1,720,986 research outputs found
A barycentric trigonometric Hermite interpolant via an iterative approach
Full text linkIn this work we construct an Hermite interpolant starting from basis
functions that satisfy a Lagrange property. In fact, we extend and generalise
an iterative approach, introduced by Cirillo and Hormann (2018) for the
Floater-Hormann family of interpolants. Secondly, we apply this scheme to
produce an effective barycentric rational trigonometric Hermite interpolant at
general ordered nodes using as basis functions the ones of the trigonometric
interpolant introduced by Berrut (1988). For an easy computational
construction, we calculate analytically the differentation matrix. Finally, we
conclude with various examples and a numerical study of the rate of convergence
at equidistant nodes and conformally mapped nodes
Bounding the Lebesgue constant for a barycentric rational trigonometric interpolant at periodic well-spaced nodes
No full textA well-known result in linear approximation theory states that the norm of the operator, known as the Lebesgue constant, of polynomial interpolation on an interval grows only logarithmically with the number of nodes, when these are Chebyshev points. Results like this are important for studying the conditioning of the approximation. A cosine change of variable shows that polynomial interpolation at Chebyshev points is just the special case for even functions of trigonometric interpolation (on the circle) at equidistant points. The Lebesgue constant of the latter grows logarithmically, also for functions with no particular symmetry. In the present work, we show that a linear rational generalization of the trigonometric interpolant enjoys a logarithmically growing Lebesgue constant for more general sets of nodes, namely periodic well-spaced ones, patterned after those introduced for an interval by Bos et al. (2013) few years ago. An important special case are conformally shifted equispaced points, for which the rational trigonometric interpolant is known to converge exponentially
A linear barycentric rational interpolant on starlike domains
Full text linkWhen an approximant is accurate on an interval, it is only natural to try to extend it to multidimensional domains. In the present article we make use of the fact that linear rational barycentric interpolants converge rapidly toward analytic and several-times differentiable functions to interpolate them on two-dimensional starlike domains parametrized in polar coordinates. In the radial direction, we engage interpolants at conformally shifted Chebyshev nodes, which converge exponentially for analytic functions. In the circular direction, we deploy linear rational trigonometric barycentric interpolants, which converge similarly rapidly for periodic functions but now for conformally shifted equispaced nodes. We introduce a variant of a tensor-product interpolant of the above two schemes and prove that it converges exponentially for two-dimensional analytic functions—up to a logarithmic factor—and with an order limited only by the order of differentiability for real functions (provided th..
A periodic map for linear barycentric rational trigonometric interpolation
No full textConsider the set of equidistant nodes in [0, 2π), θk:=k·2πn,k=0,⋯,n−1. For an arbitrary 2π–periodic function f(θ), the barycentric formula for the corresponding trigonometric interpolant between the θk’s is [Formula presented] where cst(·):=ctg(·) if the number of nodes n is even, and cst(·):=csc(·) if n is odd. Baltensperger [3] has shown that the corresponding barycentric rational trigonometric interpolant given by the right-hand side of the above equation for arbitrary nodes introduced in [9] converges exponentially toward f when the nodes are the images of the θk’s under a periodic conformal map. In the present work, we introduce a simple periodic conformal map which accumulates nodes in the neighborhood of an arbitrarily located front, as well as its extension to several fronts. Despite its simplicity, this map allows for a very accurate approximation of smooth periodic functions with steep gradients
Tchakaloff-like compression of QMC volume and surface integration on the union of balls
Full text linkWe present an algorithm for Tchakaloff-like compression of quasi-Monte Carlo volume and surface integration on an arbitrary union of balls, via non-negative least squares. We also provide the corresponding Matlab codes together with several numerical tests
CQMC: an improved code for low-dimensional Compressed Quasi-MonteCarlo cubature
Full text linkAn improved version of Compressed Quasi-MonteCarlo cubature (CQMC) on large low-discrepancy
samples is implemented, on 2D and 3D regions with complex shape. The algorithms rests on the concept
of Tchakaloff set and on NNLS solution of a sequence of “small” moment-matching systems. Examples of
area and volume integration are provided
Stable discontinuous mapped bases: the Gibbs–Runge-Avoiding Stable Polynomial Approximation (GRASPA) method
Full text linkThe 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
Electrically-tunable active metamaterials for damped elastic wave propagation control
Full text linkAn electrically-tunable metamaterial is herein designed for the active control of damped elastic waves. The periodic device is conceived including both elastic phases and a piezoelectric phase, shunted by a dissipative electric circuit whose impedance/admittance can be adjusted on demand. As a consequence, the frequency band structure of the metamaterial can be modified to meet design requirements, possibly changing over time. A significant issue is that in the presence of a dissipative circuit, the frequency spectra are obtained by solving eigen-problems with rational terms. This circumstance makes the problem particularly difficult to treat, either resorting to analytical or numerical techniques. In this context, a new derationalization strategy is proposed to overcome some limitations of standard approaches. The starting point is an infinite-dimensional rational eigen-problem, obtained by expanding in their Fourier series the periodic terms involved in the governing dynamic equations. A special derationalization is then applied to the truncated eigen-problem. The key idea is exploiting a LU factorization of the matrix collecting the rational terms. The method allows to considerably reduce the size of the problem to solve with respect to available techniques in literature. This strategy is successfully applied to the case of a three-phase metamaterial shunted by a series RLC circuit with rational admittance
Polynomial mapped bases: theory and applications
Full text linkIn this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial mapped bases allowing, for instance, to incorporate data or function discontinuities in a suitable mapping function. The new technique substantially mitigates the Runge's and Gibbs effects
Homogenization of composite materials reinforced with unidirectional fibres with complex curvilinear cross section: a virtual element approachy
Full text linkThe paper presents an augmented curvilinear virtual element method to determine homogenized in-plane shear material moduli of long-fibre reinforced composites in the framework of asymptotic homogenization method. The new virtual element combine an exact representation of the curvilinear computational geometry for complex fibre cross section shapes through an innovative two-dimensional cubature suite for NURBS-like polygonal domains. A selection of representative numerical tests supports the accuracy and efficiency of the proposed approach for both doubly periodic and random fibre arrangement with matrix domain
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