Abstract
We consider differential Lyapunov and Riccati equations, and generalized versions thereof. Such equations arise in many different areas and are especially important within the field of optimal control. In order to approximate their solution, one may use several different kinds of numerical methods. Of these, splitting schemes are often a very competitive choice. In this article, we investigate the use of graphical processing units (GPUs) to parallelize such schemes and thereby further increase their effectiveness. According to our numerical experiments, large speed-ups are often observed for sufficiently large matrices. We also provide a comparison between different splitting strategies, demonstrating that splitting the equations into a moderate number of subproblems is generally optimal.
Highlights
We are interested in differential matrix equations of Lyapunov or Riccati type, or generalized versions of these
We address questions that naturally arise while solving these equations by splitting methods
We have considered several different splitting schemes based on Leja point interpolation for the computation of matrix exponential actions
Summary
We are interested in differential matrix equations of Lyapunov or Riccati type, or generalized versions of these. For the generalized DLE and DRE versions, an additional linear term SP ST appears in G(P ), where S ∈ Rn×n is a given matrix Such equations arise in the LQR setting, when a stochastic perturbation of multiplicative type is included in the state equation. The hypothesis to be investigated in this paper is that utilizing a graphical processing unit (GPU) to parallelize the schemes may further greatly increase the efficiency Such speed-ups have already been observed for other related methods for DREs [11,12,13] as well as for their steady-state versions: the algebraic Lyapunov and Riccati equations [13, 16].
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