From: couturie Date: Wed, 6 Feb 2013 08:06:40 +0000 (+0100) Subject: add ch16 X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_gpu.git/commitdiff_plain/4932967a1c684dc3f7ed04c19144101278b79972?ds=inline;hp=-c add ch16 --- 4932967a1c684dc3f7ed04c19144101278b79972 diff --git a/BookGPU/BookGPU.tex b/BookGPU/BookGPU.tex index edd313e..fab2a67 100755 --- a/BookGPU/BookGPU.tex +++ b/BookGPU/BookGPU.tex @@ -3,6 +3,7 @@ \usepackage{amssymb} \usepackage{amsfonts,amssymb} \usepackage{amsmath} +\usepackage{amscd} \usepackage{graphicx} \usepackage{subfigure} \usepackage{makeidx} @@ -27,6 +28,12 @@ \usepackage{pgfplots} \usepackage{pgf} \usepackage{tikz} +\usepackage{url} +\usepackage{dsfont} +\usepackage{commath} +\usepackage{stmaryrd} +\usepackage{moreverb} +\usepackage{commath} %% \usepackage{listings} %% \usepackage{subfigure} @@ -98,6 +105,11 @@ \captionsetup[lstlisting]{format=listing,labelfont=white,textfont=white, singlelinecheck=false, margin=0pt, font={bf,footnotesize}} +\DeclareGraphicsRule{.pdftex}{pdf}{*}{} + +\DeclareGraphicsRule{.pstex}{eps}{*}{} + + \makeindex @@ -130,17 +142,19 @@ \setcounter{page}{1} \part{This is a Part} -\include{Chapters/chapter1/ch1} -\include{Chapters/chapter2/ch2} -\include{Chapters/chapter3/ch3} -\include{Chapters/chapter5/ch5} -\include{Chapters/chapter6/ch6} -\include{Chapters/chapter7/ch7} -\include{Chapters/chapter8/ch8} -\include{Chapters/chapter9/ch9} -\include{Chapters/chapter11/ch11} -\include{Chapters/chapter14/ch14} -\include{Chapters/chapter15/ch15} +%% \include{Chapters/chapter1/ch1} +%% \include{Chapters/chapter2/ch2} +%% \include{Chapters/chapter3/ch3} +%% \include{Chapters/chapter5/ch5} +%% \include{Chapters/chapter6/ch6} +%% \include{Chapters/chapter7/ch7} +%% \include{Chapters/chapter8/ch8} +%% \include{Chapters/chapter9/ch9} +%% \include{Chapters/chapter11/ch11} +%% \include{Chapters/chapter14/ch14} +%% \include{Chapters/chapter15/ch15} + \include{Chapters/chapter16/ch16} +%% \include{Chapters/chapter18/ch18} \bibliographystyle{hep} %%%\bibliography{biblio} diff --git a/BookGPU/Chapters/chapter16/bdf.tex b/BookGPU/Chapters/chapter16/bdf.tex new file mode 100644 index 0000000..dcf6313 --- /dev/null +++ b/BookGPU/Chapters/chapter16/bdf.tex @@ -0,0 +1,179 @@ +\subsection{Gear-2 based sensitivity calculation} +\label{sec:gear} + +The Gear-2 integration method is a backward differentiation +formula (BDF), which approximates the derivative of a function +using information from past few steps. Gear-2 method has been +shown to be more suitable for many practical problems such as +stiff problems where circuit behavior is affected by different time constants +(fast ones with large poles and slow ones with small poles)~\cite{Vlach:Book'94}. +Switching power converters and RF systems are typically stiff systems +as waveforms changing in two different time scales are involved. + +% Although on one time step, Gear-2 +% involves more computation than backward-Euler and trapezoidal method, +% it has better convergence and stability, and usually allows larger +% time step. As a result, fewer time points are needed for a given +% transient sweep range, and the simulation is speeded up with the same +% accuracy + +Given the DAE~(\ref{eq:dae}), at the $i$-th time step, +Gear-2 integration\index{Gear-2 integration} +approximates the derivative $\ud q(x(t))/\ud t$ +by a two-step backward finite difference, +\[ + \frac{\ud q(x(t_i))}{\ud t} + %\ud q(x(t_i)) / \ud t + = \frac{1}{h_i} + \left[ q(x(t_{i})) - \alpha_1^i q(x(t_{i-1})) - \alpha_2^i q(x(t_{i-2}))\right], +\] +where the coefficients $\alpha$'s are chosen to satisfy Gear's +backward differentiation formula~\cite{Gear:book'71}. +For uniformly discretised time steps, +the index $i$ in $h_i$, $\alpha_1^i$ and $\alpha_2^i$ can be omitted. + +For the first step $t_1$, +only the initial condition at $t_0$ is available. +Therefore backward-Euler is used, i.e., +\begin{equation}\label{eq:BE} + \tfrac{1}{h_1}[q(x_1)-q(x_0)] + f(x_1) = b_1. +\end{equation} +Since $x_0$ directly affects the solution of $x_1$, +the sensitivity matrix\index{sensitivity matrix} up to now is +\begin{equation}\label{eq:sens1} + \frac{\ud x_1}{\ud x_0} + = \left[\frac{1}{h_1}C_1+G_1\right]^{-1} \frac{C_0}{h_1} + = J_1^{-1} \frac{C_0}{h_1}. +\end{equation} +%For the sake of simplicity, +Let $J_i$ denote $(1/h_i)C_i + G_i$ +in the remaining part of this chapter. +In addition, the LU\index{LU} factorizations of $J_i$ are already calculated +since they are used to solve the circuit state at +each time step before we calculate the sensitivity. + +Next, for the solution at $t_2$, +with the previous two steps information available, +Gear-2 integration can be applied, +\begin{equation}\label{eq:Gear_t2} + \tfrac{1}{h_2}[q(x_2) - \alpha_1 q(x_1) - \alpha_2 q(x_0)] + f(x_2) = b_2. +\end{equation} +In view of the sensitivity of $x_2$ with respect to the changes of $x_0$, +\eqref{eq:Gear_t2} indicates that $x_2$'s perturbation can be +traced back to $x_0$ along two paths: +$x_2$ is directly affected by $x_0$, +and $x_2$ is also affected indirectly by $x_0$ via $x_1$. +That is, +\[ %begin{equation} + \frac{\ud x_2}{\ud x_0} = \frac{\partial x_2}{\partial x_1}\frac{\ud x_1}{\ud x_0} + + \frac{\partial x_2}{\partial x_0}, +\] %end{equation} +where the two partial derivatives are +\[ + \frac{\partial x_2}{\partial x_1} = J_2^{-1} \frac{\alpha_1}{h_2} C_1, \qquad + \frac{\partial x_2}{\partial x_0} = J_2^{-1} \frac{\alpha_2}{h_2} C_0, +\] +and $\ud x_1/\ud x_0$ %, the sensitivity of $x_1$ with respect to $x_0$, +is calculated previously in \eqref{eq:sens1}. +Thus, the sensitivity matrix deduced from \eqref{eq:Gear_t2} is +\begin{equation}\label{eq:sens2} + \frac{\ud x_2}{\ud x_0} + = J_2^{-1} \frac{\alpha_1}{h_2} C_1 \cdot J_1^{-1} \frac{C_0}{h_1} + + J_2^{-1} \frac{\alpha_2}{h_2} C_0. +\end{equation} + +Likewise, for the third time step $t_3$, +apply the Gear-2 integration formula to DAE~(\ref{eq:dae}), +\begin{equation}\label{eq:Gear_t3} + \tfrac{1}{h_3} [q(x_3) - \alpha_1 q(x_2) - \alpha_2 q(x_1)] + f(x_3) = b_3, +\end{equation} +and the chain rule for sensitivity is +\[ %begin{equation}\label{eq:sens3} +% \begin{split} +% \frac{\ud x_3}{\ud x_0} &= \frac{\partial x_3}{\partial x_2}\frac{\ud x_2}{\ud x_0} +% + \frac{\partial x_3}{\partial x_1}\frac{\ud x_1}{\ud x_0} \\ +% &= J_3^{-1} \frac{\alpha_1}{h_3} C_2 \frac{\ud x_2}{\ud x_0} +% + J_3^{-1} \frac{\alpha_2}{h_3} C_1 \frac{\ud x_1}{\ud x_0}. +% \end{split} + \frac{\ud x_3}{\ud x_0} \!=\! \frac{\partial x_3}{\partial x_2}\frac{\ud x_2}{\ud x_0} + + \frac{\partial x_3}{\partial x_1}\frac{\ud x_1}{\ud x_0} + \!=\! J_3^{-1} \!\left[ \frac{\alpha_1}{h_3} C_2 \frac{\ud x_2}{\ud x_0} + \!+\!\frac{\alpha_2}{h_3} C_1 \frac{\ud x_1}{\ud x_0}\right], +\] %end{equation} +where both $\ud x_2/\ud x_0$ and $\ud x_1/\ud x_0$ +are ready from the previous two time steps, i.e., Eqs.~(\ref{eq:sens2}) and~(\ref{eq:sens1}). +Follow this chain rule in the aforementioned recursive style, +the sensitivity matrix for Gear-2 integration is computed +along the remaining time steps up to the $M$-th step, +\begin{equation}\label{eq:sensM} +% \begin{split} +% J = \frac{\ud x_M}{\ud x_0} +% & = J_M^{-1} \frac{\alpha_1}{h_M} C_{M-1} \frac{\ud x_{M-1}}{\ud x_0} \\ +% &\quad + J_M^{-1} \frac{\alpha_2}{h_M} C_{M-2} \frac{\ud x_{M-2}}{\ud x_0}. +% \end{split} +J\! =\! \frac{\ud x_M}{\ud x_0} + \! =\! J_M^{-1} \! \left[ \frac{\alpha_1}{h_M} C_{M-1} \frac{\ud x_{M-1}}{\ud x_0} + \! + \! \frac{\alpha_2}{h_M} C_{M-2} \frac{\ud x_{M-2}}{\ud x_0}\right]. +\end{equation} + +%% \begin{algorithm} +%% \caption{The matrix-free\index{matrix-free} method for +%% Krylov subspace\index{Krylov subspace} construction.} +%% \label{alg:mf_Gear} +%% \begin{algorithmic}[1] +%% \REQUIRE current Krylov subspace basis vector $v$, +%% time step lengths $h_i$, +%% saved $C_i$ matrices and LU\index{LU} factors of $J_i$, $i=0,\ldots,M$ +%% \ENSURE matrix-vector product $r$, such that $r = A v$ +%% \STATE solve $J_1 p_2 = h_1^{-1} C_0 v$ for $p_2$ +%% \STATE solve $J_2 p_1 = h_2^{-1} \left( \alpha_1 C_1 p_2 + \alpha_2 C_0 v \right)$ for $p_1$ +%% \FOR{$i=3 \ldots M$} +%% \STATE solve +%% $J_i p_0 = \alpha_1 h_i^{-1} C_{i-1} p_1 + \alpha_2 h_i^{-1} C_{i-2} p_2$ +%% for $p_0$ \label{line:mf_Gear_loop} +%% \STATE $p_2 = p_1$ +%% \STATE $p_1 = p_0$ +%% \ENDFOR +%% \STATE $r = (k-1) p_0 - k v$ \label{line:shift} +%% \end{algorithmic} +%% \end{algorithm} + + +%It is easy to judge that, as $J$ is dense, explicitly +%computing $J$ is expensive and impractical for large problems. +%In addition, with a dense matrix, +%both direct or iterative solvers will be inefficient. +%Fortunately, one iterative solver, GMRES, +%only requires $J x$ (actually $Ax$), +%instead of an explicit $J$, to fulfill the solving process. +%This virtue greatly saves computation cost. +Note that as the matrix-free\index{matrix-free} GMRES method is applied, +which only requires matrix-vector multiplication and no explicit $J$ is required, +as explained in Algorithm~\ref{alg:mf_Gear}. +% Next section will explain how to use this technique for +% our proposed Gear-2 envelope-following. + +With matrix-free method, the matrix-vector multiplication +in Line~\ref{line:mvp} of Algorithm~\ref{alg:GMRES} +is replaced by the iteration shown in Algorithm~\ref{alg:mf_Gear}, +which calculates the multiplication product of the Gear-2 sensitivity we +encounter in envelope-following and a basis vector in the Krylov +subspace. +Note that the scaling and shift of $J$ in $A=(k-1)J-kI$, +as described in Eq.~\eqref{eq:A}, +is taken into consideration by Line~\ref{line:shift} +of Algorithm~\ref{alg:mf_Gear}. + +For the matrix-free generation of new basis vectors, it is +straightforward to apply CUBLAS and CUSPARSE routines, and some +customized GPU kernel functions to implement +Algorithm~\ref{alg:mf_Gear} with the stored LU matrices of $J_i$ and +$C_i$ mentioned earlier. For example in Line~\ref{line:mf_Gear_loop}, +as the iteration index $i$ traverses all the $M$ time points' +matrix information, +sparse matrix-vector multiplication \verb|csrmv| +is first called to calculate $C_{i-1} p_1$ and $C_{i-2} p_2$. +And after the two resulted vectors are combined by \verb|axpy| of CUBLAS, +$p_0$ is solved for by \verb|trsv|, since as we noted before, +$J_i$ is already in LU factorization form when transient simulation +at step $i$ finished. diff --git a/BookGPU/Chapters/chapter16/biblio16.bib b/BookGPU/Chapters/chapter16/biblio16.bib new file mode 100644 index 0000000..1252282 --- /dev/null +++ b/BookGPU/Chapters/chapter16/biblio16.bib @@ -0,0 +1,171 @@ + +@Book{Trzynadlowski:Book'10, + author = "A. M. Trzynadlowski", + title = "Introduction to Modern Power Electronics", + publisher = "Wiley", + edition = "Second", + year = "2010" +} + +@InProceedings{Kundert:ICCAD'88, + author = {K. Kundert and J. White and A. Sangiovanni-Vincentelli}, + title = {An envelope following method for the efficient transient simulation + of switching power and filter circuits}, + booktitle = {Proc. ICCAD}, + pages = {446--449}, + year = 1988, + month = {Oct.} +} + + +@Article{White:TPE'91, + author = {J. White and S. Leeb}, + title = {An envelope-following approach to switching + power converter simulation}, + journal = {IEEE Trans. Power Electron.}, + year = 1991, + volume = 6, + number = 2, + pages = {303--307}, + month = {Apr.} +} + +@InProceedings{Feldmann:ICCAD'96, + author = {P. Feldmann and J. Roychowdhury}, + title = {Computation of circuit waveform envelopes using an efficient, + matrix-decomposed harmonic balance algorithm}, + booktitle = {Proc. ICCAD}, + pages = {295--300}, + year = 1996, + month = {Nov.} +} + +@Book{Krein:book'97, + author = {P. Krein}, + title = {Elements of Power Electronics}, + publisher = {Oxford University Press}, + year = 1997 +} + +@InProceedings{Telichevesky:DAC'95, + author = {R. Telichevesky and K. Kundert and J. White}, + title = {Efficient steady-state analysis based on matrix-free {K}rylov-subspace methods}, + booktitle = dac, + pages = {480--484}, + year = 1995, +} + +@Misc{IntelMC'06, + author = {{Intel Corporation}}, + title = {Intel Multi-Core Processors, Making the move to quad-core and beyond ({W}hite {P}aper)}, + year = 2006, + note = {\url{http://www.intel.com/multi-core}} +} + +@Misc{AMDMC'06, + author = {{AMD Inc.}}, + title = {Multi-Core Processors---The Next Evolution in Computing ({W}hite {P}aper)}, + year = 2006, + note = {\url{http://multicore.amd.com}} +} + +@MISC{nvidia, + author = {{NVIDIA Corporation}}, + key = {openmp}, + note = {{h}ttp://www.nvidia.com}, + year = 2011 +} + +@Book{Kirk:Book'10, + Author="D. B. Kirk and W.-M. Hwu", + Title="Programming Massively Parallel Processors: A Hands-on Approach", + Year="2010", + Publisher="Morgan Kaufmann Publishers Inc.", + Address="San Francisco, CA", +} + +@MISC{gpgpu, + author = {D. G\"{o}ddeke}, + title = {General-Purpose Computation Using Graphics Harware}, + howpublished = {\url{http://www.gpgpu.org/}}, + year = 2011 +} + +@misc{cuda, + author = {{{NVIDIA} Corporation}}, + title = {{CUDA} ({C}ompute {U}nified {D}evice {A}rchitecture)}, + note = {\url{http://www.nvidia.com/object/cuda_home.html}}, + year = 2011 +} + +@MISC{streamSDK, + author = {{AMD Inc.}}, + title = {{AMD} {S}team {SDK}}, + howpublished = {\url{http://developer.amd.com/gpu/ATIStreamSDK}}, + year = 2011 +} + +@MISC{openCL, + author = {{Khronos Group}}, + title = {Open {C}omputing {L}anguage ({O}pen{CL})}, + howpublished = {\url{http://www.khronos.org/opencl}}, + year = 2011 +} + +@Misc{CUDA_Zone, + title = {{CUDA} Community Showcase}, + howpublished = {\url{http://www.nvidia.com/}} +} + +@BOOK{Golub:Book'96, + AUTHOR = "G. H. Golub and C. Van Loan", + TITLE = "Matrix Computations", + PUBLISHER = "The Johns Hopkins University Press", + YEAR = "1996", + edition = "3rd", +} + +@Misc{CUDASDK, + author = {{NVIDIA}}, + year = 2011, + note = {\url{http://developer.nvidia.com/cuda-toolkit-40}} +} + +@Book{Gear:book'71, + author = {C. W. Gear}, + title = {Numerical Initial Value Problems in Ordinary Differential Equations}, + publisher = {Prentice-Hall}, + year = 1971, + address = {Englewood Cliffs, NJ} +} + +@Misc{ngspice, + title = {{NGSPICE}}, + howpublished = {\url{http://ngspice.sourceforge.net/}} +} + +@INPROCEEDINGS{Kato:COMPEL'06, + author={Kato, T. and others}, + booktitle={IEEE COMPEL'06}, + title={Envelope Following Analysis of an Autonomous Power Electronic System}, + year={2006}, + month={July}, + pages={29--33} +} + +@Book{Vlach:Book'94, + author = {J. Vlach and K. Singhal}, + title = {Computer Methods for Circuit Analysis and Design}, + publisher = {Van Nostrand Reinhold}, + year = {1995}, + address = {New York, NY}, +} + +@conference{LiuTan1:DATE'12, +author={X.-X. Liu and S. X.-D. Tan and H. Wang and H. Yu}, +title={A {GPU}-accelerated envelope-following method for switching power converter simulation”}, +booktitle= date, +year={2012}, +pages = {1349-1354}, +month ={March}, +} diff --git a/BookGPU/Chapters/chapter16/ch16.aux b/BookGPU/Chapters/chapter16/ch16.aux new file mode 100644 index 0000000..7d1293a --- /dev/null +++ b/BookGPU/Chapters/chapter16/ch16.aux @@ -0,0 +1,101 @@ +\relax +\@writefile{toc}{\author{X.-X. Liu}{}} +\@writefile{toc}{\author{S. X.-D. Tan}{}} +\@writefile{toc}{\author{H. Wang}{}} +\@writefile{toc}{\author{H. Yu}{}} +\@writefile{loa}{\addvspace {10\p@ }} +\@writefile{toc}{\contentsline {chapter}{\numberline {1}GPU-Accelerated Envelope-Following Method}{3}} +\@writefile{lof}{\addvspace {10\p@ }} +\@writefile{lot}{\addvspace {10\p@ }} +\@writefile{toc}{\contentsline {section}{\numberline {1.1}Introduction}{3}} +\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} +\newlabel{fig:ef1}{{1.1(a)}{5}} +\newlabel{sub@fig:ef1}{{(a)}{5}} +\newlabel{fig:ef2}{{1.1(b)}{5}} +\newlabel{sub@fig:ef2}{{(b)}{5}} +\@writefile{lof}{\contentsline {figure}{\numberline {1.1}{\ignorespaces Transient envelope-following analysis. (Both two figures reflect backward-Euler style envelope-following.)\relax }}{5}} +\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {Illustration of one envelope skip.}}}{5}} +\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {The envelope changes in a slow time scale.}}}{5}} +\newlabel{fig:ef_intro}{{1.1}{5}} +\@writefile{toc}{\contentsline {section}{\numberline {1.2}The envelope-following method in a nutshell}{6}} +\newlabel{sec:ef}{{1.2}{6}} +\newlabel{eq:dae}{{1.1}{6}} +\newlabel{eq:Newton}{{1.2}{7}} +\newlabel{eq:A}{{1.3}{7}} +\@writefile{toc}{\contentsline {section}{\numberline {1.3}New parallel envelope-following method}{8}} +\newlabel{sec:gmres}{{1.3}{8}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.1}GMRES solver for Newton update equation}{8}} +\@writefile{lof}{\contentsline {figure}{\numberline {1.2}{\ignorespaces The flow of envelope-following method.\relax }}{9}} +\newlabel{fig:ef_flow}{{1.2}{9}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.2}Parallelization on GPU platforms}{10}} +\newlabel{sec:gpu}{{1.3.2}{10}} +\@writefile{lof}{\contentsline {figure}{\numberline {1.3}{\ignorespaces GPU parallel solver for envelope-following update.\relax }}{11}} +\newlabel{fig:gmres}{{1.3}{11}} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.3}Gear-2 based sensitivity calculation}{12}} +\newlabel{sec:gear}{{1.3.3}{12}} +\newlabel{eq:BE}{{1.4}{12}} +\newlabel{eq:sens1}{{1.5}{12}} +\newlabel{eq:Gear_t2}{{1.6}{13}} +\newlabel{eq:sens2}{{1.7}{13}} +\newlabel{eq:Gear_t3}{{1.8}{13}} +\newlabel{eq:sensM}{{1.9}{13}} +\@writefile{toc}{\contentsline {section}{\numberline {1.4}Numerical examples}{14}} +\newlabel{sec:exp}{{1.4}{14}} +\@writefile{lof}{\contentsline {figure}{\numberline {1.4}{\ignorespaces Diagram of a zero-voltage quasi-resonant flyback converter.\relax }}{15}} +\newlabel{fig:flyback}{{1.4}{15}} +\@writefile{lof}{\contentsline {figure}{\numberline {1.5}{\ignorespaces Illustration of power/ground network model.\relax }}{15}} +\newlabel{fig:pg}{{1.5}{15}} +\@writefile{lot}{\contentsline {table}{\numberline {1.1}{\ignorespaces CPU and GPU time comparisons (in seconds) for solving Newton update equation with the proposed Gear-2 sensitivity. \relax }}{15}} +\newlabel{table:circuit}{{1.1}{15}} +\newlabel{fig:flybackWhole}{{1.6(a)}{16}} +\newlabel{sub@fig:flybackWhole}{{(a)}{16}} +\newlabel{fig:flybackZoom}{{1.6(b)}{16}} +\newlabel{sub@fig:flybackZoom}{{(b)}{16}} +\@writefile{lof}{\contentsline {figure}{\numberline {1.6}{\ignorespaces Flyback converter solution calculated by envelope-following. The red curve is traditional SPICE simulation result, and the back curve is the envelope-following output with simulation points marked.\relax }}{16}} +\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {The whole plot}}}{16}} +\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {Detail of one EF simulation period}}}{16}} +\newlabel{fig:flyback_wave}{{1.6}{16}} +\@writefile{lof}{\contentsline {figure}{\numberline {1.7}{\ignorespaces Buck converter solution calculated by envelope-following.\relax }}{17}} +\newlabel{fig:buck_wave}{{1.7}{17}} +\@writefile{toc}{\contentsline {section}{\numberline {1.5}Summary}{17}} +\newlabel{sec:summary}{{1.5}{17}} +\@writefile{toc}{\contentsline {section}{\numberline {1.6}Glossary}{18}} +\@writefile{toc}{\contentsline {section}{Bibliography}{18}} +\@setckpt{Chapters/chapter16/ch16}{ +\setcounter{page}{20} +\setcounter{equation}{9} +\setcounter{enumi}{0} +\setcounter{enumii}{0} +\setcounter{enumiii}{0} +\setcounter{enumiv}{22} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{part}{1} +\setcounter{chapter}{1} +\setcounter{section}{6} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{figure}{7} +\setcounter{table}{1} +\setcounter{numauthors}{0} +\setcounter{parentequation}{0} +\setcounter{subfigure}{0} +\setcounter{lofdepth}{1} +\setcounter{subtable}{0} +\setcounter{lotdepth}{1} +\setcounter{lstnumber}{1} +\setcounter{ContinuedFloat}{0} +\setcounter{AlgoLine}{0} +\setcounter{algocfline}{0} +\setcounter{algocfproc}{0} +\setcounter{algocf}{0} +\setcounter{proposition}{0} +\setcounter{theorem}{0} +\setcounter{exercise}{0} +\setcounter{example}{0} +\setcounter{definition}{0} +\setcounter{proof}{0} +\setcounter{lstlisting}{0} +} diff --git a/BookGPU/Chapters/chapter16/ch16.tex b/BookGPU/Chapters/chapter16/ch16.tex new file mode 100644 index 0000000..bfa5d6c --- /dev/null +++ b/BookGPU/Chapters/chapter16/ch16.tex @@ -0,0 +1,85 @@ +\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}} + +%\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} + +\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} diff --git a/BookGPU/Chapters/chapter16/date12envelope.tex b/BookGPU/Chapters/chapter16/date12envelope.tex new file mode 100644 index 0000000..67f9016 --- /dev/null +++ b/BookGPU/Chapters/chapter16/date12envelope.tex @@ -0,0 +1,139 @@ +\documentclass[conference]{IEEEtran} + +\usepackage{graphicx} +\usepackage{algorithm,algorithmic} +\usepackage{multirow} +\usepackage{subfigure} +\usepackage{booktabs} +\usepackage{epstopdf} +\usepackage{graphicx} +\usepackage[dvips]{color} +\usepackage{psfrag} +\usepackage{pstricks} +\usepackage{amssymb,amsmath} +%\usepackage{authblk} +\usepackage{cite} +\usepackage{url} + +\DeclareGraphicsRule{.pdftex}{pdf}{*}{} +\graphicspath{{matlab/}} + + +\newtheorem{myformulation}{Formulation} +\newtheorem{mytheorem}{Theorem} +\newtheorem{mylemma}{Lemma} +\newtheorem{mydefinition}{Definition} +%\ninept + +% correct bad hyphenation here +%\hyphenation{op-tical net-works semi-conduc-tor IEEEtran} +%\renewcommand{\baselinestretch}{0.99} + + +\renewcommand{\algorithmicrequire}{\textbf{Input:}} +\renewcommand{\algorithmicensure}{\textbf{Output:}} +\renewcommand{\algorithmiccomment}[1]{\% \textit{#1}} + + +\newcommand{\setRn}{\mathbb{R}_{}^N} +\newcommand{\setR}{\mathbb{R}} +\newcommand{\ud}{\,\mathrm{d}} +\newcommand{\res}[1]{\mathbf{r}^{(#1)}} +\newcommand{\vtr}[1]{\mathbf{#1}} +\newcommand{\mtx}[1]{\mathbf{#1}} +\newcommand{\mtxPhi}{\boldsymbol{\phi}} +\newcommand{\mT}{\mathrm{T}} + +\IEEEoverridecommandlockouts + + +\renewcommand{\baselinestretch}{0.97} + +\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 \& 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 +\input ef.tex +%\input bdf.tex % now inside gpu.tex now +\input gpu.tex +\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} diff --git a/BookGPU/Chapters/chapter16/ef.tex b/BookGPU/Chapters/chapter16/ef.tex new file mode 100644 index 0000000..b0967a9 --- /dev/null +++ b/BookGPU/Chapters/chapter16/ef.tex @@ -0,0 +1,146 @@ +\section{The envelope-following method in a nutshell} +\label{sec:ef} + +In transient analysis\index{transient analysis}, +a nonlinear circuit's behavior can be represented +by a coupled set of nonlinear first-order +differential algebraic equations\index{differential algebraic equations} +(DAE) of the form +\begin{equation}\label{eq:dae} + \frac{\ud}{\ud t} q(x(t)) + f(x(t)) = b(t), +\end{equation} +where $x(t)\in \mathbb{R}^N$ is the vector of circuit variables, +usually comprising node voltages and possibly branch currents, +$f(\cdot)$ is a nonlinear function that maps $x(t)$ to a vector +of $N$ entries most of +which are sums of resistive currents at a node, +$q(\cdot)$ %is also a nonlinear function that +maps $x(t)$ to a vector of $N$ entries of +capacitor charges or inductor fluxes, +and $b(t)$ contains the input source values. +For many power electronics circuits, the input +switching signal is known %as a circuit parameter +and is periodic with clock period $T$. + +Assume the time inside one period $T$ has been discretised +into $M$ time steps, $0=t_0 < t_1 < t_2 < \cdots < t_M=T$, and the +$i$-th step length is $h_i = t_i-t_{i-1}$. +In practice, the discretisation is nonuniform to +control truncation error\index{truncation error} and convergence. +Then, given an initial condition $x(0)$ at $t=0$, +numerical integration is applied to find the time domain solution of +circuit state $x(t)$ at each time step till $t=T$. +%, in a consequent time series. + +For those circuits whose carrier waveforms\index{carrier waveforms} +vary slowly in a large number of periods, +the envelope-following\index{envelope-following} method only integrates the +DAE in several selected periods and then jumps over to a new time point. +By repeating this ``simulate and skip'' action, +envelope-following method attains its efficiency +compared to conventional transient analysis, but still can accurately +obtain the slow varying envelope. + +For example, if the clock period is $T$, and the suitable jump +interval for the envelope is $k$ periods, then the envelope step is +$kT$. Suppose the state at time $t=mT$ is known as $x(mT)$, and the +envelope-following wants to obtain the state at the next envelope +step, $x((m+k)T)$. If the envelope step is much larger than the clock +period ($k$ is much bigger than one), envelope-following will lead to +significant savings in simulation time. + +% One simple way to estimate $x((m+k)T)$ is to use the information at +% $mT$ and $(m-1)T$, which leads to the forward-Euler method as +To estimate $x((m+k)T)$, +a forward-Euler\index{forward-Euler} style jumping relies only on $x(mT)$ and $x((m-1)T)$, +i.e., +\[ + x((m+k)T) + = + x(mT) + k\left[ x(mT)-x((m-1)T) \right]. +\] +However, this approach is inefficient due to its restriction on +envelope step $k$, since larger $k$ usually causes instability. +Instead, backward-Euler\index{backward-Euler} jumping, +%and the equation becomes +\[ + x((m+k)T)-x(mT) = k\left[x((m+k)T)-x((m+k-1)T)\right], +\] +allows larger envelope steps. +Here $x((m+k-1)T)$ is the unknown variable to be solved +by Newton iteration\index{Newton iteration}, +and $x((m+k)T)$ is dependent on $x((m+k-1)T)$ +in each iteration. +% Forward-Euler may be used to generate the initial guess. +The Newton update equation in each iteration is thus expressed as +% x^{j+1}((m+k-1)T) = x^{j}((m+k-1)T) - A^{-1} \left[(k-1)x^j((m+k)T) - kx^j((m+k-1)T)+ x(mT)\right], +%\begin{equation} \label{eq:Newton} +%\begin{split} +\begin{multline} \label{eq:Newton} + \Delta x((m+k-1)T)= A^{-1} \left[(k-1)x^j((m+k)T) \right. \\ + \left. - kx^j((m+k-1)T)+ x(mT)\right], +\end{multline} +%\end{split} +%\end{equation} +where the Jacobian matrix\index{Jacobian matrix} $A$ is computed as +a combination of circuit sensitivity matrix\index{sensitivity matrix} +and identity matrix, as +\begin{equation} \label{eq:A} + A = (k-1)\frac{\ud x((m+k)T)}{\ud x((m+k-1)T)} -kI + = (k-1)J-kI. +\end{equation} +In each Newton iteration, +$x((m+k-1)T)$ is used as initial condition to calculate +$x((m+k)T)$ by integrating the DAE in one clock period. +Meanwhile, %circuit matrices, for example, +the conductance matrix $G_i=\ud f(x(t_i))/\ud x(t_i)$ +and the capacitance matrix $C_i=\ud q(x(t_i))/\ud x(t_i)$ +at each time step %in this period, +are used to derive the %circuit +sensitivity matrix $J= \ud x((m+k)T) / \ud x((m+k-1)T)$. +% i.e., +% \[ +% x((m+k)T) = \Phi(x((m+k-1)T),(m+k-1)T,(m+k)T). +% \] + +Different integration rules can be applied to +the computation of sensitivity matrix. +It can be easily derived that, +%for one signal period with $M$ time steps, +if the DAE is integrated with backward-Euler rule, +the sensitivity is +\[ + J = \frac{\ud x_M}{\ud x_0} + = \prod_{i=1}^{M} + \left( \frac{1}{h_i} C_i + G_i \right)^{-1} + \frac{1}{h_i}C_{i-1} +% \frac{\ud x_n}{\ud x_0} +% = \left(\frac{1}{h_n} C_n + G_n\right)^{-1} +% \frac{1}{h_n}C_{n-1} \frac{\ud x_{n-1}}{\ud x_0} +\] +%is the sensitivity matrix +% In the next section, we will formulate the sensitivity in +% Gear-2 scheme. +% and for trapezoidal integration, +% \[ +% \frac{\ud x_n}{\ud x_0} +% = \left(\frac{1}{h_n} C_n + \frac{1}{2} G_n\right)^{-1} +% \left(\frac{1}{h_n} C_{n-1} - \frac{1}{2} G_{n-1}\right) +% \frac{\ud x_{n-1}}{\ud x_0}. +% \] + +In summary, the envelope-following method is fundamentally +a boosted version of traditional transient analysis, +with certain skips over several periods and a Newton iteration +to update or correct the errors brought by the skips, +as is exhibited by Fig.~\ref{fig:ef_flow}. +\begin{figure}[!tb] +\centering +\resizebox{.7\textwidth}{!}{\input{./Chapters/chapter16/figures/ef_flow.pdf_t}} +\caption{The flow of envelope-following method.} +\label{fig:ef_flow} +%\end{center} +\end{figure} + + + diff --git a/BookGPU/Chapters/chapter16/exp.tex b/BookGPU/Chapters/chapter16/exp.tex new file mode 100644 index 0000000..6fd3eb9 --- /dev/null +++ b/BookGPU/Chapters/chapter16/exp.tex @@ -0,0 +1,121 @@ +\section{Numerical examples} +\label{sec:exp} + +The presented algorithm has been prototyped and numerical experiments +have been carried out on a server which has an Intel Xeon quad-core +CPU with 2.0~GHz clock speed, and 24~GBytes memory. The GPU card +mounted on this server is NVIDIA's Tesla~C2070 (Fermi), which contains +448 cores (14 MPs $\times$ 32 cores per MP) running at a 1.30~GHz and +has 4~GBytes on-chip memory. Some initial results have been published +in~\cite{LiuTan1:DATE'12}. + +The envelope-following method with the proposed Gear-2 +sensitivity matrix computation is added to an open-source +SPICE\index{SPICE}, implemented in C~\cite{ngspice}. +Our envelope-following program is implemented by following +the algorithm mentioned in~\cite{Kato:COMPEL'06}. +To solve the Newton update equation, +different methods are used to compare the computation time, +such as direct LU, GMRES with explicitly formed matrix, +and GMRES with implicit matrix-vector multiplication (matrix-free). +Moreover, the matrix-free method is also incorporated to the same SPICE +simulator using CUDA C programming interface, as described in +Section~\ref{sec:gpu}. + +\begin{figure} + \centering + \resizebox{.8\textwidth}{!}{\input{./Chapters/chapter16/figures/resonant_flyback.pdf_t}} + \caption{Diagram of a zero-voltage quasi-resonant flyback converter.} + \label{fig:flyback} +\end{figure} + + +\begin{figure}%[hbfp] + \centering + \resizebox{.6\textwidth}{!}{\input{./Chapters/chapter16/figures/pgMesh.pdf_t}} + \caption{Illustration of power/ground network model.} + \label{fig:pg} +\end{figure} + +\begin{figure} +\centering +\subfigure[The whole plot]{ + \includegraphics[width=.6\textwidth]{./Chapters/chapter16/figures/flyback_wave_emb.eps} + \label{fig:flybackWhole} +} +\subfigure[Detail of one EF simulation period]{ + \includegraphics[width=.6\textwidth]{./Chapters/chapter16/figures/flyback_zoomin_emb.eps} + \label{fig:flybackZoom} +} +\caption{Flyback converter solution calculated by envelope-following. +The red curve is traditional SPICE simulation result, and +the back curve is the envelope-following output with simulation points +marked.} +\label{fig:flyback_wave} +\end{figure} + + +\begin{figure}%[hbfp] + \centering + \includegraphics[width=.6\textwidth]{./Chapters/chapter16/figures/buck_wave_emb.eps} + \caption{Buck converter solution calculated by envelope-following.} + \label{fig:buck_wave} +\end{figure} + +We use several integrated on-chip converters as simulation examples +to measure running time and speedup. They include a Buck converter, +a quasi-resonant flyback converter (shown in Fig.~\ref{fig:flyback}), +and two boost converters. +Each converter is directly integrated with on-chip power grid networks, +since the performance of converters should be studied with their loads and +we can easily observe the waveforms at different nodes in a power +grid (see Fig.~\ref{fig:pg} for a simplified power grid structure). + +Fig.~\ref{fig:flyback_wave} +and Fig.~\ref{fig:buck_wave} +shows the waveform at output node of the resonant flyback converter +and the Buck converter. +Note that on the envelope curve, the darker +dots in separated segments indicate the real simulation points were +calculated in those cycles, and the segments without dots are the +envelope jumps where no simulation were done. +It can be verified that the proposed Gear-2 envelope-following method +produces a envelope matching the original waveform well. + +\begin{table} +\centering +\caption{CPU and GPU time comparisons (in seconds) for solving Newton update equation + with the proposed Gear-2 sensitivity. +} +\vspace{0.1in} +\label{table:circuit} +{%\normalsize +\begin{tabular}{@{}c|c|c|c|c|c|c@{}} +\hline\hline +Circuit & Nodes & Direct & Explicit & \multicolumn{3}{c}{Implicit GMRES}\\ \cline{5-7} + & & LU & GMRES & CPU & GPU & X \\ +\hline +Buck & 910 & 423.8 & 420.3 & 36.8 & 3.9 & 9.4$\times$ \\ +Flyback & 941 & 462.4 & 459.6 & 64.5 & 7.4 & 8.7$\times$ \\ +Boost-1 & 976 & 695.1 & 687.7 & 73.2 & 6.2 & 11.8$\times$ \\ +Boost-2 & 1093 & 729.5 & 720.8 & 71.0 & 8.5 & 9.9$\times$ \\ +\hline\hline +\end{tabular} +} +\end{table} + +For the comparison of running time spent in solving +Newton update equation, Table~\ref{table:circuit} lists the time +costed by direct method, explicit GMRES, matrix-free GMRES, +and GPU matrix-free GMRES. All methods carry out the Gear-2 based +envelope-following method, but they handle the sensitivity and +equation solving in different implementation steps. +It is obvious that as long as the sensitivity matrix is explicitly formed, +such as the cases in direct method and explicitly GMRES, +the cost is much higher than the implicit methods. +When matrix-free technique is applied to generate matrix-vector +products implicitly, the computation cost is greatly reduced. +Thus, for the same example, implicit GMRES would be one order +of magnitude faster than explicit GMRES. Furthermore, our GPU parallel +implementation of implicit GMRES makes this method even faster, +with a further 10$\times$ speedup. diff --git a/BookGPU/Chapters/chapter16/figures/buck_wave.eps b/BookGPU/Chapters/chapter16/figures/buck_wave.eps new file mode 100755 index 0000000..4ff6a1a --- /dev/null +++ b/BookGPU/Chapters/chapter16/figures/buck_wave.eps @@ -0,0 +1,1992 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: MATLAB, The Mathworks, Inc. Version 7.8.0.347 (R2009a). 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839 1331 mt 5843 1331 L +5843 1331 mt 5843 1331 L + 839 363 mt 5843 363 L +5843 363 mt 5843 363 L +SO +12 w + 839 4237 mt 5843 4237 L + 839 4237 mt 839 363 L + 839 4237 mt 839 4186 L +%%IncludeResource: font Helvetica +/Helvetica /ISOLatin1Encoding 240 FMSR + + 773 4494 mt +(0) s +2090 4237 mt 2090 4186 L +1924 4494 mt +(0.5) s +3341 4237 mt 3341 4186 L +3275 4494 mt +(1) s +4592 4237 mt 4592 4186 L +4426 4494 mt +(1.5) s +5843 4237 mt 5843 4186 L +5777 4494 mt +(2) s +5208 4844 mt +(x 10) s +%%IncludeResource: font Helvetica +/Helvetica /ISOLatin1Encoding 160 FMSR + +5661 4696 mt +(-3) s + 839 4237 mt 889 4237 L +%%IncludeResource: font Helvetica +/Helvetica /ISOLatin1Encoding 240 FMSR + + 671 4326 mt +(4) s + 839 3268 mt 889 3268 L + 671 3357 mt +(6) s + 839 2300 mt 889 2300 L + 671 2389 mt +(8) s + 839 1331 mt 889 1331 L + 538 1420 mt +(10) s + 839 363 mt 889 363 L + 538 452 mt +(12) s +gs 839 363 5005 3875 rc +36 w +/c8 { 1.000000 0.000000 0.000000 sr} bdef +c8 +1 0 1 0 1 1 1 0 1 0 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pdfTextClipPath aload pop + {/moveto cvx} + {/lineto cvx} + {/curveto cvx} + {/closepath cvx} + pathforall ] def + currentpoint newpath moveto + } { + 8 {pop} repeat + } ifelse + 0 pdfTextRise neg pdfTextMat dtransform rmoveto +} def +/TJm { pdfFontSize 0.001 mul mul neg 0 + pdfTextMat dtransform rmoveto } def +/TJmV { pdfFontSize 0.001 mul mul neg 0 exch + pdfTextMat dtransform rmoveto } def +/Tclip { pdfTextClipPath cvx exec clip newpath + /pdfTextClipPath [] def } def +% Level 2 image operators +/pdfImBuf 100 string def +/pdfIm { + image + { currentfile pdfImBuf readline + not { pop exit } if + (%-EOD-) eq { exit } if } loop +} def +/pdfImM { + fCol imagemask + { currentfile pdfImBuf readline + not { pop exit } if + (%-EOD-) eq { exit } if } loop +} def +/pdfImClip { + gsave + 0 2 4 index length 1 sub { + dup 4 index exch 2 copy + get 5 index div put + 1 add 3 index exch 2 copy + get 3 index div put + } for + pop pop rectclip +} def +/pdfImClipEnd { grestore } def +% shading operators +/colordelta { + false 0 1 3 index length 1 sub { + dup 4 index exch get 3 index 3 2 roll get sub abs 0.004 gt { + pop true + } if + } for + exch pop exch pop +} def +/funcCol { func n array astore } def +/funcSH { + dup 0 eq { + true + } { + dup 6 eq { + false + } { + 4 index 4 index funcCol dup + 6 index 4 index funcCol dup + 3 1 roll colordelta 3 1 roll + 5 index 5 index funcCol dup + 3 1 roll colordelta 3 1 roll + 6 index 8 index funcCol dup + 3 1 roll colordelta 3 1 roll + colordelta or or or + } ifelse + } ifelse + { + 1 add + 4 index 3 index add 0.5 mul exch 4 index 3 index add 0.5 mul exch + 6 index 6 index 4 index 4 index 4 index funcSH + 2 index 6 index 6 index 4 index 4 index funcSH + 6 index 2 index 4 index 6 index 4 index funcSH + 5 3 roll 3 2 roll funcSH pop pop + } { + pop 3 index 2 index add 0.5 mul 3 index 2 index add 0.5 mul + funcCol sc + dup 4 index exch mat transform m + 3 index 3 index mat transform l + 1 index 3 index mat transform l + mat transform l pop pop h f* + } ifelse +} def +/axialCol { + dup 0 lt { + pop t0 + } { + dup 1 gt { + pop t1 + } { + dt mul t0 add + } ifelse + } ifelse + func n array astore +} def +/axialSH { + dup 0 eq { + true + } { + dup 8 eq { + false + } { + 2 index axialCol 2 index axialCol colordelta + } ifelse + } ifelse + { + 1 add 3 1 roll 2 copy add 0.5 mul + dup 4 3 roll exch 4 index axialSH + exch 3 2 roll axialSH + } { + pop 2 copy add 0.5 mul axialCol sc + exch dup dx mul x0 add exch dy mul y0 add + 3 2 roll dup dx mul x0 add exch dy mul y0 add + dx abs dy abs ge { + 2 copy yMin sub dy mul dx div add yMin m + yMax sub dy mul dx div add yMax l + 2 copy yMax sub dy mul dx div add yMax l + yMin sub dy mul dx div add yMin l + h f* + } { + exch 2 copy xMin sub dx mul dy div add xMin exch m + xMax sub dx mul dy div add xMax exch l + exch 2 copy xMax sub dx mul dy div add xMax exch l + xMin sub dx mul dy div add xMin exch l + h f* + } ifelse + } ifelse +} def +/radialCol { + dup t0 lt { + pop t0 + } { + dup t1 gt { + pop t1 + } if + } ifelse + func n array astore +} def +/radialSH { + dup 0 eq { + true + } { + dup 8 eq { + false + } { + 2 index dt mul t0 add radialCol + 2 index dt mul t0 add radialCol colordelta + } ifelse + } ifelse + { + 1 add 3 1 roll 2 copy add 0.5 mul + dup 4 3 roll exch 4 index radialSH + exch 3 2 roll radialSH + } { + pop 2 copy add 0.5 mul dt mul t0 add axialCol sc + exch dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add + 0 360 arc h + dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add + 0 360 arc h f* + } ifelse +} def +end +/CIDInit /ProcSet findresource begin +10 dict begin + begincmap + /CMapType 1 def + /CMapName /Identity-H def + /CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def + end def + 1 begincodespacerange + <0000> + endcodespacerange + 0 usefont + 1 begincidrange + <0000> 0 + endcidrange + endcmap + currentdict CMapName exch /CMap defineresource pop +end +10 dict begin + begincmap + /CMapType 1 def + /CMapName /Identity-V def + /CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def + end def + /WMode 1 def + 1 begincodespacerange + <0000> + endcodespacerange + 0 usefont + 1 begincidrange + <0000> 0 + endcidrange + endcmap + currentdict CMapName exch /CMap defineresource pop +end +end +xpdf begin +%!FontType1-1.0: Helvetica +12 dict begin +/FontInfo 10 dict dup begin +/Notice (Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development; Cyrillic glyphs added by Valek Filippov (C) 2001-2004) readonly def +/FullName (Nimbus Sans L Regular) readonly def +/FamilyName (Nimbus Sans L) readonly def +/isFixedPitch false def +/ItalicAngle 0 def +/UnderlinePosition 0 def +/UnderlineThickness 0 def +end readonly def +/FontName /Helvetica def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/FontBBox [0 -218 762 741] readonly def +/StrokeWidth 0 def +/UniqueID 4095404 def 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7 0 0 5 + 655 3808 3088 3808 3088 5116 655 5116 655 3808 +2 2 3 1 4 7 100 -1 -1 4.000 0 0 7 0 0 5 + 600 3750 3150 3750 3150 5175 600 5175 600 3750 +-6 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 7 1 0 2 + 2 1 1.00 60.00 120.00 + 1800 2325 1200 2625 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2 + 2 1 1.00 60.00 120.00 + 1200 3075 1800 3375 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 4 + 2 1 1.00 60.00 120.00 + 750 5400 375 5400 375 1725 975 1725 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 4 + 2 1 1.00 60.00 120.00 + 1200 3525 525 3525 525 2175 1350 2175 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 + 2 1 1.00 60.00 120.00 + 1800 1875 1800 2100 +2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5 + 975 1650 2625 1650 2625 1875 975 1875 975 1650 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 + 2 1 1.00 60.00 120.00 + 1800 1425 1800 1650 +2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2 + 2 1 1.00 60.00 120.00 + 1800 975 1800 1200 +2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5 + 1200 750 2475 750 2475 975 1200 975 1200 750 +2 1 0 1 0 7 50 -1 -1 0.000 0 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300 3150 300 3150 525 600 525 600 300 +2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5 + 675 1200 3000 1200 3000 1425 675 1425 675 1200 +2 2 0 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 5 + 600 2625 1650 2625 1650 3075 600 3075 600 2625 +2 2 0 1 0 -1 50 -1 -1 0.000 0 0 -1 0 0 5 + 900 3900 2775 3900 2775 4125 900 4125 900 3900 +2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5 + 750 4350 2925 4350 2925 4575 750 4575 750 4350 +3 0 3 1 1 7 50 -1 -1 4.000 0 1 0 2 + 2 0 1.00 60.00 120.00 + 1650 2850 1950 2850 + 0.000 0.000 +4 0 0 50 -1 0 9 0.0000 2 165 1440 1050 1800 set $t=(m+k-1)T$\001 +4 0 0 50 -1 0 9 0.0000 2 180 2610 750 1350 make a guess of $x((m+k-1)T)$\001 +4 0 0 50 -1 0 9 0.0000 2 165 1800 1275 900 select jump size $k$\001 +4 0 0 50 -1 0 9 0.0000 2 165 3240 675 450 circuit state $x(mT)$ at time $t=mT$\001 +4 0 0 50 -1 0 9 0.0000 2 120 270 750 3450 yes\001 +4 0 0 50 -1 0 9 0.0000 2 165 2520 825 4500 solve Newton update equation\001 +4 0 0 50 -1 0 9 0.0000 2 90 180 1950 3750 no\001 +4 0 0 50 -1 0 9 0.0000 2 150 2340 975 4050 compute sensitivity matrix\001 +4 0 0 50 -1 0 9 0.0000 2 165 990 1425 5400 converged ?\001 +4 0 0 50 -1 0 9 0.0000 2 90 180 1125 5325 no\001 +4 0 0 50 -1 0 9 0.0000 2 120 270 1950 5625 yes\001 +4 0 0 50 -1 0 9 0.0000 2 135 630 1425 5850 $m=m+k$\001 +4 0 0 50 -1 0 9 0.0000 2 165 1080 1275 3525 $t<(m+k)T$ ?\001 +4 0 0 50 -1 0 9 0.0000 2 165 900 675 3000 for $x(t)$\001 +4 0 0 50 -1 0 9 0.0000 2 165 1170 675 2775 integrate DAE\001 diff --git a/BookGPU/Chapters/chapter16/figures/ef_flow.pdf b/BookGPU/Chapters/chapter16/figures/ef_flow.pdf new file mode 100644 index 0000000..d7dc1e8 Binary files /dev/null and b/BookGPU/Chapters/chapter16/figures/ef_flow.pdf differ diff --git a/BookGPU/Chapters/chapter16/figures/ef_flow.pdf_t b/BookGPU/Chapters/chapter16/figures/ef_flow.pdf_t new file mode 100644 index 0000000..7bd7063 --- /dev/null +++ b/BookGPU/Chapters/chapter16/figures/ef_flow.pdf_t @@ -0,0 +1,53 @@ +\begin{picture}(0,0)% 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#e2c8a8 +0 80 #e1e1e1 +0 81 #d2d2d2 +0 82 #ededed +0 83 #da7a1a +0 84 #f1e41a +0 85 #887dc2 +0 86 #d6d6d6 +0 87 #8c8ca5 +0 88 #4a4a4a +0 89 #8c6b6b +0 90 #5a5a5a +0 91 #636363 +0 92 #b79b73 +0 93 #4193ff +0 94 #bf703b +0 95 #db7700 +0 96 #dab800 +0 97 #006400 +0 98 #5a6b3b +0 99 #d3d3d3 +0 100 #8e8ea4 +0 101 #f3b95d +0 102 #89996b +0 103 #646464 +0 104 #b7e6ff +0 105 #86c0ec +0 106 #bdbdbd +0 107 #d39552 +0 108 #98d2fe +0 109 #8c9c6b +0 110 #f76b00 +0 111 #5a6b39 +0 112 #8c9c6b +0 113 #8c9c7b +0 114 #184a18 +0 115 #adadad +0 116 #f7bd5a +0 117 #636b9c +0 118 #f7f7f7 +0 119 #de0000 +0 120 #adadad +0 121 #f7bd5a +0 122 #adadad +0 123 #f7bd5a +0 124 #636b9c +0 125 #526b29 +0 126 #949494 +0 127 #006300 +0 128 #00634a +0 129 #7b844a +0 130 #e7bd7b +0 131 #a5b5c6 +0 132 #6b6b94 +0 133 #846b6b +0 134 #529c4a +0 135 #d6e7e7 +0 136 #526363 +0 137 #186b4a +0 138 #9ca5b5 +0 139 #ff9400 +0 140 #ff9400 +0 141 #00634a +0 142 #7b844a +0 143 #63737b +0 144 #e7bd7b +0 145 #184a18 +0 146 #f7bd5a +0 147 #dedede +0 148 #f3eed3 +0 149 #f5ae5d +0 150 #95ce99 +0 151 #b5157d +0 152 #eeeeee +0 153 #848484 +0 154 #7b7b7b +0 155 #005a00 +0 156 #e77373 +0 157 #ffcb31 +0 158 #29794a +0 159 #de2821 +0 160 #2159c6 +0 161 #f8f8f8 +0 162 #e6e6e6 +0 163 #21845a +0 164 #c9c9c9 +0 165 #dfd8df +0 166 #f7f3f7 +0 167 #cdcdcd +0 168 #6c6c6c +# Brace under +6 1500 3975 2400 4425 +3 4 0 1 0 7 50 0 -1 0.000 0 0 0 7 + 1573 4291 1598 4358 1980 4218 2004 4285 1980 4218 2363 4079 + 2339 4012 + 0.000 1.000 1.000 0.000 1.000 1.000 0.000 +-6 +# Brace under +6 3000 3600 3900 3825 +3 4 0 1 0 7 50 0 -1 0.000 0 0 0 7 + 3070 3747 3082 3818 3484 3747 3496 3816 3484 3747 3885 3676 + 3873 3606 + 0.000 1.000 1.000 0.000 1.000 1.000 0.000 +-6 +# Brace under +6 4575 3525 5625 3900 +3 4 0 1 0 7 50 0 -1 0.000 0 0 0 7 + 4639 3531 4622 3600 5089 3734 5072 3802 5089 3734 5556 3868 + 5574 3799 + 0.000 1.000 1.000 0.000 1.000 1.000 0.000 +-6 +2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2 + 7500 4800 7500 5100 +2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2 + 6225 4200 6225 5100 +2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2 + 3000 3825 3000 5100 +2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2 + 900 4575 900 5100 +2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2 + 1500 4350 1500 5100 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2 + 0 0 1.00 60.00 120.00 + 0 0 1.00 60.00 120.00 + 900 5100 1500 5100 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2 + 0 0 1.00 60.00 120.00 + 0 0 1.00 60.00 120.00 + 1500 5100 3000 5100 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2 + 0 0 1.00 60.00 120.00 + 0 0 1.00 60.00 120.00 + 3075 5100 4575 5100 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2 + 0 0 1.00 60.00 120.00 + 0 0 1.00 60.00 120.00 + 4575 5100 6225 5100 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2 + 0 0 1.00 60.00 120.00 + 0 0 1.00 60.00 120.00 + 6300 5100 7500 5100 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 1500 3225 1050 4200 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3150 3075 3900 3225 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 2250 3150 2475 3525 +2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2 + 4575 3525 4575 5100 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 6225 3150 6675 4200 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3750 4200 3525 3825 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 4200 4275 5025 3900 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2 + 0 0 1.00 60.00 120.00 + 3075 4275 2175 4275 +2 2 0 1 7 7 50 -1 -1 0.000 0 0 -1 0 0 5 + 3300 2475 4500 2475 4500 2625 3300 2625 3300 2475 +3 0 0 3 4 7 50 -1 -1 0.000 0 0 0 3 + 900 4500 1050 4200 1425 4275 + 0.000 1.000 0.000 +3 0 0 3 4 7 50 -1 -1 0.000 0 0 0 3 + 2400 3900 2550 3525 3000 3675 + 0.000 1.000 0.000 +3 0 0 3 4 7 50 -1 -1 0.000 0 0 0 3 + 3978 3436 4203 3136 4575 3450 + 0.000 1.000 0.000 +3 0 0 3 4 7 50 -1 -1 0.000 0 0 0 3 + 5622 3764 5847 3464 6225 4050 + 0.000 1.000 0.000 +3 0 0 3 4 7 50 -1 -1 0.000 0 0 0 3 + 7050 4425 7275 4125 7500 4725 + 0.000 1.000 0.000 +3 0 1 2 1 7 50 -1 -1 6.000 0 0 0 5 + 900 4500 3000 3600 4500 3300 6300 3975 7800 4950 + 0.000 1.000 1.000 1.000 0.000 +4 0 0 50 -1 0 12 0.0000 2 150 540 2025 5025 $m_1T$\001 +4 0 0 50 -1 0 12 0.0000 2 150 540 3600 5025 $m_2T$\001 +4 0 0 50 -1 0 12 0.0000 2 150 540 5175 5025 $m_3T$\001 +4 0 0 50 -1 0 12 0.0000 2 150 540 6600 5025 $m_4T$\001 +4 0 4 50 -1 0 12 0.0000 2 165 1530 1425 3075 simulated periods\001 +4 0 1 50 -1 0 12 0.0000 2 165 720 5850 3075 envelope\001 +4 0 0 50 -1 0 12 0.0000 2 165 1350 3150 4425 skipped periods\001 +4 0 0 50 -1 0 12 0.0000 2 135 270 1125 5025 $T$\001 diff --git a/BookGPU/Chapters/chapter16/figures/envelope.pdf b/BookGPU/Chapters/chapter16/figures/envelope.pdf new file mode 100644 index 0000000..fbe16f0 Binary files /dev/null and b/BookGPU/Chapters/chapter16/figures/envelope.pdf differ diff --git a/BookGPU/Chapters/chapter16/figures/envelope.pdf_t b/BookGPU/Chapters/chapter16/figures/envelope.pdf_t new file mode 100644 index 0000000..e62b914 --- /dev/null +++ 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control +/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 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sCol stroke } def +/Sf { fCol stroke } def +/f { fCol fill } def +/f* { fCol eofill } def +% clipping operators +/W { clip newpath } def +/W* { eoclip newpath } def +% text state operators +/Tc { /pdfCharSpacing exch def } def +/Tf { dup /pdfFontSize exch def + dup pdfHorizScaling mul exch matrix scale + pdfTextMat matrix concatmatrix dup 4 0 put dup 5 0 put + exch findfont exch makefont setfont } def +/Tr { /pdfTextRender exch def } def +/Ts { /pdfTextRise exch def } def +/Tw { /pdfWordSpacing exch def } def +/Tz { /pdfHorizScaling exch def } def +% text positioning operators +/Td { pdfTextMat transform moveto } def +/Tm { /pdfTextMat exch def } def +% text string operators +/cshow where { + pop + /cshow2 { + dup { + pop pop + 1 string dup 0 3 index put 3 index exec + } exch cshow + pop pop + } def +}{ + /cshow2 { + currentfont /FontType get 0 eq { + 0 2 2 index length 1 sub { + 2 copy get exch 1 add 2 index exch get + 2 copy exch 256 mul add + 2 string dup 0 6 5 roll put dup 1 5 4 roll put + 3 index exec + } for + } { + dup { + 1 string dup 0 3 index put 3 index exec + } forall + } ifelse + pop pop + } def +} ifelse +/awcp { + exch { + false charpath + 5 index 5 index rmoveto + 6 index eq { 7 index 7 index rmoveto } if + } exch cshow2 + 6 {pop} repeat +} def +/Tj { + fCol + 1 index stringwidth pdfTextMat idtransform pop + sub 1 index length dup 0 ne { div } { pop pop 0 } ifelse + pdfWordSpacing pdfHorizScaling mul 0 pdfTextMat dtransform 32 + 4 3 roll pdfCharSpacing pdfHorizScaling mul add 0 + pdfTextMat dtransform + 6 5 roll Tj1 +} def +/Tj16 { + fCol + 2 index stringwidth pdfTextMat idtransform pop + sub exch div + pdfWordSpacing pdfHorizScaling mul 0 pdfTextMat dtransform 32 + 4 3 roll pdfCharSpacing pdfHorizScaling mul add 0 + pdfTextMat dtransform + 6 5 roll Tj1 +} def +/Tj16V { + fCol + 2 index stringwidth pdfTextMat idtransform exch pop + sub exch div + 0 pdfWordSpacing pdfTextMat dtransform 32 + 4 3 roll pdfCharSpacing add 0 exch + pdfTextMat dtransform + 6 5 roll Tj1 +} def +/Tj1 { + 0 pdfTextRise pdfTextMat dtransform rmoveto + currentpoint 8 2 roll + pdfTextRender 1 and 0 eq { + 6 copy awidthshow + } if + pdfTextRender 3 and dup 1 eq exch 2 eq or { + 7 index 7 index moveto + 6 copy + currentfont /FontType get 3 eq { fCol } { sCol } ifelse + false awcp currentpoint stroke moveto + } if + pdfTextRender 4 and 0 ne { + 8 6 roll moveto + false awcp + /pdfTextClipPath [ pdfTextClipPath aload pop + {/moveto cvx} + {/lineto cvx} + {/curveto cvx} + {/closepath cvx} + pathforall ] def + currentpoint newpath moveto + } { + 8 {pop} repeat + } ifelse + 0 pdfTextRise neg pdfTextMat dtransform rmoveto +} def +/TJm { pdfFontSize 0.001 mul mul neg 0 + pdfTextMat dtransform rmoveto } def +/TJmV { pdfFontSize 0.001 mul mul neg 0 exch + pdfTextMat dtransform rmoveto } def +/Tclip { pdfTextClipPath cvx exec clip newpath + /pdfTextClipPath [] def } def +% Level 2 image operators +/pdfImBuf 100 string def +/pdfIm { + image + { currentfile pdfImBuf readline + not { pop exit } if + (%-EOD-) eq { exit } if } loop +} def +/pdfImM { + fCol imagemask + { currentfile pdfImBuf readline + not { pop exit } if + (%-EOD-) eq { exit } if } loop +} def +/pdfImClip { + gsave + 0 2 4 index length 1 sub { + dup 4 index exch 2 copy + get 5 index div put + 1 add 3 index exch 2 copy + get 3 index div put + } for + pop pop rectclip +} def +/pdfImClipEnd { grestore } def +% shading operators +/colordelta { + false 0 1 3 index length 1 sub { + dup 4 index exch get 3 index 3 2 roll get sub abs 0.004 gt { + pop true + } if + } for + exch pop exch pop +} def +/funcCol { func n array astore } def +/funcSH { + dup 0 eq { + true + } { + dup 6 eq { + false + } { + 4 index 4 index funcCol dup + 6 index 4 index funcCol dup + 3 1 roll colordelta 3 1 roll + 5 index 5 index funcCol dup + 3 1 roll colordelta 3 1 roll + 6 index 8 index funcCol dup + 3 1 roll colordelta 3 1 roll + colordelta or or or + } ifelse + } ifelse + { + 1 add + 4 index 3 index add 0.5 mul exch 4 index 3 index add 0.5 mul exch + 6 index 6 index 4 index 4 index 4 index funcSH + 2 index 6 index 6 index 4 index 4 index funcSH + 6 index 2 index 4 index 6 index 4 index funcSH + 5 3 roll 3 2 roll funcSH pop pop + } { + pop 3 index 2 index add 0.5 mul 3 index 2 index add 0.5 mul + funcCol sc + dup 4 index exch mat transform m + 3 index 3 index mat transform l + 1 index 3 index mat transform l + mat transform l pop pop h f* + } ifelse +} def +/axialCol { + dup 0 lt { + pop t0 + } { + dup 1 gt { + pop t1 + } { + dt mul t0 add + } ifelse + } ifelse + func n array astore +} def +/axialSH { + dup 0 eq { + true + } { + dup 8 eq { + false + } { + 2 index axialCol 2 index axialCol colordelta + } ifelse + } ifelse + { + 1 add 3 1 roll 2 copy add 0.5 mul + dup 4 3 roll exch 4 index axialSH + exch 3 2 roll axialSH + } { + pop 2 copy add 0.5 mul axialCol sc + exch dup dx mul x0 add exch dy mul y0 add + 3 2 roll dup dx mul x0 add exch dy mul y0 add + dx abs dy abs ge { + 2 copy yMin sub dy mul dx div add yMin m + yMax sub dy mul dx div add yMax l + 2 copy yMax sub dy mul dx div add yMax l + yMin sub dy mul dx div add yMin l + h f* + } { + exch 2 copy xMin sub dx mul dy div add xMin exch m + xMax sub dx mul dy div add xMax exch l + exch 2 copy xMax sub dx mul dy div add xMax exch l + xMin sub dx mul dy div add xMin exch l + h f* + } ifelse + } ifelse +} def +/radialCol { + dup t0 lt { + pop t0 + } { + dup t1 gt { + pop t1 + } if + } ifelse + func n array astore +} def +/radialSH { + dup 0 eq { + true + } { + dup 8 eq { + false + } { + 2 index dt mul t0 add radialCol + 2 index dt mul t0 add radialCol colordelta + } ifelse + } ifelse + { + 1 add 3 1 roll 2 copy add 0.5 mul + dup 4 3 roll exch 4 index radialSH + exch 3 2 roll radialSH + } { + pop 2 copy add 0.5 mul dt mul t0 add axialCol sc + exch dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add + 0 360 arc h + dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add + 0 360 arc h f* + } ifelse +} def +end +/CIDInit /ProcSet findresource begin +10 dict begin + begincmap + /CMapType 1 def + /CMapName /Identity-H def + /CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def + end def + 1 begincodespacerange + <0000> + endcodespacerange + 0 usefont + 1 begincidrange + <0000> 0 + endcidrange + endcmap + currentdict CMapName exch /CMap defineresource pop +end +10 dict begin + begincmap + /CMapType 1 def + /CMapName /Identity-V def + /CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def + end def + /WMode 1 def + 1 begincodespacerange + <0000> + endcodespacerange + 0 usefont + 1 begincidrange + <0000> 0 + endcidrange + endcmap + currentdict CMapName exch /CMap defineresource pop +end +end +xpdf begin +%!FontType1-1.0: Helvetica +12 dict begin +/FontInfo 10 dict dup begin +/Notice (Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development; Cyrillic glyphs added by Valek Filippov (C) 2001-2004) readonly def +/FullName (Nimbus Sans L Regular) readonly def +/FamilyName (Nimbus Sans L) readonly def +/isFixedPitch false def +/ItalicAngle 0 def +/UnderlinePosition 0 def +/UnderlineThickness 0 def +end readonly def +/FontName /Helvetica def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/FontBBox [0 -218 762 741] readonly def +/StrokeWidth 0 def +/UniqueID 4095404 def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 32 /space put +dup 40 /parenleft put +dup 41 /parenright put +dup 45 /minus put +dup 46 /period put +dup 48 /zero put +dup 49 /one put +dup 50 /two put +dup 51 /three put +dup 52 /four put +dup 67 /C put +dup 69 /E put +dup 73 /I put +dup 80 /P put +dup 83 /S put +dup 84 /T put +dup 86 /V put +dup 97 /a put +dup 99 /c put +dup 101 /e put +dup 102 /f put +dup 103 /g put +dup 105 /i put +dup 108 /l put +dup 109 /m put +dup 110 /n put +dup 111 /o put +dup 112 /p put +dup 114 /r put +dup 115 /s put +dup 116 /t put +dup 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+/Q { + end grestore + /pdfLastFill where { + pop + pdfLastFill { + pdfFillOP setoverprint + } { + pdfStrokeOP setoverprint + } ifelse + } if +} def +/cm { concat } def +/d { setdash } def +/i { setflat } def +/j { setlinejoin } def +/J { setlinecap } def +/M { setmiterlimit } def +/w { setlinewidth } def +% path segment operators +/m { moveto } def +/l { lineto } def +/c { curveto } def +/re { 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto + neg 0 rlineto closepath } def +/h { closepath } def +% path painting operators +/S { sCol stroke } def +/Sf { fCol stroke } def +/f { fCol fill } def +/f* { fCol eofill } def +% clipping operators +/W { clip newpath } def +/W* { eoclip newpath } def +% text state operators +/Tc { /pdfCharSpacing exch def } def +/Tf { dup /pdfFontSize exch def + dup pdfHorizScaling mul exch matrix scale + pdfTextMat matrix concatmatrix dup 4 0 put dup 5 0 put + exch findfont exch makefont setfont } def +/Tr { /pdfTextRender exch def } def +/Ts { /pdfTextRise exch def } def +/Tw { /pdfWordSpacing exch def } def +/Tz { /pdfHorizScaling exch def } def +% text positioning operators +/Td { pdfTextMat transform moveto } def +/Tm { /pdfTextMat exch def } def +% text string operators +/cshow where { + pop + /cshow2 { + dup { + pop pop + 1 string dup 0 3 index put 3 index exec + } exch cshow + pop pop + } def +}{ + /cshow2 { + currentfont /FontType get 0 eq { + 0 2 2 index length 1 sub { + 2 copy get exch 1 add 2 index exch get + 2 copy exch 256 mul add + 2 string dup 0 6 5 roll put dup 1 5 4 roll put + 3 index exec + } for + } { + dup { + 1 string dup 0 3 index put 3 index exec + } forall + } ifelse + pop pop + } def +} ifelse +/awcp { + exch { + false charpath + 5 index 5 index rmoveto + 6 index eq { 7 index 7 index rmoveto } if + } exch cshow2 + 6 {pop} repeat +} def +/Tj { + fCol + 1 index stringwidth pdfTextMat idtransform pop + sub 1 index length dup 0 ne { div } { pop pop 0 } ifelse + pdfWordSpacing pdfHorizScaling mul 0 pdfTextMat dtransform 32 + 4 3 roll pdfCharSpacing pdfHorizScaling mul add 0 + pdfTextMat dtransform + 6 5 roll Tj1 +} def +/Tj16 { + fCol + 2 index stringwidth pdfTextMat idtransform pop + sub exch div + pdfWordSpacing pdfHorizScaling mul 0 pdfTextMat dtransform 32 + 4 3 roll pdfCharSpacing pdfHorizScaling mul add 0 + pdfTextMat dtransform + 6 5 roll Tj1 +} def +/Tj16V { + fCol + 2 index stringwidth pdfTextMat idtransform exch pop + sub exch div + 0 pdfWordSpacing pdfTextMat dtransform 32 + 4 3 roll pdfCharSpacing add 0 exch + pdfTextMat dtransform + 6 5 roll Tj1 +} def +/Tj1 { + 0 pdfTextRise pdfTextMat dtransform rmoveto + currentpoint 8 2 roll + pdfTextRender 1 and 0 eq { + 6 copy awidthshow + } if + pdfTextRender 3 and dup 1 eq exch 2 eq or { + 7 index 7 index moveto + 6 copy + currentfont /FontType get 3 eq { fCol } { sCol } ifelse + false awcp currentpoint stroke moveto + } if + pdfTextRender 4 and 0 ne { + 8 6 roll moveto + false awcp + /pdfTextClipPath [ pdfTextClipPath aload pop + {/moveto cvx} + {/lineto cvx} + {/curveto cvx} + {/closepath cvx} + pathforall ] def + currentpoint newpath moveto + } { + 8 {pop} repeat + } ifelse + 0 pdfTextRise neg pdfTextMat dtransform rmoveto +} def +/TJm { pdfFontSize 0.001 mul mul neg 0 + pdfTextMat dtransform rmoveto } def +/TJmV { pdfFontSize 0.001 mul mul neg 0 exch + pdfTextMat dtransform rmoveto } def +/Tclip { pdfTextClipPath cvx exec clip newpath + /pdfTextClipPath [] def } def +% Level 2 image operators +/pdfImBuf 100 string def +/pdfIm { + image + { currentfile pdfImBuf readline + not { pop exit } if + (%-EOD-) eq { exit } if } loop +} def +/pdfImM { + fCol imagemask + { currentfile pdfImBuf readline + not { pop exit } if + (%-EOD-) eq { exit } if } loop +} def +/pdfImClip { + gsave + 0 2 4 index length 1 sub { + dup 4 index exch 2 copy + get 5 index div put + 1 add 3 index exch 2 copy + get 3 index div put + } for + pop pop rectclip +} def +/pdfImClipEnd { grestore } def +% shading operators +/colordelta { + false 0 1 3 index length 1 sub { + dup 4 index exch get 3 index 3 2 roll get sub abs 0.004 gt { + pop true + } if + } for + exch pop exch pop +} def +/funcCol { func n array astore } def +/funcSH { + dup 0 eq { + true + } { + dup 6 eq { + false + } { + 4 index 4 index funcCol dup + 6 index 4 index funcCol dup + 3 1 roll colordelta 3 1 roll + 5 index 5 index funcCol dup + 3 1 roll colordelta 3 1 roll + 6 index 8 index funcCol dup + 3 1 roll colordelta 3 1 roll + colordelta or or or + } ifelse + } ifelse + { + 1 add + 4 index 3 index add 0.5 mul exch 4 index 3 index add 0.5 mul exch + 6 index 6 index 4 index 4 index 4 index funcSH + 2 index 6 index 6 index 4 index 4 index funcSH + 6 index 2 index 4 index 6 index 4 index funcSH + 5 3 roll 3 2 roll funcSH pop pop + } { + pop 3 index 2 index add 0.5 mul 3 index 2 index add 0.5 mul + funcCol sc + dup 4 index exch mat transform m + 3 index 3 index mat transform l + 1 index 3 index mat transform l + mat transform l pop pop h f* + } ifelse +} def +/axialCol { + dup 0 lt { + pop t0 + } { + dup 1 gt { + pop t1 + } { + dt mul t0 add + } ifelse + } ifelse + func n array astore +} def +/axialSH { + dup 0 eq { + true + } { + dup 8 eq { + false + } { + 2 index axialCol 2 index axialCol colordelta + } ifelse + } ifelse + { + 1 add 3 1 roll 2 copy add 0.5 mul + dup 4 3 roll exch 4 index axialSH + exch 3 2 roll axialSH + } { + pop 2 copy add 0.5 mul axialCol sc + exch dup dx mul x0 add exch dy mul y0 add + 3 2 roll dup dx mul x0 add exch dy mul y0 add + dx abs dy abs ge { + 2 copy yMin sub dy mul dx div add yMin m + yMax sub dy mul dx div add yMax l + 2 copy yMax sub dy mul dx div add yMax l + yMin sub dy mul dx div add yMin l + h f* + } { + exch 2 copy xMin sub dx mul dy div add xMin exch m + xMax sub dx mul dy div add xMax exch l + exch 2 copy xMax sub dx mul dy div add xMax exch l + xMin sub dx mul dy div add xMin exch l + h f* + } ifelse + } ifelse +} def +/radialCol { + dup t0 lt { + pop t0 + } { + dup t1 gt { + pop t1 + } if + } ifelse + func n array astore +} def +/radialSH { + dup 0 eq { + true + } { + dup 8 eq { + false + } { + 2 index dt mul t0 add radialCol + 2 index dt mul t0 add radialCol colordelta + } ifelse + } ifelse + { + 1 add 3 1 roll 2 copy add 0.5 mul + dup 4 3 roll exch 4 index radialSH + exch 3 2 roll radialSH + } { + pop 2 copy add 0.5 mul dt mul t0 add axialCol sc + exch dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add + 0 360 arc h + dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add + 0 360 arc h f* + } ifelse +} def +end +/CIDInit /ProcSet findresource begin +10 dict begin + begincmap + /CMapType 1 def + /CMapName /Identity-H def + /CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def + end def + 1 begincodespacerange + <0000> + endcodespacerange + 0 usefont + 1 begincidrange + <0000> 0 + endcidrange + endcmap + currentdict CMapName exch /CMap defineresource pop +end +10 dict begin + begincmap + /CMapType 1 def + /CMapName /Identity-V def + /CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def + end def + /WMode 1 def + 1 begincodespacerange + <0000> + endcodespacerange + 0 usefont + 1 begincidrange + <0000> 0 + endcidrange + endcmap + currentdict CMapName exch /CMap defineresource pop +end +end +xpdf begin +%!FontType1-1.0: Helvetica +12 dict begin +/FontInfo 10 dict dup begin +/Notice (Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development; Cyrillic glyphs added by Valek Filippov (C) 2001-2004) readonly def +/FullName (Nimbus Sans L Regular) readonly def +/FamilyName (Nimbus Sans L) readonly def +/isFixedPitch false def +/ItalicAngle 0 def +/UnderlinePosition 0 def +/UnderlineThickness 0 def +end readonly def +/FontName /Helvetica def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/FontBBox [0 -218 762 741] readonly def +/StrokeWidth 0 def +/UniqueID 4095404 def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for 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restore +%%EOF diff --git a/BookGPU/Chapters/chapter16/figures/gmres_flow.fig b/BookGPU/Chapters/chapter16/figures/gmres_flow.fig new file mode 100644 index 0000000..16555aa --- /dev/null +++ b/BookGPU/Chapters/chapter16/figures/gmres_flow.fig @@ -0,0 +1,133 @@ +#FIG 3.2 +Landscape +Center +Inches +Letter +100.00 +Single +-2 +1200 2 +0 32 #9c0000 +0 33 #8c8c8c +0 34 #8c8c8c +0 35 #424242 +0 36 #8c8c8c +0 37 #424242 +0 38 #8c8c8c +0 39 #424242 +0 40 #8c8c8c +0 41 #424242 +0 42 #8c8c8c +0 43 #424242 +0 44 #636363 +0 45 #cdcdcd +0 46 #6c6c6c +6 3375 1050 4575 1575 +6 3375 1275 4575 1575 +2 3 0 1 0 7 100 0 20 0.000 0 0 7 0 0 8 + 3376 1370 3376 1481 4060 1481 4060 1570 4528 1426 4060 1281 + 4060 1370 3376 1370 +-6 +4 0 0 50 -1 0 12 0.0000 2 120 630 3600 1200 memcopy\001 +-6 +6 3300 5625 4500 6150 +6 3300 5850 4500 6150 +2 3 0 1 0 7 100 0 20 0.000 0 0 7 0 0 8 + 4499 5945 4499 6056 3815 6056 3815 6145 3347 6001 3815 5856 + 3815 5945 4499 5945 +-6 +4 0 0 50 -1 0 12 0.0000 2 120 630 3600 5775 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0 0 5 + 5550 3975 5550 3675 5025 3675 5025 3975 5550 3975 +2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2 + 2 1 1.00 60.00 120.00 + 5175 3975 5175 4200 +2 1 1 1 0 7 50 -1 -1 4.000 0 0 7 0 0 2 + 3900 1650 3900 3825 +2 1 1 1 0 7 50 -1 -1 4.000 0 0 7 0 0 2 + 3900 4500 3900 5550 +4 0 0 50 -1 0 12 0.0000 2 165 1710 1425 1725 right hand side $b$\001 +4 0 0 50 -1 0 12 0.0000 2 165 1350 2250 300 Host side (CPU)\001 +4 0 0 50 -1 0 12 0.0000 2 165 2430 675 675 set up parameters for GMRES\001 +4 0 0 50 -1 0 12 0.0000 2 165 1710 1425 1500 initial guess $x_0$\001 +4 0 0 50 -1 0 12 0.0000 2 150 2250 1425 1275 saved $J_i$, $C_i$, $h_i$\001 +4 0 0 50 -1 0 12 0.0000 2 165 1530 4200 300 Device side (GPU)\001 +4 0 0 50 -1 0 12 0.0000 2 150 1350 4725 2775 $v_m=r/\\|r\\|_2$\001 +4 0 0 50 -1 0 12 0.0000 2 165 990 6000 3225 $V=[V|v_m]$\001 +4 0 1 50 -1 0 12 0.0000 2 135 2340 675 2325 Arnoldi iteration in GMRES\001 +4 0 0 50 -1 0 12 0.0000 2 150 720 4725 3900 $r=Av_m$\001 +4 0 0 50 -1 0 12 0.0000 2 165 2070 4650 4350 orthogonalize $r$ w.r.t\001 +4 0 0 50 -1 0 12 0.0000 2 180 2250 1200 4200 triangularize $\\tilde{H}$\001 +4 0 0 50 -1 0 12 0.0000 2 90 180 675 5025 no\001 +4 0 0 50 -1 0 12 0.0000 2 165 1710 4650 4800 save by-product $h$\001 +4 0 0 50 -1 0 12 0.0000 2 120 630 3600 4050 memcopy\001 +4 0 0 50 -1 0 12 0.0000 2 150 1350 4650 4575 vectors in $V$,\001 +4 0 0 50 -1 0 12 0.0000 2 150 450 5400 1125 $x_0$\001 +4 0 0 50 -1 0 12 0.0000 2 135 2250 4650 5925 linear combination of $V$\001 +4 0 0 50 -1 0 12 0.0000 2 135 1440 4650 6150 to form solution\001 +4 0 0 50 -1 0 12 0.0000 2 165 1890 1200 6000 converged approximate\001 +4 0 0 50 -1 0 12 0.0000 2 150 1260 1200 6225 solution $x_m$\001 +4 0 0 50 -1 0 12 0.0000 2 120 270 1650 5550 yes\001 +4 0 0 50 -1 0 12 0.0000 2 180 16920 675 3375 $\\tilde{H}=\\begin{bmatrix}h_{1,1} & h_{1,2} & \\cdots & h_{1,m}\\\\h_{2,1} & h_{2,2} & \\cdots & h_{2,m} \\\\& h_{3,2} & \\cdots & h_{3,m} \\\\&& \\ddots & \\vdots \\\\ & & & h_{m+1,m} \\end{bmatrix}$\001 +4 0 0 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x=b$) with dense matrix $A$. +The standard GMRES\index{GMRES} is given in Algorithm~\ref{alg:GMRES}. +It constructs a Krylov subspace\index{Krylov subspace} with order $m$, +\[ \mathcal{K}_m = \mathrm{span}( b, A^{} b, A^2 b,\ldots, A^{m-1} b ),\] +where the approximate solution $x_m$ resides. +In practice, an orthonormal basis $V_m$ that spans the +subspace $\mathcal{K}_{m}$ can be generated by +the Arnoldi iteration\index{Arnoldi iterations}. +The goal of GMRES is to search for an optimal coefficient $y$ +such that the linear combination $x_m = V_m y$ will minimize +its residual $\| b-Ax_m \|_2$. +The Arnoldi iteration also creates a by-product, +an upper Hessenberg matrix\index{Hessenberg matrix} $\tilde{H}_m \in R^{(m+1)\times m}$. +Thus, the projection of $A$ on the orthonormal basis $V_m$ is described by +the Arnoldi decomposition +$ + AV_m = V_{m+1} \tilde{H}_{m}, +$ +which is useful to check the residual at each iteration +without forming $x_m$, +and to solve for coefficient $y$ when residual is smaller than +a preset tolerance~\cite{Golub:Book'96}. + +% Givens rotations are generated to triangularize +% the Hessenberg matrix $\tilde{H}_m$ in order to solve the overdetermined system. +% These same Givens rotations are also applied to the right-hand-side $h_{1,0}e_1$, +% whose last element provides residual information on whether the tolerance +% has been met in the current iteration. + +%$\left\| h_{1,0}e_1-\tilde{H}_m y_m \right\|_2$. + + +%% \begin{algorithm} +%% \caption{Standard GMRES\index{GMRES} algorithm.} \label{alg:GMRES} +%% \begin{algorithmic}[1] +%% \REQUIRE $ A \in \mathbb{R}^{N \times N}$, $b \in \mathbb{R}^N$, +%% and initial guess $x_0 \in \mathbb{R}^N$ +%% \ENSURE $x \in \mathbb{R}^N$: $\| b - A x\|_2 < tol$ + +%% \STATE $r = b - A x_0$ +%% \STATE $h_{1,0}=\left \| r \right \|_2$ +%% \STATE $m=0$ + +%% \WHILE{$m < max\_iter$} +%% \STATE $m = m+1$ +%% \STATE $v_{m} = r / h_{m,m-1}$ +%% \STATE \label{line:mvp} $r = A v_m$ +%% \FOR{$i = 1\ldots m$} +%% \STATE $h_{i,m} = \langle v_i, r \rangle$ +%% \STATE $r = r - h_{i,m} v_i$ +%% \ENDFOR +%% \STATE $h_{m+1,m} = \left\| r \right\|_2$\label{line:newnorm} +%% %\STATE Generate Givens rotations to triangularize $\tilde{H}_m$ +%% %\STATE Apply Givens rotations on $h_{1,0}e_1$ to get residual $\epsilon$ +%% \STATE Compute the residual $\epsilon$ +%% \IF{$\epsilon < tol$} +%% \STATE Solve the problem: minimize $\|b-Ax_m\|_2$ +%% \STATE Return $x_m = x_0 + V_m y_m$ +%% \ENDIF +%% \ENDWHILE +%% \end{algorithmic} +%% \end{algorithm} + +At a first glance, the cost of using standard GMRES directly to +solve the Newton update in Eq.~\eqref{eq:Newton} +seems to come mainly from two parts: the +formulation of the sensitivity matrix $J = \ud x_M/\ud x_0$ +in Eq.~\eqref{eq:sensM} in Section~\ref{sec:gear}, +and the iteration inside the standard GMRES, +especially the matrix-vector multiplication and the orthonormal +basis construction (Line~\ref{line:mvp} through +Line~\ref{line:newnorm} in Algorithm~\ref{alg:GMRES}). +Based on the observation that only the matrix-vector product is +required in GMRES, the work in~\cite{Telichevesky:DAC'95} introduces +an efficient matrix-free algorithm in the shooting-Newton method, +where the equation solving part also involves with a sensitivity +matrix. +The matrix-free method does not take an explicit matrix as +input, but directly passes the saved capacitance matrices $C_i$ and +the LU factorizations of $J_i$, $i=0,\ldots,M$, into the Arnoldi +iteration for Krylov subspace generation. +Therefore, it avoids the +cost of forming the dense sensitivity matrix and focuses on +subspace construction. +Briefly speaking, Line~\ref{line:mvp} will be replaced +by a procedure without explict $A$, and we will talk +about the flow of matrix-free generation of new basis vectors +in later sections. + + +\subsection{Parallelization on GPU platforms} +\label{sec:gpu} +% In this subsection, we describe how to parallelize +% the proposed matrix-free GMRES solver with Gear-2 sensitivity +% on GPU platforms. +There exist many GPU-friendly computing operations +in GMRES, such as the vector addition (\verb|axpy|), +2-norm of vector (\verb|nrm2|), and +sparse matrix-vector multiplication (\verb|csrmv|), +which have been parallelized in the CUDA Basic Linear +Algebra Subroutine (CUBLAS) Library and the CUDA Sparse Linear Algebra +Library (CUSPARSE)~\cite{CUDASDK}. + +GPU programming is typically limited by the data transfer bandwidth as +GPU favors computationally intensive algorithms~\cite{Kirk:Book'10}. +So how to efficiently transfer the data and wisely partition +the data between CPU memory and GPU memory +is crucial for GPU programming. +In the following, we discuss these two issues in our implementation. + +As noted in Section~\ref{sec:ef}, the envelope-following method +requires the matrices gathered from all the time steps in a +period in order to solve a Newton update. +At each time step, SPICE\index{SPICE} has +to linearize device models, stamp matrix elements +into MNA (short for modified nodal analysis\index{modified nodal analysis, or MNA}) matrices, +and solve circuit equations in its inner Newton iteration\index{Newton iteration}. +When convergence is attained, +circuit states are saved and then next time step begins. +This is also the time when we store the needed matrices +for the envelope-following computation. +Since these data are involved in the calculation of Gear-2 sensitivity +matrix in the generation of Krylov subspace vectors in +Algorithm~\ref{alg:mf_Gear}, it is desirable that all of these +matrices are transferred to GPU for its data parallel capability. + +To efficiently transfer those large data, we explore asynchronous +memory copy between host and device in the recent GPUs (Fermi architecture), +so that the copying overlaps with the host's computing of +the next time step's circuit state. The implementation of asynchronous matrices +copy includes two parts: allocating page-locked memory, also known as +pinned memory, where we save matrices for one time step, and using +asynchronous memory transfer to copy these data to GPU memory. While +it is known that page-locked host memory is a scarce resource and +should not be overused, the demand of memory size of the data +generated at one time step can be generously accommodated by today's +mainstream server configurations. + +\begin{figure}[htbp] + \centering + \resizebox{.9\textwidth}{!}{\input{./Chapters/chapter16/figures/gmres_flow.pdf_t}} + \caption{GPU parallel solver for envelope-following update.} + \label{fig:gmres} +\end{figure} + + +%When SPICE simulation reaches the end of the period, all data needed by +%Newton update equation are ready in GPU memory, and the matrix-free +%GMRES can be applied directly on them to solve for the update vector +%in \eqref{eq:Newton}. + +% Knowing the dimensions or the sizes of the +%matrices in a computation helps to boost GPU computing efficiency. +The second issue is to decide the location of data between CPU and GPU memories. +Therefore let us first make a rough sketch of the quantities in the +GMRES Algorithm~\ref{alg:GMRES}. Although GMRES tends to converge +quickly for most circuit examples, i.e., the iteration number $m \ll +N$, the space for storing all the subspace basis $V_m$ of $N$-by-$m$, +i.e., $m$ column vectors with $N$-length, is still big. In addition, +every newly generated matrix-vector product needs to be orthogonalized +with respect to all its previous basis vectors in +the Arnoldi iterations\index{Arnoldi iterations}. +Hence, keeping all the vectors of $V_m$ in GPU global +memory allows GPU to handle those operations, such as inner-product of +basis vectors (\verb|dot|) and vector subtraction (\verb|axpy|), in +parallel. + +On the other hand, it is better to keep the Hessenberg matrix +$\tilde{H}$, where intermediate results of +the orthogonalization are stored, at the host side. +This comes with the following reasons. +First, its size is $(m+1)$-by-$m$ at most, rather small +if compared to circuit matrices and Krylov basis. +Besides, it is also necessary to triangularize $\tilde{H}$ +and check the residual regularly in +each iteration so the GMRES can return the approximate solution as +soon as the residual is below a preset tolerance. +Hence, in consideration of the serial nature of the trianularization, +the small size of Hessenberg matrix, +and the frequent inspection of values by host, it is +preferable to allocate $\tilde{H}$ in CPU (host) memory. +As shown in Fig.~\ref{fig:gmres}, the memory copy from device to host +is called each time when Arnoldi iteration generates a new vector +and the orthogonalization produces the vector $h$. + +% \begin{figure}[!tb] +% \centering +% \resizebox{.45\textwidth}{!}{\input{./matlab/drawing.ps_tex}} +% \caption{The flow of envelope-following method.} +% \label{fig:gmres} +% %\end{center} +% \end{figure} + +% talk about gear 2 method here. +\input{Chapters/chapter16/bdf.tex} + diff --git a/BookGPU/Chapters/chapter16/intro.tex b/BookGPU/Chapters/chapter16/intro.tex new file mode 100644 index 0000000..cbed054 --- /dev/null +++ b/BookGPU/Chapters/chapter16/intro.tex @@ -0,0 +1,163 @@ +\section{Introduction} + + +Power converters\index{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 present a novel +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\index{GMRES} solving for Newton update equation in the +envelope-following method, we apply the matrix-free\index{matrix-free} +Krylov subspace\index{Krylov subspace} 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. + +Over the past decades, power electronics\index{power electronics} and especially switching +power converters have seen a surge of new trends and novel applications +--- from the growing significance of PWM (pulse-width modulation) +rectifiers and multilevel inverters to their widespread use in +renewable (like solar and wind) energy systems, smart grids and emerging +electric and hybrid vehicles~\cite{Trzynadlowski:Book'10}. + +This requires more efficient simulation techniques for power +electronics to meet the new application and demanding design +constraints. To facilitate the design of typical power electronics +circuits, many special-purpose simulation algorithms and +tools were developed. Among them is the envelope-following +method~\cite{Kundert:ICCAD'88,White:TPE'91,Feldmann:ICCAD'96}, %,Bose:TIE'09 +which is able to calculate the slowly changing contour, or +envelope, of a carrier waveform with a much higher +switching frequency\index{switching frequency}. +This is the case for switching power converters, +which have fast switching currents to convert powers +from one level to another level. +In those switching power converters, it is the envelope, +which is the power voltage delivered, +not the fast switching waves in every cycle, +that is of interest to the designers. +As shown in Fig.~\ref{fig:ef1}, the solid line is +the waveform of the output node in a Buck +converter~\cite{Krein:book'97}, the dots are the simulation points +of SPICE\index{SPICE}, and the appended dash line is the envelope. + +Obviously, traditional SPICE will incur an extraordinarily high +simulation time for this task, since it has to integrate the +circuit's differential equation with many time points in every +clock cycle to get the accurate details of the carrier. For +switching power converters, the waveform of the carrier in +consequent cycles does not change much, envelope-following method +is an approximation analysis method, which skips over several +cycles (the dash line in Fig.~\ref{fig:ef2}), the so called +envelope step, without simulating them, and then carries out a +correction, which usually contains a sensitivity-based Newton +iteration or shooting until convergence, in order to begin the +next envelope step. + +% \begin{figure}[!b] +% \resizebox{.5\textwidth}{!}{\input{./matlab/ef_intro.pstex_t}} +% %\includegraphics[width=.45\textwidth]{./matlab/ef_intro_ppt.eps} +% \caption{Transient envelope-following analysis. +% (This figure reflects a backward-Euler style envelope-following.)} +% \label{fig:ef_intro} +% \end{figure} +\begin{figure} + \centering + \subfigure[Illustration of one envelope skip.] + {\resizebox{.8\textwidth}{!}{\input{./Chapters/chapter16/figures/ef_intro.pdf_t}} + \label{fig:ef1} } + \subfigure[The envelope changes in a slow time scale.] + {\resizebox{.9\textwidth}{!}{\input{./Chapters/chapter16/figures/envelope.pdf_t}} + \label{fig:ef2} } + \caption{Transient envelope-following\index{envelope-following} analysis. + (Both two figures reflect backward-Euler\index{backward-Euler} style envelope-following.)} + \label{fig:ef_intro} +\end{figure} + +%\IEEEpubidadjcol + +Also, iterative GMRES\index{GMRES} solver is typically used in the +envelope-following method to compute the solution of Newton update +due to its efficiency compared to direct LU\index{LU} method. +However, as the Jacobian matrix\index{Jacobian matrix} +or the sensitivity matrix\index{sensitivity matrix} +in the equation to be solved is dense, +explicit computing of the Jacobian is a very expense process. +Recently, the matrix-free\index{matrix-free} GMRES +was proposed~\cite{Telichevesky:DAC'95} +for RF shooting based simulation. +The new method leads to significant +savings due to its implicit calculation of new basis vectors +without the explicit formulation of the sensitivity matrix. + + +Modern computer architecture has shifted towards designs that employ +so called multi-core processor or chip-multiprocessors (CMP)~\cite{IntelMC'06,AMDMC'06}. +The family of graphic processing units (GPU) are among the most +powerful many-core computing systems in mass-market use~\cite{nvidia}. +For instance, the state-of-the-art NVIDIA Tesla T10 chip has a peak +performance of over 1~TFLOPS versus about 80--100~GFLOPS of Intel i5 +series Quad-core CPUs~\cite{Kirk:Book'10}. +In addition to the primary +use of GPUs in accelerating graphics rendering operations, there has +been considerable interest in exploiting GPUs for general purpose +computation (GPGPU)~\cite{gpgpu}. The introduction of new parallel +programming interfaces for general purpose computation, such as +Computer Unified Device Architecture (CUDA), +Stream SDK, and OpenCL~\cite{cuda,streamSDK,openCL}, +has made GPUs an attractive choice +for developing high-performance scientific computation tools +and solving practical engineering problems. +Hence, the applications with GPGPUs are rapidly growing in a broad +variety of parallel numerical computation works~\cite{CUDA_Zone}. + +%For this correction to be efficient and precise, it is critical +%for the Newton iteration to converge fast and yield accurate +%result. + +%Switching power converters and RF (radio-frequency) circuits are +%typically stiff systems where +%the fast signals (switching signals or carrier signals) and +%the slow moving signals (the output voltage waveforms and user signals) +%coexist. +%However, it is challenging to simulate the stiff systems +%using the traditional integration method such as +%Backward Euler and Trapezoidal method~\cite{Iserles:Book'96}. +%It has been shown that Gear-2 is more suitable +%for many practical problems such as stiff problems +%where one have different time constants +%(fast ones with large poles and slow ones with small poles)~\cite{Vlach:Book'94}. + +%GMRES is a Krylov-subspace based method for solving linear +%equation, which guarantees convergence and is efficient if the +%eigenvalues are tightly clustered together~\cite{Golub:Book'96}. +%Fortunately, most problems arising from the shooting method and +%the envelope-following method satisfy naturally the convergence +%criterion of GMRES~\cite{Nastov:ProcIEEE'07} (or after +%preconditioners applied if not). + + +%\item Newton update equation for envelope-following is formed with +% the sensitivity matrix derived from the Gear-2 integration rule for +% better stability. +% \item Detailed mathematical derivations to show +% the relationship between the sensitivity matrix and +% the information extracted from transient steps. + +After the basic algorithm of envelope-following is briefly reviewed +and Newton update equation is derived in Section~\ref{sec:ef}. In +Section~\ref{sec:gmres} presents the new parallel envelope method +where the CPU parallelization, matrix-free GMRES and Gear-2 +integration will be discussed. We will show the parallelization of the +Newton update, which is the most time consuming step, in the +envelop-follow method, in GPU platforms. We will also present the new +GMRES solver using matrix-free Jacobian-vector multiplication and the +Gear-2 integration for sensitivity based Newton update equation. +Numerical examples are shown in Section~\ref{sec:exp}, and finally, +this chapter is summarized in Section~\ref{sec:summary}. diff --git a/BookGPU/Makefile b/BookGPU/Makefile index 9280d0b..31bb997 100644 --- a/BookGPU/Makefile +++ b/BookGPU/Makefile @@ -5,16 +5,18 @@ all: pdflatex ${BOOK} pdflatex ${BOOK} bibtex bu1 - bibtex bu2 - bibtex bu3 - bibtex bu4 - bibtex bu5 - bibtex bu6 - bibtex bu7 - bibtex bu8 - bibtex bu9 -#don't put chapter 14, refs are included - bibtex bu11 +# bibtex bu2 +# bibtex bu3 +# bibtex bu4 +# bibtex bu5 +# bibtex bu6 +# bibtex bu7 +# bibtex bu8 +# bibtex bu9 +# #don't put chapter 14, refs are included +# bibtex bu11 +# # bibtex bu12 + makeindex ${BOOK}.idx pdflatex ${BOOK} pdflatex ${BOOK}