Research reports

A mixed-precision algorithm for the solution of Lyapunov equations on hybrid CPU-GPU platforms

by P. Benner and P. Ezzatti and D. Kressner and E. S. Quintana-Orti and A. Remón

(Report number 2009-40)

Abstract
We describe a hybrid Lyapunov solver based on the matrix sign function that accelerates the intensive parts of the computation using a graphics processor (GPU) while executing the remaining operations in a general-purpose multicore processor. The initial stage of the iterative solver operates in singleprecision arithmetic, to exploit the many-core parallelism of current GPUs, returning a full-rank factor to the solution of the equation. To improve this approximate solution, the second stage consists of an efficient iterative refinement procedure that allows to cheaply recover full double-precision accuracy. The combination of these two stages results in a mixed-precision algorithm, that exploits the capabilities of both general-purpose multi-core processors and many-core GPUs, overlapping critical computations. Experiments using a platform equipped with two Intel Xeon QuadCore processors and an Nvidia Tesla C1060 show the efficiency of this approach to solve Lyapunov equations arising in practical model reduction applications: compared with a classical implementation that exploits the parallelism of a general-purpose processor using a multi-threaded implementation of BLAS and operates in double-precision, our hybrid algorithm delivers 4.24–6.46 speed-ups while attaining the same accuracy in the solution.

Keywords: Lyapunov equations, matrix sign function, graphics processors, multi-core processors, control theory

BibTeX
@Techreport{BEKQR09_421,
  author = {P. Benner and P. Ezzatti and D. Kressner and E. S. Quintana-Orti and A. Remón},
  title = {A mixed-precision algorithm for the solution of Lyapunov equations on hybrid CPU-GPU platforms},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-40},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-40.pdf },
  year = {2009}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).

JavaScript has been disabled in your browser