schemes like the [2/2]-scheme and the Merrien scheme introduced recently. An error analysis shows that the [3/2]-scheme is better than the [2/2]-scheme and the Merrien scheme. Further, it is observed that in case w 0 = w 1 = w 2 , the [3/2]-scheme reduces to the Merrien scheme. [ABSTRACT FROM AUTHOR]

Quadrature formulas (QFs) based on radial basis functions (RBFs) have become an essential tool for multivariate numerical integration of scattered data. Although numerous works have been published on RBF-QFs, their stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more mature stability theory for global and function-independent RBF-QFs. In particular, we prove stability of these for compactly supported RBFs under certain conditions on the shape parameter and the data points. As an alternative to changing the shape parameter, we demonstrate how the least-squares approach can be used to construct stable RBF-QFs by allowing the number of data points used for numerical integration to be larger than the number of centers used to generate the RBF approximation space. Moreover, it is shown that asymptotic stability of many global RBF-QFs is independent of polynomial terms, which are often included in RBF approximations. While our findings provide some novel conditions for stability of global RBF-QFs, the present work also demonstrates that there are still many gaps to fill in future investigations. [ABSTRACT FROM AUTHOR]

Digitized handwritten documents have been used for various purposes, including age detection, a crucial area of research in fields like forensic investigation and medical diagnosis. Automated age recognition is deemed to be a difficult task, due to the great degree of similarity and overlap across people's handwriting. Consequently, the efficiency of the classification system is determined by extracting pertinent features from handwritten documents. This research proposes a set of age-related features suggested by a graphologist to classify handwritten documents into two age groups: youth adult and mature adult. The extracted features are: slant irregularity (SI), pen pressure irregularity (PPI), text lines irregularity (TLI) and the percentage of black and white pixels (PWB). Support Vector Machines (SVM) and Neural Network (NN) classifiers have been used to train, validate and test the proposed approach using two different datasets: the FSHS and the Khatt datasets. When applied to the FSHS dtaset using SVM and NN approaches, the proposed method resulted in a classification accuracy of 71% and 63.5%, respectively. Meanwhile, when applied to the Khatt dataset, our method outperformed state-of-the-art methods with a classification accuracy of 65.2% and 67% utilising SVM and NN classifiers, respectively. These are the best rates available right now in this field. [ABSTRACT FROM AUTHOR]

Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples. [ABSTRACT FROM AUTHOR]

We construct new families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons that are H1 and H(div) conforming, respectively, and possess optimal order of accuracy for any order. They have a minimal number of degrees of freedom subject to the conformity and accuracy constraints. The name arises because the shape functions are defined directly on the physical elements, i.e., without using a mapping from a reference element. The finite element shape functions are defined to be the full spaces of scalar or vector polynomials plus a space of supplemental functions. The direct serendipity elements are the precursors of the direct mixed elements in a de Rham complex. The convergence properties of the finite elements are shown under a regularity assumption on the shapes of the polygons in the mesh, as well as some mild restrictions on the choices one can make in the construction of the supplemental functions. Numerical experiments on various meshes exhibit the performance of these new families of finite elements. [ABSTRACT FROM AUTHOR]

When one solves differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients an to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, u(±1) = 0, popular choices include the "Chebyshev difference basis" ςn(x) ≡ Tn+2(x)−Tn(x) with coefficients here denoted by bn and the "quadratic factor basis" ϱn(x) ≡ (1 − x2)Tn(x) with coefficients cn. If u(x) is weakly singular at the boundary, then the coefficients an decrease proportionally to O (A (n) / n κ) for some positive constant κ, where A(n) is a logarithm or a constant. We prove that the Chebyshev difference coefficients bn decrease more slowly by a factor of 1/n while the quadratic factor coefficients cn decrease more slowly still as O (A (n) / n κ − 2 ) . The error for the unconstrained Chebyshev series, truncated at degree n = N, is O (| A (N) | / N κ) in the interior, but is worse by one power of N in narrow boundary layers near each of the endpoints. Despite having nearly identical error norms in interpolation, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x. Meanwhile, for Chebyshev polynomials, the values of their derivatives at the endpoints are O (N 2) , but only O (N) for the difference basis. Furthermore, we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases, solved by the least squares method. We also find an interesting fact that on the face of it, the aliasing error is regarded as a bad thing; actually, the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation. But the premise is under the same basis, and when involving different bases, it may not be established yet. [ABSTRACT FROM AUTHOR]

We present precise anisotropic interpolation error estimates for smooth functions using a new geometric parameter and derive inverse inequalities on anisotropic meshes. In our theory, the interpolation error is bounded in terms of the diameter of a simplex and the geometric parameter. Imposing additional assumptions makes it possible to obtain anisotropic error estimates. This paper also includes corrections to an error in Theorem 2 of our previous paper, "General theory of interpolation error estimates on anisotropic meshes" (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163–191). [ABSTRACT FROM AUTHOR]

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Independently and under the milder hypothesis that f is even and monotone over [0,π], matrix-less algorithms have been developed for the fast eigenvalue computation of large Toeplitz matrices, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions predicted by the simple-loop theory, combined with the extrapolation idea. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we adapt the matrix-less algorithm to the considered new setting. Numerical experiments show a higher precision (till machine precision) and the same linear computation cost, when compared with the matrix-less procedures already presented in the relevant literature. Among the advantages, we concisely mention the following: (a) when the coefficients of the simple-loop function are analytically known, the algorithm computes them perfectly; (b) while the proposed algorithm is better or at worst comparable to the previous ones for computing the inner eigenvalues, it is vastly better for the computation of the extreme eigenvalues; a mild deterioration in the quality of the numerical experiments is observed when dense Toeplitz matrices are considered, having generating function of low smoothness and not satisfying the simple-loop assumptions. [ABSTRACT FROM AUTHOR]

In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker–Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations. [ABSTRACT FROM AUTHOR]

The barycentric forms of polynomial and rational interpolation have recently gained popularity, because they can be computed with simple, efficient, and numerically stable algorithms. In this paper, we show more generally that the evaluation of any function that can be expressed as r (x) = ∑ i = 0 n a i (x) f i / ∑ j = 0 m b j (x) in terms of data values f i and some functions a i and b j for i = 0 , ... , n and j = 0 , ⋯ , m with a simple algorithm that first sums up the terms in the numerator and the denominator, followed by a final division, is forward and backward stable under certain assumptions. This result includes the two barycentric forms of rational interpolation as special cases. Our analysis further reveals that the stability of the second barycentric form depends on the Lebesgue constant associated with the interpolation nodes, which typically grows with n, whereas the stability of the first barycentric form depends on a similar, but different quantity, that can be bounded in terms of the mesh ratio, regardless of n. We support our theoretical results with numerical experiments. [ABSTRACT FROM AUTHOR]

A numerical method is applied for the 2D sine-Gordon equation in this paper, which is named bivariate multiquadrics quasi-interpolation (MQQI). More specifically, the spatial derivative is approximated by MQQI and time derivative is approximated by forward difference method. The main merits of this scheme are its simple structure and easy implementation. Meanwhile, truncation and total errors are presented. Furthermore, Numerical examples verify the effectiveness and high accuracy of the method. In addition, the optimal value of parameters are investigated in this article based on Luh (2016, 2019). [ABSTRACT FROM AUTHOR]

We present precise Raviart–Thomas interpolation error estimates on anisotropic meshes. The novel aspect of our theory is the introduction of a new geometric parameter of simplices. It is possible to obtain new anisotropic Raviart–Thoma error estimates using the parameter. We also include corrections to an error in “General theory of interpolation error estimates on anisotropic meshes” (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191), in which Theorem 3 was incorrect. [ABSTRACT FROM AUTHOR]

In this work, a class of interpolation and smoothing exponential splines with respect to a sixth order differential operator with two parameters is constructed. All the square matrices involved in the construction are proved to be tridiagonal symmetric and diagonally dominant, which results in algorithms for computing this class of exponential splines. The obtained splines have C 4 continuity and are the minimum solution of the combination of interpolation and a generalized smoothing energy integral. The performances of the resulting splines in financial data from the S &P500 index and the effect of fitting multi-exponential decay data are given. Numerical experiments show that the resulting splines have more freedom to adjust the shape and control the energy of the curves and perform better than previous methods in fitting multi-exponential decay data. And cross validation and generalized cross validation for determining an appropriate smoothing parameter are also developed. [ABSTRACT FROM AUTHOR]

We propose to extend results on the interpolation theory for scalar functions to the case of differential k-forms. More precisely, we consider the interpolation of fields in P r - Λ k (T) , the finite element spaces of trimmed polynomial k-forms of arbitrary degree r ≥ 1 , from their weights, namely their integrals on k-chains. These integrals have a clear physical interpretation, such as circulations along curves, fluxes across surfaces, densities in volumes, depending on the value of k. In this work, for k = 1 , we rely on the flexibility of the weights with respect to their geometrical support, to study different sets of 1-chains in T for a high order interpolation of differential 1-forms, constructed starting from "good" sets of nodes for a high order multi-variate polynomial representation of scalar fields, namely 0-forms. We analyse the growth of the generalized Lebesgue constant with the degree r and preliminary numerical results for edge elements support the nonuniform choice, in agreement with the well-known nodal case. [ABSTRACT FROM AUTHOR]

In this paper, the solution to a bivariate Appell interpolation problem proposed in a previous work is given. Bounds of the truncation error are considered. Ten new interpolants for real, regular, bivariate functions are constructed. Numerical examples and comparisons with bivariate Bernstein polynomials are considered. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of constructing C 2 cubic spline functions on a given arbitrary triangulation T . To this end, we endow every triangle of T with a Wang–Shi macro-structure. The C 2 cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in this space can be locally built on each of the macro-triangles independently via Hermite interpolation. We provide a simplex spline basis for the space of C 2 cubics defined on a single macro-triangle which behaves like a Bernstein/B-spline basis over the triangle. The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for C 2 joins across the edges of neighboring triangles, and they enjoy a Marsden-like identity. Also, there is a single control net to facilitate control and early visualization of a spline function over the macro-triangle. Thanks to these properties, the complex geometry of the Wang–Shi macro-structure is transparent to the user. Stable global bases for the full space of C 2 cubics on the Wang–Shi refined triangulation T are deduced from the local simplex spline basis by extending the concept of minimal determining sets. [ABSTRACT FROM AUTHOR]

This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic-like Hermite interpolant. A characterization of solutions according to the tangents at both ends and a global free shape parameter α is performed. Since this degree of freedom can be used for adjustments, we study how the curve evolves with respect to α. Several examples illustrating the construction process and a simple fitting method to determine the unique ATPH curve passing through a given point are proposed. [ABSTRACT FROM AUTHOR]

In this study, by combining the generalized pseudospectral method, which is a new numerical method, with the quasi-linearization method (QLM), an efficient hybrid method to solve nonlinear differential equations in applied sciences and engineering is presented. Given that this method requires generalized Lagrange functions and their derivative operational matrices, we first introduce them and then implement the generalized pseudospectral method at each iteration of the quasi-linearization method and obtain a system of linear algebraic equations. In the presented method, derivative operational matrices are used, and also, the nonlinear differential equation is converted into a sequence of linear differential equations using QLM, so there is no need to calculate the analytical derivative and solve nonlinear systems during the implementation of the method, which reduces computational costs and increasing the efficiency of the method. The efficiency and accuracy of the method have been demonstrated by applying it to several important applied equations; the Blasius equation, the Falkner–Skan problem over an isothermal moving wedge, and the third-grade fluid in a porous half-space. Then the obtained results are compared with other researchers. [ABSTRACT FROM AUTHOR]

The intersection of surfaces is a fundamental process in computational geometry and computer-aided design applications to build and interrogate complex shapes in the computer. This paper presents a novel and simple dual quaternion-based osculating circle DQOC algorithm to find the intersection curve of two regular surfaces based on the osculating circle concept and dual quaternions. Additionally, we expressed the natural equations of the intersection curve. We have also demonstrated the superiority of our method through numerical examples.

Facial expression generation has always been an intriguing task for scientists and researchers all over the globe. In this context, we present our novel approach for generating videos of the six basic facial expressions. Starting from a single neutral facial image and a label indicating the desired facial expression, we aim to synthesize a video of the given identity performing the specified facial expression. Our approach, referred to as FEV-GAN (Facial Expression Video GAN), is based on Spatio-temporal Convolutional GANs, that are known to model both content and motion in the same network. Previous methods based on such a network have shown a good ability to generate coherent videos with smooth temporal evolution. However, they still suffer from low image quality and low identity preservation capability. In this work, we address this problem by using a generator composed of two image encoders. The first one is pre-trained for facial identity feature extraction and the second for spatial feature extraction. We have qualitatively and quantitatively evaluated our model on two international facial expression benchmark databases: MUG and Oulu-CASIA NIR&VIS. The experimental results analysis demonstrates the effectiveness of our approach in generating videos of the six basic facial expressions while preserving the input identity. The analysis also proves that the use of both identity and spatial features enhances the decoder ability to better preserve the identity and generate high-quality videos. The code and the pre-trained model will soon be made publicly available.

We present a number of new piecewise-polynomial kernels for image interpolation. The kernels are constructed by optimizing a measure of interpolation quality based on the magnitude of anisotropic artifacts. The kernel design process is performed symbolically using the Mathematica computer algebra system. An experimental evaluation involving 14 image quality assessment methods demonstrates that our results compare favorably with the existing linear interpolators. [ABSTRACT FROM AUTHOR]

In December 2019, in Wuhan, China, a new disease was detected, and the virus easily spread throughout other nations. March 2, 2020, Morocco announced 1st infection of coronavirus. Morocco verified a total of 653,286 cases, 582,692 recovered, 60,579 active case, and 10,015 as confirmatory fatalities, as of 4 August 2021. The objective of this article is to study the mathematical modeling of undetected cases of the novel coronavirus in Morocco. The model is shown to have disease-free and an endemic equilibrium point. We have discussed the local and global stability of these equilibria. The parameters of the model and undiscovered instances of COVID-19 were assessed by the least squares approach in Morocco and have been eliminated. We utilized a Matlab tool to show developments in undiscovered instances in Morocco and to validate predicted outcomes. Like results, until August 4, 2021, the total number of infected cases of COVID-19 in Morocco is 24,663,240, including 653,286 confirmed cases, against 24,009,954 undetected. Further, our approach gives a good approximation of the actual COVID-19 data from Morocco and will be used to estimate the undetected cases of COVID-19 in other countries of the world and to study other pandemics that have the same nature of spread as COVID-19. [ABSTRACT FROM AUTHOR]

An inverse central ordering of the nodes is proposed for the Newton interpolation formula. This ordering may improve the stability for certain distributions of nodes. For equidistant nodes, an upper bound of the conditioning is provided. This bound is close to the bound of the conditioning in the Lagrange interpolation formula, whose conditioning is the lowest. This ordering is related to a pivoting strategy of a matrix elimination procedure called Neville elimination. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]

Learning mappings of data on manifolds is an important topic in contemporary machine learning, with applications in astrophysics, geophysics, statistical physics, medical diagnosis, biochemistry, and 3D object analysis. This paper studies the problem of learning real-valued functions on manifolds through filtered hyperinterpolation of input–output data pairs where the inputs may be sampled deterministically or at random and the outputs may be clean or noisy. Motivated by the problem of handling large data sets, it presents a parallel data processing approach which distributes the data-fitting task among multiple servers and synthesizes the fitted sub-models into a global estimator. We prove quantitative relations between the approximation quality of the learned function over the entire manifold, the type of target function, the number of servers, and the number and type of available samples. We obtain the approximation rates of convergence for distributed and non-distributed approaches. For the non-distributed case, the approximation order is optimal. [ABSTRACT FROM AUTHOR]

In one dimension, nodal locations that are distinct are necessary and sufficient to ensure that a unique polynomial interpolant exists for data provided at a set of nodes, i.e. that the set of nodes is unisolvent. In multiple dimensions however, unisolvency for a polynomial interpolant of degree p is not ensured even with nodal locations that are distinct and a set of n nodes, with n equal to the cardinality of a set of polynomial basis functions of at most degree p. In this paper a set of equations is derived for simplices of one to three dimensions with symmetrical nodal distributions to identify a combination of symmetry orbits that can provide a unisolvent set of nodes. The results suggest that there is a unique combination of symmetry orbits that can provide a unisolvent set of nodes for each degree of polynomial interpolant. Consequently, all other combinations of symmetry orbits cannot provide a unisolvent set of nodes for a degree p polynomial interpolant. This is verified numerically up to degree 10 for triangles and degree 7 for tetrahedra. The results suggest that the same is also true for higher-order polynomial interpolants. This significantly reduces the number of combination of symmetry orbits that needs to be considered. For example, for a tetrahedron with a degree seven interpolant, only one combination of symmetry orbits needs to be considered instead of the 161 different combinations of symmetry orbits that provide a set of nodes with n equal to the cardinality of the set of basis functions of at most degree seven. For a symmetrical nodal distribution in a simplex, the conditions presented are necessary but not sufficient to have a unisolvent set of nodes for polynomial interpolation. [ABSTRACT FROM AUTHOR]

The Floater--Hormann family of rational interpolants do not have spuriouspoles or unattainable points, are efficient to calculate, and have arbitrarilyhigh approximation orders. One concern when using them is that theamplification of rounding errors increases with approximation order, and canmake balancing the interpolation error and rounding error difficult. Thisarticle proposes to modify the Floater--Hormann interpolants by includingadditional local polynomial interpolants at the ends of the interval. Thisappears to improve the conditioning of the interpolants and allow higherapproximation orders to be used in practice.