\chapterauthor{X.-X. Liu}{Dept. Electrical Engineering,
  University  of California, Riverside, CA 92521}
\chapterauthor{S. X.-D. Tan}{Dept. Electrical Engineering,
  University  of California, Riverside, CA 92521}
\chapterauthor{H. Wang}{Univ. of Electronics Science and Technology of China,
  Chengdu, Sichuan, China}
\chapterauthor{H. Yu}{School of Electrical \& Electronic Engineering,
  Nanyang Technological University, Singapore}

%  \thanks{
%    This research was supported in part by NSF grants under
%    No.~CCF-1017090 
%    No.~OISE-1051797, and
%    No.~OISE-0929699.
%  }

\newcommand{\ud}{\,\mathrm{d}}

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\chapter[GPU-Accelerated Envelope-Following Method]{A GPU-Accelerated Envelope-Following Method for Switching Power Converter Simulation}

% \section{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 chapter, 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.

\input{Chapters/chapter16/intro.tex}
\input{Chapters/chapter16/ef.tex}
%\input bdf.tex  % now inside gpu.tex now
\input{Chapters/chapter16/gpu.tex}
\input{Chapters/chapter16/exp.tex}

\section{Summary}
\label{sec:summary}
In this chapter, we present a new envelope-following method for
transient analysis of switching power converters.  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 envelope-following methods while still keeps the similar
accuracy.


\section{Glossary}
\begin{Glossary}
\item[Envelope-Following] In transient simulation of switching power circuits,
nodal voltage waveforms in neighboring high frequency clock cycles are similar,
but not exactly the duplicates. Envelope-following technique approximates
the slowly changing transient trend over a lot of clock cycles
without calculating waveforms in all cycles.
\end{Glossary}

\putbib[Chapters/chapter16/biblio16]
%\bibliography{./envelope,../../bib/interconnect,../../bib/architecture,../../bib/simulation}
