Related papers: Approximation, regularization and smoothing of tri…
The total generalized variation extends the total variation by incorporating higher-order smoothness. Thus, it can also suffer from similar discretization issues related to isotropy. Inspired by the success of novel discretization schemes…
I want to prove that all classical techniques of interpolation and approximation as Lagrange, Taylor, Hermite interpolations Beziers interpolants, Quasi interpolants, Box splines and others (radial splines, simplicial splines) are derived…
This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to…
Numerical computations involving rational matrices often benefit from preserving underlying matrix structures such as symmetry, Hermitian properties, or sparsity that reflect physical, geometric, or algebraic characteristics of the system.…
The problem of extrapolation and interpolation of asymptotic series is considered. Several new variants of improving the accuracy of the self-similar approximants are suggested. The methods are illustrated by examples typical of chemical…
We use discrete holomorphic polynomials to prove that, given a refining sequence of critical maps of a Riemann surface, any holomorphic function can be approximated by a converging sequence of discrete holomorphic functions.
In this paper, we investigate the problem of designing compact support interpolation kernels for a given class of signals. By using calculus of variations, we simplify the optimization problem from an infinite nonlinear problem to a finite…
In this work, we present some new integration formulas for any order of accuracy as an application of the B-spline relations obtained in [1]. The resulting rules are defined as a perturbation of the trapezoidal integration method. We prove…
In this paper, we propose a fully discrete soft thresholding trigonometric polynomial approximation on $[-\pi,\pi],$ named Lasso trigonometric interpolation. This approximation is an $\ell_1$-regularized discrete least squares approximation…
Generalizing tensor-product splines to smooth functions whose control nets outline topological polyhedra, bi-cubic polyhedral splines form a piecewise polynomial, first-order differentiable space that associates one function with each…
Detecting slender, overlapping structures remains a challenge in computational microscopy. While recent coordinate-based approaches improve detection, they often produce less accurate splines than pixel-based methods. We introduce a…
The present paper provides a general formula for the dimension of spline space over T-meshes using smoothing cofactor-conformality method. And we introduce a new notion, Diagonalizable T-mesh, over which the dimension formula is only…
Normal multi-scale transform [4] is a nonlinear multi-scale transform for representing geometric objects that has been recently investigated [1, 7, 10]. The restrictive role of the exact order of polynomial reproduction $P_e$ of the…
We present a nodal interpolation method to approximate a subdivision model. The main application is to model and represent curved geometry without gaps and preserving the required simulation intent. Accordingly, we devise the technique to…
In this paper we provide a priori error estimates with explicit constants for both the $L^2$-projection and the Ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids. This extends the results recently…
We introduce a numerical method for reconstructing a multidimensional surface using the gradient of the surface measured at some values of the coordinates. The method consists of defining a multidimensional spline function and minimizing…
The design of fast solvers for isogeometric analysis is receiving a lot of attention due to the challenge that offers to find an algorithm with a robust convergence with respect to the spline degree. Here, we analyze the application of…
A method for computing singular or nearly singular integrals on closed surfaces was presented by J. T. Beale, W. Ying, and J. R. Wilson [Comm. Comput. Phys. 20 (2016), 733--753, arXiv:1508.00265] and applied to single and double layer…
We obtain the asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials with the equidistant nodes $x_k^{(n-1)}=\frac{2k\pi}{2n-1},\ k\in\mathbb{Z},$ in metrics of the spaces $L_p$ on…
The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for…