Spectral Methods

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Ruo Li - One of the best experts on this subject based on the ideXlab platform.

  • computing nearly singular solutions using pseudo Spectral Methods
    Journal of Computational Physics, 2007
    Co-Authors: Ruo Li
    Abstract:

    In this paper, we investigate the performance of pseudo-Spectral Methods in computing nearly singular solutions of fluid dynamics equations. We consider two different ways of removing the aliasing errors in a pseudo-Spectral method. The first one is the traditional 2/3 dealiasing rule. The second one is a high (36th) order Fourier smoothing which keeps a significant portion of the Fourier modes beyond the 2/3 cut-off point in the Fourier spectrum for the 2/3 dealiasing method. Both the 1D Burgers equation and the 3D incompressible Euler equations are considered. We demonstrate that the pseudo-Spectral method with the high order Fourier smoothing gives a much better performance than the pseudo-Spectral method with the 2/3 dealiasing rule. Moreover, we show that the high order Fourier smoothing method captures about 12–15% more effective Fourier modes in each dimension than the 2/3 dealiasing method. For the 3D Euler equations, the gain in the effective Fourier codes for the high order Fourier smoothing method can be as large as 20% over the 2/3 dealiasing method. Another interesting observation is that the error produced by the high order Fourier smoothing method is highly localized near the region where the solution is most singular, while the 2/3 dealiasing method tends to produce oscillations in the entire domain. The high order Fourier smoothing method is also found be very stable dynamically. No high frequency instability has been observed. In the case of the 3D Euler equations, the energy is conserved up to at least six digits of accuracy throughout the computations. � 2007 Elsevier Inc. All rights reserved.

Jie Shen - One of the best experts on this subject based on the ideXlab platform.

  • Accurate and Efficient Spectral Methods for Elliptic PDEs in Complex Domains
    Journal of Scientific Computing, 2020
    Co-Authors: Yiqi Gu, Jie Shen
    Abstract:

    We develop accurate and efficient Spectral Methods for elliptic PDEs in complex domains using a fictitious domain approach. Two types of Petrov–Galerkin formulations with special trial and test functions are constructed, one is suitable only for the Poisson equation but with a rigorous error analysis, the other works for general elliptic equations but its analysis is not yet available. Our numerical examples demonstrate that our Methods can achieve Spectral convergence, i.e., the convergence rate only depends on the smoothness of the solution.

  • Spectral Methods algorithms analysis and applications
    2011
    Co-Authors: Jie Shen, Tao Tang, Lilian Wang
    Abstract:

    Along with finite differences and finite elements, Spectral Methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic Spectral algorithms, as well as a systematical presentation of basic convergence theory and error analysis for Spectral Methods. Readers of this book will be exposed to a unified framework for designing and analyzing Spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains. The book contains a large number of figures which are designed to illustrate various concepts stressed in the book. A set of basic matlab codes has been made available online tohelp the readers to develop their own Spectral codes for their specific applications.

  • some recent advances on Spectral Methods for unbounded domains
    2008
    Co-Authors: Jie Shen, Lilian Wang
    Abstract:

    We present in this paper a unified framework for analyzing the Spectral Methods in unbounded domains using mapped Jacobi, Laguerre and Hermite functions. A detailed comparison of the convergence rates of these Spectral Methods for solutions with typical decay behaviors is carried out, both theoretically and computationally. A brief review on some of the recent advances in the Spectral Methods for unbounded domains is also presented. AMS subject classifications: 65N35, 65N22,65F05, 35J05

  • stable and efficient Spectral Methods in unbounded domains using laguerre functions
    SIAM Journal on Numerical Analysis, 2000
    Co-Authors: Jie Shen
    Abstract:

    Stable and efficient Spectral Methods using Laguerre functions are proposed and analyzed for model elliptic equations on regular unbounded domains. It is shown that Spectral-Galerkin approximations based on Laguerre functions are stable and convergent with Spectral accuracy in the usual (not weighted) Sobolev spaces. Efficient, accurate, and well-conditioned algorithms using Laguerre functions are developed and implemented. Numerical results indicating the Spectral convergence rate and effectiveness of these algorithms are presented.

Alex Townsend - One of the best experts on this subject based on the ideXlab platform.

  • Fast Poisson solvers for Spectral Methods
    Ima Journal of Numerical Analysis, 2019
    Co-Authors: Daniel Fortunato, Alex Townsend
    Abstract:

    Abstract Poisson’s equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference (FD) and finite element Methods, fast Poisson solvers for Spectral Methods have remained elusive. Here we derive Spectral Methods for solving Poisson’s equation on a square, cylinder, solid sphere and cube that have optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom used to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of FD matrices, our solver exploits a separated spectra property that holds for our carefully designed Spectral discretizations. Without parallelization we can solve Poisson’s equation on a square with 100 million degrees of freedom in under 2 min on a standard laptop.

  • fast poisson solvers for Spectral Methods
    arXiv: Numerical Analysis, 2017
    Co-Authors: Daniel Fortunato, Alex Townsend
    Abstract:

    Poisson's equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference and finite element Methods, fast Poisson solvers for Spectral Methods have remained elusive. Here, we derive Spectral Methods for solving Poisson's equation on a square, cylinder, solid sphere, and cube that have an optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom required to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of finite difference matrices, our solver exploits a separated spectra property that holds for our Spectral discretizations. Without parallelization, we can solve Poisson's equation on a square with 100 million degrees of freedom in under two minutes on a standard laptop.

Mustafa Khammash - One of the best experts on this subject based on the ideXlab platform.

  • A Spectral Methods-based solution of the Chemical Master Equation for gene regulatory networks
    2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012
    Co-Authors: João P. Hespanha, Mustafa Khammash
    Abstract:

    We present a new method to approximate the time evolution of the probability density function (PDF) for molecule counts in gene regulatory networks modeled by the Chemical Master Equation (CME). A key feature of our method is that molecular states can be aggregated to reduce the computational burden without the need for assumptions like time-scales separation. The CME is amenable to the use of various Spectral Methods adapted from partial differential equations and our method results from expanding the solution using carefully selected basis functions. The method is illustrated in the context of an example taken from the field of systems biology.

Javier Hormigo - One of the best experts on this subject based on the ideXlab platform.

  • Texture segmentation and analysis using local Spectral Methods
    Seventh International Symposium on Signal Processing and Its Applications 2003. Proceedings., 2003
    Co-Authors: Gabriel Cristóbal, M. Forero-vargas, R. Redondo, S Fischer, Javier Hormigo
    Abstract:

    In this paper we present two new Methods for texture segmentation and analysis using local Spectral Methods. The first approach to the problem is to use a modular pattern detection in textured images based on the use of a pseudo-wigner distribution (PWD) followed by a decorrelation procedure that consists of a principal component analyzer (for texture segmentation). The goal is to combine the advantages of a high Spectral resolution of a joint representation given by the pseudo-Wigner distribution (PWD) with an effective adaptive principal component analysis. The second approach is based on a modular procedure that encompasses a region of interest extraction procedure followed by a log-prolate filtering scheme (for texture classification). Performance of both Methods is evaluated in different application domains: fabric defective textures, epithelial cell cultures and a diatom's classification scenario yielding excellent results over other conventional spatial or Spectral Methods.

  • ISSPA (1) - Texture segmentation and analysis using local Spectral Methods
    Seventh International Symposium on Signal Processing and Its Applications 2003. Proceedings., 2003
    Co-Authors: Gabriel Cristóbal, M. Forero-vargas, R. Redondo, S Fischer, Javier Hormigo
    Abstract:

    In this paper we present two new Methods for texture segmentation and analysis using local Spectral Methods. The first approach to the problem is to use a modular pattern detection in textured images based on the use of a pseudo-wigner distribution (PWD) followed by a decorrelation procedure that consists of a principal component analyzer (for texture segmentation). The goal is to combine the advantages of a high Spectral resolution of a joint representation given by the pseudo-Wigner distribution (PWD) with an effective adaptive principal component analysis. The second approach is based on a modular procedure that encompasses a region of interest extraction procedure followed by a log-prolate filtering scheme (for texture classification). Performance of both Methods is evaluated in different application domains: fabric defective textures, epithelial cell cultures and a diatom's classification scenario yielding excellent results over other conventional spatial or Spectral Methods.