Search results for "Mathematics::Numerical Analysis"
showing 10 items of 57 documents
Mixed Convolutions and Zak Transforms
2015
In this chapter we introduce the mixed continuous–discrete and discrete–discrete convolutions. Important special cases of such convolutions are the polynomial and discrete splines, respectively. The Zak transforms, which are introduced in the chapter, provide integral representation of signals, which, in the following chapters, serves as a tool for the design of splines and spline-wavelets and operations over them. The exponential splines, which are the Zak transforms of polynomial and discrete B-splines are introduced. Explicit formulas for the characteristic functions of splines’ spaces are derived.
Qualitative analysis of matrix splitting methods
2001
Abstract Qualitative properties of matrix splitting methods for linear systems with tridiagonal and block tridiagonal Stieltjes-Toeplitz matrices are studied. Two particular splittings, the so-called symmetric tridiagonal splittings and the bidiagonal splittings, are considered, and conditions for qualitative properties like nonnegativity and shape preservation are shown for them. Special attention is paid to their close relation to the well-known splitting techniques like regular and weak regular splitting methods. Extensions to block tridiagonal matrices are given, and their relation to algebraic representations of domain decomposition methods is discussed. The paper is concluded with ill…
A Property on Singularities of NURBS Curves
2002
We prove that if a.n open Non Uniform Rational B-Spline curve of order k has a singular point, then it belongs to both curves of order k - 1 defined in the k - 2 step of the de Boor algorithm. Moreover, both curves are tangent at the singular point.
Quadrature Formula Based on Interpolating Polynomials: Algorithmic and Computational Aspects
2007
The aim of this article is to obtain a quadrature formula for functions in several variables and to analyze the algorithmic and computational aspects of this formula. The known information about the integrand is {λi(f)}i=1n, where λi are linearly independent linear functionals. We find a form of the coefficients of the quadrature formula which can be easy used in numerical calculations. The main algorithm we use in order to obtain the coefficients and the remainder of the quadrature formula is based on the Gauss elimination by segments method. We obtain an expression for the exactness degree of the quadrature formula. Finally, we analyze some computational aspects of the algorithm in the pa…
A brief overview on the numerical behavior of an implicit meshless method and an outlook to future challenges
2015
In this paper recent results on a leapfrog ADI meshless formulation are reported and some future challenges are addressed. The method benefits from the elimination of the meshing task from the pre-processing stage in space and it is unconditionally stable in time. Further improvements come from the ease of implementation, which makes computer codes very flexible in contrast to mesh based solver ones. The method requires only nodes at scattered locations and a function and its derivatives are approximated by means of a kernel representation. A perceived obstacle in the implicit formulation is in the second order differentiations which sometimes are eccesively sensitive to the node configurat…
Highly Accurate Conservative Finite Difference Schemes and Adaptive Mesh Refinement Techniques for Hyperbolic Systems of Conservation Laws
2007
We review a conservative finite difference shock capturing scheme that has been used by our research team over the last years for the numerical simulations of complex flows [3, 6]. This scheme is based on Shu and Osher’s technique [9] for the design of highly accurate finite difference schemes obtained by flux reconstruction procedures (ENO, WENO) on Cartesian meshes and Donat-Marquina’s flux splitting [4]. We then motivate the need for mesh adaptivity to tackle realistic hydrodynamic simulations on two and three dimensions and describe some details of our Adaptive Mesh Refinement (AMR) ([2, 7]) implementation of the former finite difference scheme [1]. We finish the work with some numerica…
Cell-average WENO with progressive order of accuracy close to discontinuities with applications to signal processing
2020
In this paper we translate to the cell-average setting the algorithm for the point-value discretization presented in S. Amat, J. Ruiz, C.-W. Shu, D. F. Y\'a\~nez, A new WENO-2r algorithm with progressive order of accuracy close to discontinuities, submitted to SIAM J. Numer. Anal.. This new strategy tries to improve the results of WENO-($2r-1$) algorithm close to the singularities, resulting in an optimal order of accuracy at these zones. The main idea is to modify the optimal weights so that they have a nonlinear expression that depends on the position of the discontinuities. In this paper we study the application of the new algorithm to signal processing using Harten's multiresolution. Se…
Cubic Local Splines on Non-uniform Grid
2015
In this chapter, two types of local cubic splines on non-uniform grids are described: 1. The simplest variation-diminishing splines and 2. The quasi-interpolating splines. The splines are computed by a simple fast computational algorithms that utilizes a relation between the splines and cubic interpolation polynomials. Those splines can serve as an efficient tool for real-time signal processing. As an input, they use either clean or noised arbitrarily-spaced samples. On the other hand, the capability to adapt the grid to the structure of an object and minimal requirements to the operating memory are great advantages for off-line processing of signals and multidimensional data arrays.
Periodic Discrete and Discrete-Time Splines
2018
Periodic discrete splines with different periods and spans are introduced in Sect. 3.4 of Volume I (Averbuch, Neittaanmaki and Zheludev, Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, Springer, Berlin, 2014) [2]. In this chapter, we regard periodic discrete splines as a base for the design of periodic discrete-time wavelets, wavelet packets and wavelet frames. Therefore, only the discrete splines whose spans are 2 are outlined. These discrete splines are linear combinations of the discrete B-splines. So also, the so-called discrete-time splines are discussed in the chapter that are linear combinations of the discrete-time B-splines. The discrete-time B-s…
Quasi-interpolating and Smoothing Local Splines
2015
In this chapter, local quasi-interpolating and smoothing splines are described. Although approximation properties of local spline are similar to properties of the global interpolating and smoothing splines, their design does not require the IIR filtering of the whole data array. The computation of a local spline value at some point utilizes only a few adjacent grid samples. Therefore, local splines can be used for real-time processing of signals and for the design of FIR filter banks generating wavelets and wavelet frames (Chaps. 12 and 14). In the chapter, local splines of different orders are designed and their approximation properties are established which are compared with the propertie…