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We extend the classical theory of variational interpolating splines to the case of compact Riemannian manifolds. Our consideration includes in particular such problems as interpolation of a function by its values on a discrete set of points…
Approximation of scattered geometric data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for large scattered (unordered) datasets in d-dimensional space. This method is useful for…
A method based on orthogonal function series interpolation of the square root probability density to analyze higher dimensional scattered data is presented. The method is targeted for the use-case when the model and/or data are available…
In this paper we address the problem of interpolating a spline developable patch bounded by a given spline curve and the first and the last rulings of the developable surface. In order to complete the boundary of the patch a second spline…
Numerically obtaining the inverse of a function is a common task for many scientific problems, often solved using a Newton iteration method. Here we describe an alternative scheme, based on switching variables followed by spline…
We introduce remarkable upper bounds for the interpolation error constants on triangles, which are sharp and given by simple formulas. These constants are crucial in analyzing interpolation errors, particularly those associated with the…
This paper aims at developing new shape functions adapted to smooth vanishing coefficients for scalar wave equation. It proposes the numerical analysis of their interpolation properties. The interpolation is local but high order convergence…
Solutions of partial differential equations can often be written as surface integrals having a kernel related to a singular fundamental solution. Special methods are needed to evaluate the integral accurately at points on or near the…
The double exponential formula was introduced for calculating definite integrals with singular point oscillation functions and Fourier integral. The double exponential transformation is not only useful for numerical computations but it is…
Modern shape design and capture techniques often lead to the geometric data presented in the form of serial rows of data points. In general, the number of data points varies from row to row. Lofted or skinned B-spline surface interpolation…
Using a deterministic framework allows us to estimate a function with the purpose of interpolating data in spatial statistics. Radial basis functions are commonly used for scattered data interpolation in a d-dimensional space, however,…
We describe a two-level method for computing a function whose zero-level set is the surface reconstructed from given points scattered over the surface and associated with surface normal vectors. The function is defined as a linear…
The masked projection techniques are popular in the area of non-linear model reduction. Quantifying and minimizing the error in model reduction, particularly from masked projections, is important. The exact error expressions are often…
In this contribution we introduce a mixed interpolation-regression operator for functions defined in some domains of the plane. We focus the attention on the ellipse, an annulus and a polygon. An upper bound for such an operator is…
We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the…
The paper is devoted to the construction of an optimal interpolation formula in $K_2(P_2)$ Hilbert space. Here the interpolation formula consists of a linear combination $\sum_{\beta=0}^NC_{\beta}(z)\varphi(x_\beta)$ of given values of a…
We adapt Stein's method of diffusion approximations, developed by Barbour, to the study of chaotic dynamical systems. We establish an error bound in the functional central limit theorem with respect to an integral probability metric of…
We obtain new Fourier interpolation and -uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa, and by the second author. We show that the only Schwartz function which, together with its…
Let $X(D,1) =\Gamma(D,1) \backslash \mathbb{H}$ denote the Shimura curve of level $N=1$ arising from an indefinite quaternion algebra of fixed discriminant $D$. We study the discrete average of the error term in the hyperbolic circle…
Alternative expressions for calculating the oblate spheroidal radial functions of both kinds R1ml and R2ml are shown to provide accurate values over very large parameter ranges using 64 bit arithmetic, even where the traditional expressions…