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High-dimensional Lagrange interpolation plays a pivotal role in finite element methods, where ensuring the unisolvence and symmetry of its interpolation space and nodes set is crucial. In this paper, we leverage group action and group representation theories to precisely delineate the conditions for unisolvence. We establish a necessary condition for unisolvence: the symmetry of the interpolation nodes set is determined by the given interpolation space. Our findings not only contribute to a deeper theoretical understanding but also promise practical benefits by reducing the computational overhead associated with identifying appropriate interpolation nodes., Comment: 18 pages

The Sinc approximation applied to double-exponentially decaying functions is referred to as the DE-Sinc approximation. Because of its high efficiency, this method has been used in various applications. In the Sinc approximation, the mesh size and truncation numbers should be optimally selected to achieve its best performance. However, the standard selection formula has only been "near-optimally" selected because the optimal formula of the mesh size cannot be expressed in terms of elementary functions of truncation numbers. In this study, we propose two improved selection formulas. The first one is based on the concept by an earlier research that resulted in a better selection formula for the double-exponential formula. The formula performs slightly better than the standard one, but is still not optimal. As a second selection formula, we introduce a new parameter to propose truly optimal selection formula. We provide explicit error bounds for both selection formulas. Numerical comparisons show that the first formula gives a better error bound than the standard formula, and the second formula gives a much better error bound than the standard and first formulas., Comment: Keywords: Sinc approximation, double-exponential transformation, error bound, mesh size, truncation number

In the present paper, we consider one-hidden layer ANNs with a feedforward architecture, also referred to as shallow or two-layer networks, so that the structure is determined by the number and types of neurons. The determination of the parameters that define the function, called training, is done via the resolution of the approximation problem, so by imposing the interpolation through a set of specific nodes. We present the case where the parameters are trained using a procedure that is referred to as Extreme Learning Machine (ELM) that leads to a linear interpolation problem. In such hypotheses, the existence of an ANN interpolating function is guaranteed. The focus is then on the accuracy of the interpolation outside of the given sampling interpolation nodes when they are the equispaced, the Chebychev, and the randomly selected ones. The study is motivated by the well-known bell-shaped Runge example, which makes it clear that the construction of a global interpolating polynomial is accurate only if trained on suitably chosen nodes, ad example the Chebychev ones. In order to evaluate the behavior when growing the number of interpolation nodes, we raise the number of neurons in our network and compare it with the interpolating polynomial. We test using Runge's function and other well-known examples with different regularities. As expected, the accuracy of the approximation with a global polynomial increases only if the Chebychev nodes are considered. Instead, the error for the ANN interpolating function always decays and in most cases we observe that the convergence follows what is observed in the polynomial case on Chebychev nodes, despite the set of nodes used for training.

Many applications of geometry modeling and computer graphics necessite accurate curvature estimations of curves on the plane or on manifolds. In this paper, we define the notion of the discrete geodesic curvature of a geodesic polygon on a smooth surface. We show that, when a geodesic polygon P is closely inscribed on a $C^2$-regular curve, the discrete geodesic curvature of P estimates the geodesic curvature of C. This result allows us to evaluate the geodesic curvature of discrete curves on surfaces. In particular, we apply such result to planar and spherical 4-point angle-based subdivision schemes. We show that such schemes cannot generate in general $G^2$-continuous curves. We also give a novel example of $G^2$-continuous subdivision scheme on the unit sphere using only points and discrete geodesic curvature called curvature-based 6-point spherical scheme.

An important component of a number of computational modeling algorithms is an interpolation method that preserves the positivity of the function being interpolated. This report describes the numerical testing of a new positivity-preserving algorithm that is designed to be used when interpolating from a solution defined on one grid to different spatial grid. The motivating application for this work was a numerical weather prediction (NWP) code that uses a spectral element mesh discretization for its dynamics core and a cartesian tensor product mesh for the evaluation of its physics routines. This coupling of spectral element mesh, which uses nonuniformly spaced quadrature/collocation points, and uniformly-spaced cartesian mesh combined with the desire to maintain positivity when moving between these meshes necessitates our work. This new approach is evaluated against several typical algorithms in use on a range of test problems in one or more space dimensions. The results obtained show that the new method is competitive in terms of observed accuracy while at the same time preserving the underlying positivity of the functions being interpolated., Comment: 57 pages, 15 figures

A collection of algorithms in object-oriented MATLAB is described for numerically computing with smooth functions defined on the unit ball in the Chebfun software. Functions are numerically and adaptively resolved to essentially machine precision by using a three-dimensional analogue of the double Fourier sphere method to form "ballfun" objects. Operations such as function evaluation, differentiation, integration, fast rotation by an Euler angle, and a Helmholtz solver are designed. Our algorithms are particularly efficient for vector calculus operations, and we describe how to compute the poloidal-toroidal and Helmholtz--Hodge decomposition of a vector field defined on the ball., Comment: 23 pages, 9 figures

In this paper, we propose an interpolation formula for periodic functions. This formula can be regarded as an analog of the Sinc approximation, which is an interpolation formula for functions defined on the entire infinite interval. Theoretical error analysis and numerical examples show that the proposed formula converges exponentially for analytic periodic functions., Comment: 9 pages, 1 figure I have decided to withdraw the paper because the contents of the paper were already proposed by other researchers

We propose a novel approach to allocating resources for expensive simulations of high fidelity models when used in a multifidelity framework. Allocation decisions that distribute computational resources across several simulation models become extremely important in situations where only a small number of expensive high fidelity simulations can be run. We identify this allocation decision as a problem in optimal subset selection, and subsequently regularize this problem so that solutions can be computed. Our regularized formulation yields a type of group lasso problem that has been studied in the literature to accomplish subset selection. Our numerical results compare performance of algorithms that solve the group lasso problem for algorithmic allocation against a variety of other strategies, including those based on classical linear algebraic pivoting routines and those derived from more modern machine learning-based methods. We demonstrate on well known synthetic problems and more difficult real-world simulations that this group lasso solution to the relaxed optimal subset selection problem performs better than the alternatives., Comment: 27 pages, 10 figures

The Floater--Hormann family of rational interpolants do not have spurious poles or unattainable points, are efficient to calculate, and have arbitrarily high approximation orders. One concern when using them is that the amplification of rounding errors increases with approximation order, and can make balancing the interpolation error and rounding error difficult. This article proposes to modify the Floater--Hormann interpolants by including additional local polynomial interpolants at the ends of the interval. This appears to improve the conditioning of the interpolants and allow higher approximation orders to be used in practice.

The Floater--Hormann family of rational interpolants do not have spurious poles or unattainable points, are efficient to calculate, and have arbitrarily high approximation orders. One concern when using them is that the amplification of rounding errors increases with approximation order, and can make balancing the interpolation error and rounding error difficult. This article proposes to modify the Floater--Hormann interpolants by including additional local polynomial interpolants at the ends of the interval. This appears to improve the conditioning of the interpolants and allow higher approximation orders to be used in practice.

In this paper, we consider methods to compute the coefficients of interpolants relative to a basis of polynomials satisfying a three-term recurrence relation. Two new algorithms are presented: the first constructs the coefficients of the interpolation incrementally and can be used to update the coefficients whenever a nodes is added to or removed from the interpolation. The second algorithm, which constructs the interpolation coefficients by decomposing the Vandermonde-like matrix iteratively, can not be used to update or downdate an interpolation, yet is more numerically stable than the first algorithm and is more efficient when the coefficients of multiple interpolations are to be computed over the same set of nodes., Comment: 18 pages, submitted to the Journal of Scientific Computing.