GMRES for the Differentiation Operator

@article{Olver2009GMRESFT,
  title={GMRES for the Differentiation Operator},
  author={Sheehan Olver},
  journal={SIAM J. Numer. Anal.},
  year={2009},
  volume={47},
  pages={3359-3373}
}
  • S. Olver
  • Published 1 October 2009
  • Computer Science, Mathematics
  • SIAM J. Numer. Anal.
We investigate using the gmres method with the differentiation operator. This operator is unbounded and thus does not fall into the framework of existing Krylov subspace theory. We establish conditions under which a function can be approximated by its own derivatives in a domain of the complex plane. These conditions are used to determine when gmres converges. This algorithm outperforms traditional quadrature schemes for a large class of highly oscillatory integrals, even when the kernel of… 

Figures from this paper

GMRES for Oscillatory Matrix-Valued Differential Equations

  • S. Olver
  • Computer Science, Mathematics
  • 2011
This work investigates the use of Krylov subspace methods to solve linear, oscillatory ODEs and demonstrates the effectiveness of this method by computing error and Mathieu functions.

Continuous analogues of Krylov methods for differential operators

The developed Krylov methods are practical iterative BVP solvers that are particularly efficient when a fast operator-function product is available and an operator preconditioner that ensures that an approximate solution is computed after a finite number of iterations.

Shifted GMRES for oscillatory integrals

  • S. Olver
  • Computer Science
    Numerische Mathematik
  • 2010
A new method is presented that satisfies all of the following properties of the oscillatory integral: high asymptotic order, stability, avoiding deformation into the complex plane and insensitivity to oscillations in f.

Fast, numerically stable computation of oscillatory integrals with stationary points

We present a numerically stable way to compute oscillatory integrals. For each additional frequency, only a small, well-conditioned linear system with a Hessenberg matrix must be solved, and the

Convergence of the conjugate gradient method with unbounded operators

It is proved the convergence of the conjugate gradient iterates to an exact solution to the inverse problem in the most general case where the self-adjoint, non-negative operator is unbounded and with minimal, technically unavoidable assumptions on the initial guess of the iterative algorithm.

Computing highly oscillatory integrals

Two classes of composite moment-free numerical quadratures for computing highly oscillatory integrals having integrable singularities and stationary points are developed and results show that the proposed methods outperform methods published most recently.

Continuous Analogues of Krylov Subspace Methods for Differential Operators

Analogues of the conjugate gradient method, minimum residual method, and generalized minimum residual method are derived for solving boundary value problems (BVPs) involving ordinary differential e...

' s personal copy Automatic spectral collocation for integral , integro-differential , and integrally reformulated differential equations q

Automatic Chebyshev spectral collocation methods for Fredholm and Volterra integral and integro-differential equations have been implemented as part of the chebfun software system. This system

References

SHOWING 1-10 OF 17 REFERENCES

Moment-free numerical integration of highly oscillatory functions

The aim of this paper is to derive new methods for numerically approximating the integral of a highly oscillatory function. We begin with a review of the asymptotic and Filon-type methods developed

Efficient quadrature of highly oscillatory integrals using derivatives

  • A. IserlesS. P. Nørsett
  • Mathematics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2005
In this paper, we explore quadrature methods for highly oscillatory integrals. Generalizing the method of stationary phase, we expand such integrals into asymptotic series in inverse powers of the

A method for numerical integration on an automatic computer

A new method for the numerical integration of a “well-behaved” function over a finite range of argument is described. It consists essentially of expanding the integrand in a series of Chebyshev

Moment-free numerical approximation of highly oscillatory integrals with stationary points

  • S. Olver
  • Mathematics
    European Journal of Applied Mathematics
  • 2007
This article presents a method for the numerical quadrature of highly oscillatory integrals with stationary points. We begin with the derivation of a new asymptotic expansion, which has the property

GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems

We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from t...

Restarted GMRES for Shifted Linear Systems

This work develops a variant of the restarted GMRES method exhibiting the same advantage and investigates its convergence for positive real matrices in some detail and applies it to speed up "multiple masses" calculations arising in lattice gauge computations in quantum chromodynamics, one of the most time-consuming supercomputer applications.

An Extension of MATLAB to Continuous Functions and Operators

About eighty MATLAB functions from plot and sum to svd and cond have been overloaded so that one can work with "chebfun" objects using almost exactly the usual MATLAB syntax.

Applied asymptotic analysis

Fundamentals: Themes of asymptotic analysis The nature of asymptotic approximations Asymptotic analysis of exponential integrals: Fundamental techniques for integrals Laplace's method for asymptotic

Convergence of Iterations for Linear Equations

1. Motivation, problem and notation.- 1.1 Motivation.- 1.2 Problem formulation.- 1.3 Usual tools.- 1.4 Notation for polynomial acceleration.- 1.5 Minimal error and minimal residual.- 1.6