Abstract

Rational Krylov methods are applicable to a wide range of scientific computing problems, and the rational Arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs, which gives rise to a near-optimal parallelization strategy that allows us to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using several numerical examples from different application areas.

Keywords

  1. rational Krylov
  2. orthogonalization
  3. parallelization

MSC codes

  1. 68W10
  2. 65F25
  3. 65G50
  4. 65F15

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Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: S197 - S221
ISSN (online): 1095-7197

History

Submitted: 9 June 2016
Accepted: 10 January 2017
Published online: 26 October 2017

Keywords

  1. rational Krylov
  2. orthogonalization
  3. parallelization

MSC codes

  1. 68W10
  2. 65F25
  3. 65G50
  4. 65F15

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