910 research outputs found
Optimal Chebyshev Smoothers and One-sided V-cycles
No full textThe solution to the Poisson equation arising from the spectral element
discretization of the incompressible Navier-Stokes equation requires robust
preconditioning strategies. One such strategy is multigrid. To realize the
potential of multigrid methods, effective smoothing strategies are needed.
Chebyshev polynomial smoothing proves to be an effective smoother. However,
there are several improvements to be made, especially at the cost of symmetry.
For the same cost per iteration, a full V-cycle with order Chebyshev
polynomial smoothing may be substituted with a half V-cycle with order
Chebyshev polynomial smoothing, wherein the smoother is omitted on the up-leg
of the V-cycle. The choice of omitting the post-smoother in favor of higher
order Chebyshev pre-smoothing is shown to be advantageous in cases where the
multigrid approximation property constant, , is large. Results utilizing
Lottes's fourth-kind Chebyshev polynomial smoother are shown. These methods
demonstrate substantial improvement over the standard Chebyshev polynomial
smoother. The authors demonstrate the effectiveness of this scheme in
-geometric multigrid, as well as a 2D model problem with finite differences.Comment: 28 pages, 13 figures, 6 tables (including supplementary materials
Projeto de filtros digitais transicionais Cauer-Chebyshev inverso
No full textDissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico. Programa de Pós-Graduação em Engenharia Elétrica.O presente trabalho apresenta uma metodologia de projeto de filtros transicionais a partir de aproximações não-polinomiais. A implementação desses filtros é realizada com base em técnicas de síntese de filtros digitais IIR, com o objetivo de obter o melhor desempenho de respostas de magnitude, fase e tempo visando uma específica aplicação. A utilização de filtros transicionais não-polinomiais, mais especificamente filtros transicionais Cauer-Chebyshev Inverso, deve-se ao fato de a aproximação Cauer apresentar a menor ordem dentre todas as funções de filtros seletores e de a aproximação Chebyshev Inverso ser também não-polinomial e apresentar melhores características de fase e de tempo em relação à aproximação Cauer. Os exemplos de aplicação mostrados são avaliados através de seis técnicas de projeto de filtros digitais utilizando-se uma abordagem de projeto indireta. Na tentativa de obter o melhor desempenho de cada uma delas são consideradas algumas estratégias de projeto, tais como pré-distorção e principalmente transformação espectral, cujo estudo resultou em procedimentos que melhoram a aplicabilidade dessa última. Assim, é possível compará-las entre si, possibilitando a escolha da melhor estratégia de filtragem para cada problema. Para auxiliar no projeto de filtros digitais como também viabilizar algumas medidas de linearidade de fase consideradas, um software em ambiente Matlab foi desenvolvido
OPTION PRICING WITH V. G. MARTINGALE COMPONENTS
No full textEuropean call options are priced when the uncertainty driving the stock price follows the V. G. stochastic process (Madan and Seneta 1990). The incomplete markets equilibrium change of measure is approximated and identified using the log return mean, variance, and kurtosis. An exact equilibrium interpretation is also provided, allowing inference about relative risk aversion coefficients from option prices. Relative to Black-Scholes, V. G. option values are higher, particularly so for out of the money options with long maturity on stocks with high means. low variances, and high kurtosis.Option, pricing, Variance Gamma, martingale
CHEBYSHEV SPECTRAL METHOD FOR GAS TRANSIENTS IN PIPELINES
No full textA Chebyshev pseudospectral method is proposed for slow transients simulation of gas-transportation systems. The scheme is theoretically investigated. Comparisons with the Lax-Wendroff scheme prove that the Chebyshev method is more efficient in terms of storage and computer time. Infinite-order accuracy for smooth solutions is predicted by the theory and observed in the experiments
A Dual-Band Chebyshev Impedance Transformer
No full textAbstract: This paper presents a simple extension of the Chebyshev quarter-wave multisection transformer synthesis to dual-bond operation. The method is a variation of the classical Chebyshev transformer design procedure, using a suitable 2(nd) order trigonometric polynomial as the argument of the Chebyshev polynomial. As compared to a single-band Chebyshev transformer encompassing both required passbands, the proposed design yields significantly better performance
Chebyshev centers in hyperplanes of
No full textsummary:We give a full characterization of the closed one-codimensional subspaces of , in which every bounded set has a Chebyshev center. It turns out that one can consider equivalently only finite sets (even only three-point sets) in our case, but not in general. Such hyperplanes are exactly those which are either proximinal or norm-one complemented
Chebyshev constants, linear algebra and computation on algebraic curves
No full textIn previous work of the author, directional Chebyshev constants were studied on a complex algebraic curve. Using linear algebra, we study further properties of these constants. We show that Chebyshev constants in different directions are proportional if their corresponding polynomial classes satisfy certain algebraic relations. If V is a quadratic curve, this condition is equivalent to V being irreducible. We conjecture that it is equivalent to irreducibility in general.AM - Accepted Manuscrip
Graeffe's, Chebyshev, and Cardinal's processes for splitting a polynomial into factors,
No full textAbstractNumerical splitting of a real or complex univariate polynomial into factors is the basic step of the divide-and-conquer algorithms for approximating complex polynomial zeros. Such algorithms are optimal (up to polylogarithmic factors) and are quite promising for practical computations. In this paper, we develop some new techniques, which enable us to improve numerical analysis, performance, and computational cost bounds of the known splitting algorithms. In particular, we study a Chebyshev-like modification of Graeffe's lifting iteration (which is a basic block of the splitting algorithms, as well as of several other known algorithms for approximating polynomial zeros), analyze its numerical performance, compare it with Graeffe's, prove some results on numerical stability of both lifting processes (that is, Graeffe's and Chebyshev-like), study their incorporation into polynomial root-finding algorithms, and propose some improvements of Cardinal's recent effective technique for numerical splitting of a polynomial into factors. Our improvement relies, in particular, on a modification of the matrix sign iteration, based on the analysis of some conformal mappings of the complex plane and of techniques of recursive lifting/recursive descending. The latter analysis reveals some otherwise hidden correlations among Graeffe's, Chebyshev-like, and Cardinal's iterative processes, and we exploit these correlations in order to arrive at our improvement of Cardinal's algorithm. Our work may also be of some independent interest for the study of applications of conformal maps of the complex plane to polynomial root-finding and of numerical properties of the fundamental techniques for polynomial root-finding such as Graeffe's and Chebyshev-like iterations
Convexity of Chebyshev sets revisited
No full textA subset C of a normed vector space V is called a Chebyshev set if every point in V admits a unique nearest point in C. In this article we give a novel proof that every Chebyshev set in n-dimensional Euclidean space is convex. This statement is sometimes referred to as the “Bunt–Motzkin Theorem.
Chebyshev polynomials and Pell equations over finite fields
No full textsummary:We shall describe how to construct a fundamental solution for the Pell equation over finite fields of characteristic . Especially, a complete description of the structure of these fundamental solutions will be given using Chebyshev polynomials. Furthermore, we shall describe the structure of the solutions of the general Pell equation
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