Published in Advances in Computational Mathematics
In this paper, we study the error behavior of the nonequispaced fast Fourier transform (NFFT). This approximate algorithm is mainly based on the convenient choice of a compactly supported window function. Here, we consider the continuous Kaiser–Bessel, continuous exp-type, sinh-type, and continuous cosh-type window functions with the same support a...
Published in Advances in Computational Mathematics
In this paper, an improved superconvergence analysis is presented for both the Crouzeix-Raviart element and the Morley element. The main idea of the analysis is to employ a discrete Helmholtz decomposition of the difference between the canonical interpolation and the finite element solution for the first-order mixed Raviart–Thomas element and the m...
Published in Advances in Computational Mathematics
We investigate the use of the so-called variably scaled kernels (VSKs) for learning tasks, with a particular focus on support vector machine (SVM) classifiers and kernel regression networks (KRNs). Concerning the kernels used to train the models, under appropriate assumptions, the VSKs turn out to be more expressive and more stable than the standar...
Published in Advances in Computational Mathematics
We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which a...
Published in Advances in Computational Mathematics
We consider the general question of constructing a partition of unity formed by translates of a compactly supported function g : ℝd → ℂ. In particular, we prove that such functions have a special structure that simplifies the construction of partitions of unity with specific properties. We also prove that it is possible to modify the function g in ...
Published in Advances in Computational Mathematics
A new sampling technique for the application of proper orthogonal decomposition to a set of snapshots has been recently developed by the authors to facilitate a variety of data processing tasks (J. Comput. Phys. 335, 2017). According to it, robust modal expansions result from performing the decomposition on a limited number of relevant snapshots an...
Published in Advances in Computational Mathematics
Consider a linear elliptic PDE defined over a stochastic stochastic geometry a function of N random variables. In many application, quantify the uncertainty propagated to a quantity of interest (QoI) is an important problem. The random domain is split into large and small variations contributions. The large variations are approximated by applying a...
Published in Advances in Computational Mathematics
The design of globally Cs-smooth (s ≥ 1) isogeometric spline spaces over multi-patch geometries with possibly extraordinary vertices, i.e. vertices with valencies different from four, is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods Kapl et al. Comput. Aided Geom. D...
Published in Advances in Computational Mathematics
We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method ...
Published in Advances in Computational Mathematics
In this paper, we propose a three-level linearized implicit difference scheme for the two-dimensional spatial fractional nonlinear complex Ginzburg-Landau equation. We prove that the difference scheme is stable and convergent under mild conditions. The optimal convergence order O(τ2+hx2+hy2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepac...
