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This paper introduces a novel technique to approximate a given continuous function f defined on a real compact interval by a new class of zipper α -fractal functions which contain a scaling vector and a binary vector or signature. For specific choices of scaling and signature vectors, the corresponding zipper fractal functions simultaneously interpolate and approximate f. When signature is zero, the proposed zipper fractal functions coincide with existing α -fractal functions. Hence, the zipper approximation proposed in this manuscript generalizes the existing fractal and classical approximations. Zipper fractal analogue of some elementary results in the classical approximation theory are obtained. Using convex optimization technique, we investigate the existence of optimal zipper fractal function for a given continuous function. The parameter identification problems for zipper α -fractal approximants are investigated. We derive sufficient conditions on the parameters of zipper α -fractal functions so that these functions preserve the positivity, monotonicity and convexity of the original function f. Also, we have studied the constructions of k-times continuously differentiable zipper α -fractal functions and one sided zipper fractal approximants for f. Numerical illustrations are provided to support the proposed theoretical results on zipper α -fractal functions. [ABSTRACT FROM AUTHOR]

The paper introduces new sufficient conditions of strict positive definiteness for kernels on d-dimensional spheres which are not radially symmetric but possess specific coefficient structures. The results use the series expansion of the kernel in spherical harmonics. The kernels either have a convolutional form or are axially symmetric with respect to one axis. The given results on convolutional kernels generalise the result derived by Chen et al. (Proc Am Math Soc 131:2733–2740, 2003) for radial kernels. [ABSTRACT FROM AUTHOR]

We propose a fast solver for the variable-order (VO) time-fractional diffusion equation. Due to the impact of the time-dependent VO function, the resulting coefficient matrix of the large linear system assembling discrete equations of all time levels is a block lower triangular matrix without the block Toeplitz structure. Here, we approximate the off-diagonal blocks by low-rank matrices based on the polynomial interpolation, which can be constructed in O (M log 2 M) operations with the same number storage requirement, where M is the number of time steps. Furthermore, a divide-and-conquer method is developed to fast solve the approximated linear system. The proposed solver can be implemented in O (N M log 2 M) complexity with N being the degree of freedom in space. The accuracy of approximation is theoretically studied, and the stability and convergence of the proposed fast method are also investigated. Numerical experiments are carried out to exemplify the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to provide an efficient scheme for constructing kernel-based reduced interpolation models. In the existing literature such problems are mainly addressed via the well-established knot insertion or knot removal schemes. Such iterative strategies are usually quite demanding from a computational point of view and our goal is to study an efficient implementation for data removal approaches, namely efficient reduced basis algorithm (ERBA). Focusing on kernel-based interpolation, the algorithm makes use of two iterative rules for removing data. The former, called ERBA-r, is based on classical residual evaluations. The latter, namely ERBA-p, is independent of the function values and relies on error bounds defined by the power function. In both cases, inspired by the so-called extended Rippa’s algorithm, our ERBA takes advantage of a fast implementation. [ABSTRACT FROM AUTHOR]

Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer potentials belonging to the modified Helmholtz, modified biharmonic, and modified Stokes equations. These equations depend on a parameter, denoted α, and kernel-split quadrature loses its accuracy rapidly when this parameter grows beyond a certain threshold. This paper describes an algorithm that remedies this problem, using per-target adaptive sampling of the source geometry. The refinement is carried out through recursive bisection, with a carefully selected rule set. This maintains accuracy for a wide range of the parameter α, at an increased cost that scales as log α . Using this algorithm allows kernel-split quadrature to be both accurate and efficient for a much wider range of problems than previously possible. [ABSTRACT FROM AUTHOR]

The classical serendipity and mixed finite element spaces suffer from poor approximation on nondegenerate, convex quadrilaterals. In this paper, we develop families of direct serendipity and direct mixed finite element spaces, which achieve optimal approximation properties and have minimal local dimension. The set of local shape functions for either the serendipity or mixed elements contains the full set of scalar or vector polynomials of degree r, respectively, defined directly on each element (i.e., not mapped from a reference element). Because there are not enough degrees of freedom for global H 1 or H (div) conformity, exactly two supplemental shape functions must be added to each element when r ≥ 2 , and only one when r = 1 . The specific choice of supplemental functions gives rise to different families of direct elements. These new spaces are related through a de Rham complex. For index r ≥ 1 , the new families of serendipity spaces DS r + 1 are the precursors under the curl operator of our direct mixed finite element spaces, which can be constructed to have reduced or full H (div) approximation properties. One choice of direct serendipity supplements gives the precursor of the recently introduced Arbogast–Correa spaces (SIAM J Numer Anal 54:3332–3356, 2016. https://doi.org/10.1137/15M1013705). Other fully direct serendipity supplements can be defined without the use of mappings from reference elements, and these give rise in turn to fully direct mixed spaces. Our development is constructive, so we are able to give global bases for our spaces. Numerical results are presented to illustrate their properties. [ABSTRACT FROM AUTHOR]

In this study, the second kind Volterra integral equations (VIEs) are considered. An algorithm based on the two-point Taylor formula as a special case of the Hermite interpolation is proposed to approximate the solution of such problems. The method can be applied for solving both the linear and nonlinear VIEs and systems of nonlinear VIEs. The convergence analysis and the error estimate of the method are described. A multistep form of the algorithm which is particularly beneficial in large intervals is also presented. The main advantage of the proposed algorithm is that it gives high accurate results in acceptable computational times. In order to indicate the validity of the method, it is employed for solving several illustrative examples. The efficiency of the method is confirmed through our study respecting the absolute errors and CPU times. [ABSTRACT FROM AUTHOR]

We compute Chebyshev-like norming grids for polynomials on spherical triangles. The construction is based on a conjecture about norming grids for univariate trigonometric polynomials (supported by wide numerical testing), together with the fundamental notion of Dubiner distance for multivariate compact sets. Such grids can be used to extract Fekete-like interpolation points with slowly increasing Lebesgue constant, by basic numerical linear algebra. [ABSTRACT FROM AUTHOR]

A cubic H-Bézier geometric approximation scheme with one free parameter τis introduced for parabolic arcs. It has four control points. The control points are computed by matching the end points and end unit tangents of the two curves and provide two more free parameters. These free parameters are computed by curvature continuity constraints. Optimal value of τ is obtained by minimizing the maximum value of the distance between the cubic H-Bézier curve and the concerned parabolic arc. The method is illustrated using different numerical examples which show that the proposed method is economical, reliable and efficient.

Classical rational interpolation usually enjoys better approximation properties than polynomial interpolation because it avoids wild oscillations and exhibits exponential convergence for approximating analytic functions. We develop a rational interpolation operator, which not only preserves the advantage of classical rational interpolation, but also has a finite Lebesgue constant. In particular, it is convergent for approximating any continuous function, and the convergence rate of the interpolants approximating a function is obtained using the modulus of continuity. [ABSTRACT FROM AUTHOR]

In this paper we analyze a greedy procedure to approximate a linear functional defined in a reproducing kernel Hilbert space by nodal values. This procedure computes a quadrature rule which can be applied to general functionals. For a large class of functionals, that includes integration functionals and other interesting cases, but does not include differentiation, we prove convergence results for the approximation by means of quasi-uniform and greedy points which generalize in various ways several known results. A perturbation analysis of the weights and node computation is also discussed. Beyond the theoretical investigations, we demonstrate numerically that our algorithm is effective in treating various integration densities, and that it is even very competitive when compared to existing methods for Uncertainty Quantification. [ABSTRACT FROM AUTHOR]

In this work, we construct a class of Hermite-type interpolation basis functions based on the sixth-order ordinary differential equation S (6) (t) - τ 4 S (2) (t) = 0 . Using them, we propose a kind of C 2 tension interpolation splines with a local tension parameter τ i . For C 2 interpolation, the given interpolant has O (h 2) convergence. Some applications of the C 2 tension interpolation splines on the construction of interest rate term structure in Chinese financial market are given. Moreover, a kind of generalized non-uniform B-splines of the space spanned by span 1 , t , ... , t n - 4 , sin (τ t) , cos (τ t) , sinh (τ t) , cosh (τ t) is constructed. [ABSTRACT FROM AUTHOR]

The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes it possible to identify conditions for the existence of the sought schemes. [ABSTRACT FROM AUTHOR]

We propose an efficient, deterministic algorithm for constructing exponentially convergent deep neural network (DNN) approximations of multivariate, analytic maps f : [ - 1 , 1 ] K → R . We address in particular networks with the rectified linear unit (ReLU) activation function. Similar results and proofs apply for many other popular activation functions. The algorithm is based on collocating f in deterministic families of grid points with small Lebesgue constants, and by a-priori (i.e., “offline”) emulation of a spectral basis with DNNs to prescribed fidelity. Assuming availability of N function values of a possibly corrupted, numerical approximation f ˘ of f in [ - 1 , 1 ] K and a bound on ‖ f - f ˘ ‖ L ∞ ([ - 1 , 1 ] K) , we provide an explicit, computational construction of a ReLU DNN which attains accuracy ε (depending on N and ‖ f - f ˘ ‖ L ∞ ([ - 1 , 1 ] K) ) uniformly, with respect to the inputs. For analytic maps f : [ - 1 , 1 ] K → R , we prove exponential convergence of expression and generalization errors of the constructed ReLU DNNs. Specifically, for every target accuracy ε ∈ (0 , 1) , there exists N depending also on f such that the error of the construction algorithm with N evaluations of f ˘ as input in the norm L ∞ ([ - 1 , 1 ] K ; R) is smaller than ε up to an additive data-corruption bound ‖ f - f ˘ ‖ L ∞ ([ - 1 , 1 ] K) multiplied with a factor growing slowly with 1 / ε and the number of non-zero DNN weights grows polylogarithmically with respect to 1 / ε . The algorithmic construction of the ReLU DNNs which will realize the approximations, is explicit and deterministic in terms of the function values of f ˘ in tensorized Clenshaw–Curtis grids in [ - 1 , 1 ] K . We illustrate the proposed methodology by a constructive algorithm for (offline) computations of posterior expectations in Bayesian PDE inversion. [ABSTRACT FROM AUTHOR]

We analyze a time-delay Caputo-type fractional mathematical model containing the infection rate of Beddington–DeAngelis functional response to study the structure of a vector-borne plant epidemic. We prove the unique global solution existence for the given delay mathematical model by using fixed point results. We use the Adams–Bashforth–Moulton P-C algorithm for solving the given dynamical model. We give a number of graphical interpretations of the proposed solution. A number of novel results are demonstrated from the given practical and theoretical observations. By using 3-D plots we observe the variations in the flatness of our plots when the fractional order varies. The role of time delay on the proposed plant disease dynamics and the effects of infection rate in the population of susceptible and infectious classes are investigated. The main motivation of this research study is examining the dynamics of the vector-borne epidemic in the sense of fractional derivatives under memory effects. This study is an example of how the fractional derivatives are useful in plant epidemiology. The application of Caputo derivative with equal dimensionality includes the memory in the model, which is the main novelty of this study. [ABSTRACT FROM AUTHOR]

In this paper, we construct a piecewise interpolation method of approximate solution of the transport equation based on the Newton interpolation polynomial of two variables. We transform the polynomial to the algebraic form with numerical coefficients; this leads us to a sequence of iterations, which improves the accuracy of the approximation. The method is implemented in software and numerical experiments are performed. The possibility of generalizations to systems of partial differential equations and integro-differential equations is discussed. [ABSTRACT FROM AUTHOR]

We present an asymptotic analysis of adaptive methods for Lp approximation of functions f ∈ Cr([a, b]), where 1 ≤ p ≤ + ∞ . The methods rely on piecewise polynomial interpolation of degree r − 1 with adaptive strategy of selecting m subintervals. The optimal speed of convergence is in this case of order m−r and it is already achieved by the uniform (nonadaptive) subdivision of the initial interval; however, the asymptotic constant crucially depends on the chosen strategy. We derive asymptotically best adaptive strategies and show their applicability to automatic Lp approximation with a given accuracy ε. [ABSTRACT FROM AUTHOR]

In this article, the optimal control for nonlinear SIRC model is studied in fractional order using the Caputo fractional derivative. Graph signal flow is given of the model and simulated by Simulink/Matlab which helps in discussing the topological structure of the model. Dynamics of the system versus certain parameters are studied via bifurcation diagrams, Lyapunov exponents and Poincare maps. The existence of a uniformly stable solution is proved after control. The obtained results display the activeness and suitability of the Mittag Generalized-Leffler function method (MGLFM). The approximate solution of the fractional order SIRC model using MGLFM is explained by giving the figures of solutions before and after control. Also, we plot the 3D relationships with different alpha (fractional order) which display the originality and suitability of the results.

In this paper, we consider general cases of linear Volterra integral equations under minimal assumptions on their weakly singular kernels and introduce a new product integration method in which we involve the linear interpolation to get a better approximate solution, figure out its effect and also we provide a convergence proof. Furthermore, we apply our method to a numerical example and conclude this paper by adding a conclusion [ABSTRACT FROM AUTHOR]

Fractal approximation is a well studied concept, but the convergence of all the existing fractal approximants towards the original function follows usually if the magnitude of the corresponding scaling factors approaches zero. In this article, for a given function f ∈ C (I) , by exploiting fractal approximation theory and considering the classical q-Bernstein polynomials as base functions, we construct a sequence { f n (q , α) (x) } n = 1 ∞ of (q , α) -fractal functions that converges uniformly to f even if the norm/magnitude of the scaling functions/scaling factors does not tend to zero. The convergence of the sequence { f n (q , α) (x) } n = 1 ∞ of (q , α) -fractal functions towards f follows from the convergence of the sequence of q-Bernstein polynomials of f towards f. If we consider a sequence { f m (x) } m = 1 ∞ of positive functions on a compact real interval that converges uniformly to a function f, we develop a double sequence { { f m , n (q , α) (x) } n = 1 ∞ } m = 1 ∞ of (q , α) -fractal functions that converges uniformly to f. [ABSTRACT FROM AUTHOR]

The mapped bases or Fake Nodes Approach (FNA), introduced in De Marchi et al. (J Comput Appl Math 364:112347, 2020c), allows to change the set of nodes without the need of resampling the function. Such scheme has been successfully applied for mitigating the Runge's phenomenon, using the S-Runge map, or the Gibbs phenomenon, with the S-Gibbs map. However, the original S-Gibbs suffers of a subtle instability when the interpolant is constructed at equidistant nodes, due to the Runge'sphenomenon. Here, we propose a novel approach, termed Gibbs–Runge-Avoiding Stable Polynomial Approximation (GRASPA), where both Runge's and Gibbs phenomena are mitigated simultaneously. After providing a theoretical analysis of the Lebesgue constant associated with the mapped nodes, we test the new approach by performing various numerical experiments which confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

In order to solve Prandtl-type equations we propose a collocation-quadrature method based on de la Vallée Poussin (briefly VP) filtered interpolation at Chebyshev nodes. Uniform convergence and stability are proved in a couple of Hölder-Zygmund spaces of locally continuous functions. With respect to classical methods based on Lagrange interpolation at the same collocation nodes, we succeed in reproducing the optimal convergence rates of the L2 case and cut off the typical log factor which seemed inevitable dealing with uniform norms. Such an improvement does not require a greater computational effort. In particular, we propose a fast algorithm based on the solution of a simple 2-bandwidth linear system and prove that, as its dimension tends to infinity, the sequence of the condition numbers (in any natural matrix norm) tends to a finite limit. [ABSTRACT FROM AUTHOR]

We will consider the problem of observability inequality of the Hermite bi-cubic orthogonal spline collocation space semi-discretizations of the 2-D integro-differential equations in the square domains. We prove the uniform (with respect to mesh-size) observability inequality in a subspace of solutions generated by the low frequencies of the negative part, and the middle frequencies of the positive part. Our method uses previously known uniform observability inequalities in the 1-d case and a dyadic spectral time decomposition. Some numerical results are presented to illustrate our theoretical analysis. [ABSTRACT FROM AUTHOR]

• New model for scene text VQA that jointly reasons about textual and visual modalities. • Attending on multi-modal features is better than attending separately to each modality. • Grid features from an object detection backbone proves to work well in this task. • The proposed model is up to 5 times faster than current state of the art models. • Combined with a standard VQA model reaches state of the art on mixed datasets. This paper presents a new model for the task of scene text visual question answering. In this task questions about a given image can only be answered by reading and understanding scene text. Current state of the art models for this task make use of a dual attention mechanism in which one attention module attends to visual features while the other attends to textual features. A possible issue with this is that it makes difficult for the model to reason jointly about both modalities. To fix this problem we propose a new model that is based on an single attention mechanism that attends to multi-modal features conditioned to the question. The output weights of this attention module over a grid of multi-modal spatial features are interpreted as the probability that a certain spatial location of the image contains the answer text to the given question. Our experiments demonstrate competitive performance in two standard datasets with a model that is × 5 faster than previous methods at inference time. Furthermore, we also provide a novel analysis of the ST-VQA dataset based on a human performance study. Supplementary material, code, and data is made available through this link. [ABSTRACT FROM AUTHOR]

In this paper, we focus on an efficient convergent method to solve the linear fractional delay integro-differential equations. We first convert the equation into an equivalent system by replacing the delay function in the sections containing delay in two intervals before the delay time and after the delay time. In the next step, we discretize the new gain system at the Jacobi–Gauss collocation points and acquire a system of algebraic equations to approximate the solution. Here, we simultaneously obtain the solution and its derivative of fractional order. We exhibit the convergence analysis of approximate solution to the exact solution in L ω α , β ∞ (I) -space, and finally with several numerical examples, we show the capability of the method. [ABSTRACT FROM AUTHOR]

Let Γ be an L -shape arc consisting of 2 line segments that meet at an angle different from π in the complex z -plane C . Application of the exterior conformal map ψ from | w | > 1 onto C \ Γ , with ψ (∞) = ∞ , introduces the level curves Γ n = { z = ψ (w) : | w | = 1 + 1 n + 1 } . Let ψ ∗ denote the continuous extension of ψ from | w | > 1 to | w | ≥ 1 , so that any family { z n , k : k = 0 , 1 , ⋯ , n } of points on Γ can be written as { z n , k = ψ ∗ (w n , k) } , where | w n , k | = 1 . Let ω n (z) = Π k = 0 n (z - z n , k) . The main objective of this paper is to show that for L -shape arcs, validation of the Marcinkiewicz–Zygmund inequalities is equivalent to that of the totality of the A p -weight conditions of | ω n (z) | on Γ n and a mild separation condition of { z n , k } . Since the Marcinkiewicz–Zygmund inequalities are essential to the study of Lagrange polynomial interpolation of continuous functions at the nodes { z n , k } , another objective of this paper is to investigate the behavior of the polynomial interpolants at the Fejér points, defined by { z n , k = ψ ∗ (e i (2 k π + θ) / (n + 1)) } for any choice of θ . In this regard, we recall that for the interval [ - 1 , 1], the Fejér points { z n , k = ψ ∗ (e i (2 k + 1) π / (n + 1)) } agree with the Chebyshev points and that the Chebyshev points are most commonly used as nodes for Lagrange polynomial interpolation. On the other hand, numerical experimentation demonstrates that for a typical open L -shape arc Γ , the Lebesgue constants tend to ∞ at the rate of O ((log (n)) 2) , as the polynomial degree n increases, while the A p -weight conditions for the Fejér points { z n , k } do not carry over from [ - 1 , 1] to a truly L -shape arc. Further numerical experiments also demonstrate that the least upper bounds of the Marcinkiewicz–Zygmund inequalities for the canonical Lagrange interpolation polynomials at { z n , k } seem to grow at the rate of n β , for some β > 0 that depends on p > 1 . [ABSTRACT FROM AUTHOR]

The existence of fundamental cardinal exponential B-splines of positive real order σ is established subject to two conditions on σ and their construction is implemented. A sampling result for these fundamental cardinal exponential B-splines is also presented. [ABSTRACT FROM AUTHOR]

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

The data driven extrapolation requires the definition of a functional model depending on the available data and has the application scope of providing reliable predictions on the unknown dynamics. Since data might be scattered, we drive our attention towards kernel models that have the advantage of being meshfree. Precisely, the proposed numerical method makes use of the so-called Variably Scaled Kernels (VSKs), which are introduced to implement a feature augmentation-like strategy based on discrete data. Due to the possible uncertainty on the data and since we are interested in modelling the behaviour of the target functions, we seek for a regularized solution by ridge regression. Focusing on polyharmonic splines, we investigate their implementation in the VSK setting and we provide error bounds in Beppo–Levi spaces. The performances of the method are then tested on functions showing exponential or rational decay. Comparisons with Support Vector Regression (SVR) are also carried out and highlight that the proposed approach is effective, particularly since it does not require to train complex architecture constructions. [ABSTRACT FROM AUTHOR]

A particle-cluster treecode based on barycentric Hermite interpolation is presented for fast summation of electrostatic particle interactions in 3D. The interpolation nodes are Chebyshev points of the 2nd kind in each cluster. It is noted that barycentric Hermite interpolation is scale-invariant in a certain sense that promotes the treecode’s efficiency. Numerical results for the Coulomb and screened Coulomb potentials show that the treecode run time scales like O(N log N), where N is the number of particles in the system. The advantage of the barycentric Hermite treecode is demonstrated in comparison with treecodes based on Taylor approximation and barycentric Lagrange interpolation.