\chapterauthor{Xuexin Liu, Sheldon Xiang-Dong Tan}{Dept. Electrical Engineering,
  University  of California, Riverside, CA 92521, USA}
%\chapterauthor{Sheldon Xiang-Dong Tan}{Dept. Electrical Engineering,  University  of California, Riverside, CA 92521}
\chapterauthor{Hai Wang}{Univ. of Electronics Science and Technology of China,
  Chengdu, Sichuan, China}
\chapterauthor{Hao 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}}

%\renewcommand{\algorithmicrequire}{\textbf{Input:}}
%\renewcommand{\algorithmicensure}{\textbf{Output:}}
%\renewcommand{\algorithmiccomment}[1]{\% \textit{#1}}

\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}

\clearpage
\section{Summary}
\label{sec:summary}
In this chapter, we have presented a new envelope-following method for
transient analysis of switching power converters.  First, the
computationally expensive step, the solving of the 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 the 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 keep 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 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}
