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\begin{document}
\title{
A GPU-Accelerated Envelope-Following Method for Switching Power Converter Simulation
}
\author{
  \IEEEauthorblockN{
    Xue-Xin Liu\IEEEauthorrefmark{1},
    Sheldon X.-D. Tan\IEEEauthorrefmark{1},
    Hai Wang\IEEEauthorrefmark{1}, and
    Hao Yu\IEEEauthorrefmark{2}
  }
  \IEEEauthorblockA{
    \IEEEauthorrefmark{1}Dept. Electrical Engineering,
    University  of California, Riverside, CA 92521
  }
  \IEEEauthorblockA{
    \IEEEauthorrefmark{2}School of Electrical \&amp; Electronic Engineering,
    Nanyang Technological University
  }
  \thanks{
    This research was supported in part by NSF grants under
    No.~CCF-1017090 
    No.~OISE-1051797, and
    No.~OISE-0929699.
  }
}

%\IEEEpubid{978-3-9810801-8-6/DATE12/@2012 EDAA}

\maketitle

\begin{abstract}
% Power converters have seen a surge of new trends and novel
% applications due to their widespread use  in renewable energy
% systems and emerging  hybrid and purely-electric vehicles. More
% efficient simulation  techniques for power converters are urgently
% needed to meet more design constraints.
In this paper, we propose a new envelope-following parallel transient analysis method for 
the general switching power converters. The new method first exploits 
the parallelisim in the envelope-following method
and parallelize the Newton update solving part,
which is the most computational expensive, in GPU platforms 
to boost the simulation performance.
To further speed up the iterative GMRES
solving for Newton update equation in the envelope-following
method, we apply the matrix-free Krylov basis generation
technique, which was previously used for RF simulation. 
Last, the new method also applies more robust
Gear-2 integration to compute the sensitivity matrix instead of
traditional integration methods. 
%Furthermore, the resulted Gear-2 and matrix-free GMRES have been
Experimental results from several integrated on-chip power
converters show that the proposed GPU envelope-following algorithm leads to
about 10$\times$ speedup compared to its CPU counterpart,
and 100$\times$ faster than the traditional envelop-following methods
while still keeps the similar accuracy.
\end{abstract}

\input intro.tex
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%\input bdf.tex  % now inside gpu.tex now
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\input exp.tex

\section{Conclusion} \label{sec:conclusion}
A new envelope-following method for transient analysis of
switching power converters has been introduced.
First, the computationally expensive step,
the solving of Newton update equation,
has been parallelized on CUDA-enabled GPU platforms
with iterative GMRES solver to boost performance of the analysis method.
To further speed up the GMRES solving for Newton update equation,
we have employed the matrix-free Krylov basis generation technique.
The proposed method also applies the more robust Gear-2 integration
to compute the sensitivity matrix.
Experimental results from several integrated
on-chip power converters have shown that the proposed GPU
envelope-following algorithm can lead to about 10$\times$
speedup compared to its CPU counterpart,
and 100$\times$ faster than the traditional
envelop-following methods while still keeps the similar accuracy.


\bibliographystyle{unsrt} % IEEEtranS
\bibliography{./envelope,../../bib/interconnect,../../bib/architecture,../../bib/simulation}
% ./rfsim.bib

\end{document}
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