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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

@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}
}

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