X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_gpu.git/blobdiff_plain/1874c46934f4ba7e8c2013d3829f65309456d292..55ce7168c6e69a2462d76c95dc9a5298ceedb04f:/BookGPU/Chapters/chapter5/ch5.tex

diff --git a/BookGPU/Chapters/chapter5/ch5.tex b/BookGPU/Chapters/chapter5/ch5.tex
index fed9a80..6355b09 100644
--- a/BookGPU/Chapters/chapter5/ch5.tex
+++ b/BookGPU/Chapters/chapter5/ch5.tex
@@ -12,7 +12,8 @@
 %\item efficient and scalable iterative methods for solution of high-order numerical methods and strategies for efficient implementations on desktop architectures.
 %\end{itemize}
 
-\section{Software development for heterogeneous architectures}
+\clearpage
+\section{Software development for heterogeneous\hfill\break architectures}
 %Our library facilitates massively parallelization through GPU computing and contains components for various iterative strategies such as DC, CG, CGNR, BiCGSTAB and GMRES(m) for solution of large linear systems along with support for preconditioning strategies. The goal is to create a reusable library and framework which provide general components with performance similar to that of a dedicated solver. Preliminary results show that performance overhead can be kept minimal in this new software framework [8].
 
 
@@ -25,9 +26,9 @@
 
 % there is a price/penalty to pay when using libraries, i.e. one doesn't necessarily get the best performance, because the architectural details are not visible to the programmer (keyword: Visibility  [Berkeley dwarf paper]).  However, if the library permits to write own add-ons/kernels (flexibility), this is no longer an issue.
 
-Massively parallel processors, such as graphical processing units (GPUs), have in recent years proven to be effective for a vast amount of scientific applications. Today, most desktop computers are equipped with one or more powerful GPUs, offering heterogeneous high-performance computing to a broad range of scientific researchers and software developers. Though GPUs are now programmable and can be highly effective computing units, they still pose challenges for software developers to fully utilize their efficiency. Sequential legacy codes are not always easily parallelized, and the time spent on conversion might not pay off in the end. This is particular true for heterogenous computers, where the architectural differences between the main and coprocessor can be so significant that they require completely different optimization strategies. The cache hierarchy management of CPUs and GPUs are an evident example hereof. In the past, industrial companies were able to boost application performance solely by upgrading their hardware systems, with an overt balance between investment and performance speedup. Today, the picture is different; not only do they have to invest in new hardware, but they also must account for the adaption and training of their software developers. What traditionally used to be a hardware problem, addressed by the chip manufacturers, has now become a software problem for application developers.
+Massively parallel processors, such as graphical processing units (GPUs), have in recent years proven to be effective for a vast amount of scientific applications. Today, most desktop computers are equipped with one or more powerful GPUs, offering heterogeneous high-performance computing to a broad range of scientific researchers and software developers. Though GPUs are now programmable and can be highly effective computing units, they still pose challenges for software developers to fully utilize their efficiency. Sequential legacy codes are not always easily parallelized, and the time spent on conversion might not pay off in the end. This is particular true for heterogeneous computers, where the architectural differences between the main and coprocessor can be so significant that they require completely different optimization strategies. The cache hierarchy management of CPUs and GPUs are an evident example hereof. In the past, industrial companies were able to boost application performance solely by upgrading their hardware systems, with an overt balance between investment and performance speedup. Today, the picture is different; not only do they have to invest in new hardware, but they also must account for the adaption and training of their software developers. What traditionally used to be a hardware problem, addressed by the chip manufacturers, has now become a software problem for application developers.
 
-Software libraries\index{software library}\index{library|see{software library}} can be a tremendous help for developers as they make it easier to implement an application, without requiring special knowledge of the underlying computer architecture and hardware. A library may be referred to as \emph{opaque} when it automatically utilize the available resources, without requiring specific details from the developer\cite{ch5:Asanovic:EECS-2006-183}. The ultimate goal for a successful library is to simplify the process of writing new software and thus to increase developer productivity. Since programmable heterogeneous CPU/GPU systems are a rather new phenomenon, there is yet a limited number of established software libraries that take full advantage of such heterogeneous high performance systems, and there are no de facto design standards for such systems either. Some existing libraries for conventional homogeneous systems have already added support for offloading computationally intense operations onto coprocessing GPUs. However, this approach comes at the cost of frequent memory transfers across the low bandwidth PCIe bus.
+Software libraries\index{software library}\index{library|see{software library}} can be a tremendous help for developers as they make it easier to implement an application, without requiring special knowledge of the underlying computer architecture and hardware. A library may be referred to as \emph{opaque} when it automatically utilizes the available resources, without requiring specific details from the developer\cite{ch5:Asanovic:EECS-2006-183}. The ultimate goal for a successful library is to simplify the process of writing new software and thus to increase developer productivity. Since programmable heterogeneous CPU/GPU systems are a rather new phenomenon, there is a limited number of established software libraries that take full advantage of such heterogeneous high performance systems, and there are no de facto design standards for such systems either. Some existing libraries for conventional homogeneous systems have already added support for offloading computationally intense operations onto coprocessing GPUs. However, this approach comes at the cost of frequent memory transfers across the low bandwidth PCIe bus.
 
 In this chapter, we focus on the use of a software library to help application developers achieve their goals without spending an immense amount of time on optimization details, while still offering close-to-optimal performance. A good library provides performance-portable implementations with intuitive interfaces, that hide the complexity of underlaying hardware optimizations. Unfortunately, opaqueness sometimes comes at a price, as one does not necessarily get the best performance when the architectural details are not \emph{visible} to the programmer~\cite{ch5:Asanovic:EECS-2006-183}. If, however, the library is flexible enough and permits developers to supply their own low-level implementations as well, this does not need to be an issue. These are some of the considerations library developers should take into account, and what we will try to address in this chapter.
 
@@ -99,7 +100,7 @@ Finite differences approximate the derivative of some function $u(x)$ as a weigh
 \end{align}
 where $q$ is the order of the derivative, $c_n$ is a set of finite difference coefficients, and $\alpha$ plus $\beta$ define the number of coefficients that are used for the approximation. The total set of contributing elements is called the stencil,\index{stencil} and the size of the stencil is called the rank, given as $\alpha + \beta + 1$. The stencil coefficients $c_n$ can be derived from a Taylor expansion based on the values of $\alpha$ and $\beta$, and $q$, using the method of undetermined coefficients~\cite{ch5:LeVeque2007}.
 An example of a three-point finite difference matrix that approximates the first ($q=1$) or second ($q=2$) derivative of a one-dimensional uniformly distributed vector $u$ of length $8$ is given here:
-\begin{eqnarray}\label{ch5:eq:stencilmatrix}
+\begin{flalign}\label{ch5:eq:stencilmatrix}
 \left[\begin{array}{c c c c c c c c}
 c_{00} & c_{01} & c_{02} &      0 &      0 &      0 &      0 &      0 \\
 c_{10} & c_{11} & c_{12} &      0 &      0 &      0 &      0 &      0 \\
@@ -120,7 +121,7 @@ u_5 \\
 u_6 \\
 u_7
 \end{array}\right]
-& \approx &
+ \approx 
 \left[\begin{array}{c}
 u^{(q)}_0 \\
 u^{(q)}_1 \\
@@ -131,7 +132,7 @@ u^{(q)}_5 \\
 u^{(q)}_6 \\
 u^{(q)}_7
 \end{array}\right].
-\end{eqnarray}
+\end{flalign}
 It is clear from this example that the matrix is sparse and that the same coefficients are repeated for all centered rows. The coefficients differ only near the boundaries, where off-centered stencils are used. It is natural to pack this information into a stencil operator that stores only the unique set of coefficients:
 \begin{eqnarray}\label{ch5:eq:stencilcoeffs}
 \myvec{c} & = &
@@ -141,10 +142,10 @@ c_{10} & c_{11} & c_{12} \\
 c_{20} & c_{21} & c_{22} \\
 \end{array}\right].
 \end{eqnarray}
-Matrix components precompute these compact stencil coefficients and provides member functions that computes the finite difference approximation of input vectors. Unit scaled coefficients (assuming grid spacing is one) are computed and stored to be accessible via both CPU and GPU memory. On the GPU, the constant memory space is used for faster memory access~\cite{ch5:cudaguide}. In order to apply a stencil on a non unit-spaced grid, with grid space $\Delta x$, the scale factor $1/(\Delta x)^q$ will have to be multiplied by the finite difference sum, i.e., $(c_{00}u_0 + c_{01}u_1 + c_{02}u_2)/(\Delta x)^q \approx u^{(q)}_0$ as in the first row of \eqref{ch5:eq:stencilmatrix}.
+Matrix components precompute these compact stencil coefficients and provides member functions that computes the finite difference approximation of input vectors. Unit scaled coefficients (assuming grid spacing is one) are computed and stored to be accessible via both CPU and GPU memory. On the GPU, the constant memory space is used for faster memory access~\cite{ch5:cudaguide}. In order to apply a stencil on a nonunit-spaced grid, with grid space $\Delta x$, the scale factor $1/(\Delta x)^q$ will have to be multiplied by the finite difference sum, i.e., $(c_{00}u_0 + c_{01}u_1 + c_{02}u_2)/(\Delta x)^q \approx u^{(q)}_0$ as in the first row of \eqref{ch5:eq:stencilmatrix}.
 
 Setting up a two-dimensional grid of size $N_x \times N_y$ in the unit square and computing the first derivative hereof is illustrated in Listing~\ref{ch5:lst:stencil}. The grid is a vector component, derived from the vector class. It is by default treated as a device object and memory is automatically allocated on the device to fit the grid size. The finite difference approximation as in \eqref{ch5:eq:fdstencil}, is performed via a CUDA kernel behind the scenes during the calls to \texttt{mult} and \texttt{diff\_x}, utilizing the memory hierarchy as the CUDA guidelines prescribe~\cite{ch5:cudaguide,ch5:cudapractice}. To increase developer productivity, kernel launch configurations have default settings, based on CUDA guidelines, principles, and experiences from performance testings, such that the user does not have to explicitly specify them. For problem-specific finite difference approximations, where the built-in stencil operators are insufficient, a pointer to the coefficient matrix \eqref{ch5:eq:stencilcoeffs} can be accessed as demonstrated in Listing \ref{ch5:lst:stencil} and passed to customized kernels.
-
+\pagebreak
 \lstinputlisting[label=ch5:lst:stencil,caption={two-dimensional finite difference stencil example: computing the first derivative using five points ($\alpha=\beta=2$) per dimension, a total nine-point stencil}]{Chapters/chapter5/code/ex2.cu}
 
 In the following sections we demonstrate how to go from an initial value problem (IVP) or a boundary value problem (BVP) to a working application solver by combining existing library components along with new custom-tailored components. We also demonstrate how to apply spatial and temporal domain decomposition strategies that can make existing solvers take advantage of systems equipped with multiple GPUs. Next section demonstrates how to rapidly assemble a PDE solver using library components.
@@ -184,11 +185,11 @@ because it has a known analytic solution over the entire time span, and it satis
     \caption{Discrete solution, at times $t=0s$ and $t=0.05s$, using \eqref{ch5:eq:heatinit} as the initial condition and a small $20\times20$ numerical grid.}\label{ch5:fig:heatsolution}
 \end{figure}
 
-We use a Method of Lines (MoL)\index{method of lines} approach to solve \eqref{ch5:eq:heateq}. Thus,  the spatial derivatives are replaced with finite difference approximations, leaving only the temporal derivative as unknown. The spatial derivatives are approximated from $\myvec{u}^n$, where $\myvec{u}^n$ represents the approximate solution to $u(t_n)$ at a given time $t_n$ with time step size $\delta t$ such that $t_n=n\delta t$ for $n=0,1,\ldots$. The finite difference approximation\index{finite difference} can be interpreted as a matrix-vector product as sketched in \eqref{ch5:eq:stencilmatrix}, and so the semi-discrete heat conduction problem becomes
+We use a Method of Lines (MoL)\index{method of lines} approach to solve \eqref{ch5:eq:heateq}. Thus,  the spatial derivatives are replaced with finite difference approximations, leaving only the temporal derivative as unknown. The spatial derivatives are approximated from $\myvec{u}^n$, where $\myvec{u}^n$ represents the approximate solution to $u(t_n)$ at a given time $t_n$ with time step size $\delta t$ such that $t_n=n\delta t$ for $n=0,1,\ldots$ The finite difference approximation\index{finite difference} can be interpreted as a matrix-vector product as sketched in \eqref{ch5:eq:stencilmatrix}, and so the semi-discrete heat conduction problem becomes
 \begin{align}\label{ch5:eq:discreteheateq}
 \frac{\partial u}{\partial t} = \mymat{A}\myvec{u}, \qquad \mymat{A} \in \mathbb{R}^{N\times N}, \quad \myvec{u} \in \mathbb{R}^{N},
 \end{align}
-where $\mymat{A}$ is the sparse finite difference matrix and $N$ is the number of unknowns in the discrete system. The temporal derivative is now free to be approximated by any suitable choice of a time-integration method\index{time integration}. The most simple integration scheme would be the first-order accurate explicit forward Euler method\index{forward Euler},
+where $\mymat{A}$ is the sparse finite difference matrix and $N$ is the number of unknowns in the discrete system. The temporal derivative is now free to be approximated by any suitable choice of a time-integration method\index{time integration}. The most simple integration scheme would be the first-order accurate explicit forward Euler method\index{Euler!forward Euler},
 \begin{align}\label{ch5:eq:forwardeuler}
 \myvec{u}^{n+1} = \myvec{u}^n + \delta t\,\mymat{A}\myvec{u}^n,
 \end{align}
@@ -210,14 +211,14 @@ The basic numerical approach to solve the heat conduction problem has now been o
 \subsubsection{Assembling the heat conduction solver}
 Before we are able to numerically solve the discrete heat conduction problem \eqref{ch5:eq:heateq}, we need implementations to handle the the following items:
 \begin{description}
-\item[Grid.] A discrete numerical grid to represent the two-dimensional heat distribution domain and the arithmetical working precision ($32$-bit single precision or $64$-bit double precision).
-\item[RHS.] A right-hand side operator for \eqref{ch5:eq:discreteheateq} that approximates the second-order spatial derivatives (matrix-vector product).
-\item[Boundary conditions.] A strategy that ensures that the Dirichlet conditions are satisfied on the boundary.
-\item[Time integrator.] A time integration scheme, that approximates the time derivative from \eqref{ch5:eq:discreteheateq}.
+\item[Grid]-- A discrete numerical grid to represent the two-dimensional heat distribution domain and the arithmetical working precision ($32$-bit single-precision or $64$-bit double-precision).
+\item[RHS]-- A right-hand side operator for \eqref{ch5:eq:discreteheateq} that approximates the second-order spatial derivatives (matrix-vector product).
+\item[Boundary conditions]-- A strategy that ensures that the Dirichlet conditions are satisfied on the boundary.
+\item[Time integrator]-- A time integration scheme, that approximates the time derivative from \eqref{ch5:eq:discreteheateq}.
 \end{description}
 All items are either directly available in the library or can be designed from components herein. The built-in stencil operator may assist in implementing the matrix-vector product, but we need to explicitly ensure that the Dirichlet boundary conditions are satisfied. We demonstrated in Listing \ref{ch5:lst:stencil} how to approximate the derivative using flexible order finite difference stencils. However, from \eqref{ch5:eq:heateqbc} we know that boundary values are zero. Therefore, we extend the stencil operator with a simple kernel call that assigns zero to the entire boundary. Listing \ref{ch5:lst:laplaceimpl} shows the code for the two-dimensional Laplace right-hand side operator. The constructor takes as an argument the stencil half size $\alpha$ and assumes $\alpha=\beta$. Thus, the total two-dimensional stencil rank will be $4\alpha+1$. For simplicity we also assume that the grid is uniformly distributed, $N_x=N_y$. Performance optimizations for the stencil kernel, such as shared memory utilization, are handled in the underlying implementation, accordingly to CUDA guidelines~\cite{ch5:cudaguide,ch5:cudapractice}. The macros, \texttt{BLOCK1D} and \texttt{GRID1D}, are used to help set up kernel configurations based on grid sizes and \texttt{RAW\_PTR} is used to cast the vector object to a valid device memory pointer.
 
-\lstinputlisting[label=ch5:lst:laplaceimpl,caption={the right-hand side Laplace operator: the built-in stencil approximates the two dimensional spatial derivatives, while the custom set\_dirichlet\_bc kernel takes care of satisfying boundary conditions}]{Chapters/chapter5/code/laplacian.cu}
+\lstinputlisting[label=ch5:lst:laplaceimpl,caption={the right-hand side Laplace operator: the built-in stencil approximates the two-dimensional spatial derivatives, while the custom set\_dirichlet\_bc kernel takes care of satisfying boundary conditions}]{Chapters/chapter5/code/laplacian.cu}
 
 With the right-hand side operator in place, we are ready to implement the solver. For this simple PDE problem we compute all necessary initial data in the body of the main function and use the forward Euler time integrator to compute the solution until $t=t_{end}$. For more advanced solvers, a built-in \texttt{ode\_solver} class is defined that helps take care of initialization and storage of multiple state variables. Declaring type definitions for all components at the beginning of the main file gives a good overview of the solver composition. In this way, it will be easy to control or change solver components at later times. Listing~\ref{ch5:lst:heattypedefs} lists the type definitions\index{type definitions} that are used to assemble the heat conduction solver. %\stefan{elaborate on the choices}.
 
@@ -229,7 +230,7 @@ typedef gpulab::grid<value_type>            vector_type;
 typedef vector_type::property_type          property_type;
 typedef gpulab::integration::forward_euler  time_integrator_type;
 \end{lstlisting}
-The grid is by default treated as a device object, and memory is allocated on the GPU upon initialization of the grid. Setting up the grid can be done via the property type class. The property class holds information about the discrete and physical dimensions, along with fictitious ghost (halo) layers and periodicity conditions. For the heat conduction problem we use a non periodic domain of size $N\times N$ within the unit square with no ghost layers. Listing \ref{ch5:lst:gridsetup} illustrates the grid assembly.
+The grid is by default treated as a device object, and memory is allocated on the GPU upon initialization of the grid. Setting up the grid can be done via the property type class. The property class holds information about the discrete and physical dimensions, along with fictitious ghost (halo) layers and periodicity conditions. For the heat conduction problem we use a nonperiodic domain of size $N\times N$ within the unit square with no ghost layers. Listing \ref{ch5:lst:gridsetup} illustrates the grid assembly.
 
 \lstset{label=ch5:lst:gridsetup,caption={creating a two-dimensional grid of size \texttt{N} times \texttt{N} and physical dimension $0$ to $1$}}
 \begin{lstlisting}
@@ -244,7 +245,7 @@ vector_type                    u(props);
 \end{lstlisting}
 Hereafter the vector \texttt{u} can be initialized accordingly to \eqref{ch5:eq:heatinit}. Finally we need to instantiate the right-hand side Laplacian operator from Listing \ref{ch5:lst:laplaceimpl} and the forward Euler time integrator in order to integrate from $t_0$ until $t_{end}$.
 
-\lstset{label=ch5:lst:timeintegrator,caption=creating a time integrator and the right-hand side Laplacian operator.}
+\lstset{label=ch5:lst:timeintegrator,caption=creating a time integrator and the right-hand side Laplacian operator}
 \begin{lstlisting}
 rhs_type rhs(alpha);           // Create right-hand side operator
 time_integrator_type solver;   // Create time integrator
@@ -258,7 +259,7 @@ The last line invokes the forward Euler time integration scheme defined in Listi
 
 \subsubsection{Numerical solutions to the heat conduction problem}
 
-Solution time for the heat conduction problem is in itself not very interesting, as it is only a simple model problem. What is interesting for GPU kernels, such as the finite differences kernel, is that increased computational work often comes with a very small price, because the fast computations can be hidden by the relatively slower memory fetches. Therefore, we are able to improve the accuracy of the numerical solution via more accurate finite differences (larger stencil sizes), while improving the computational performance in terms of floating point operations per second (flops). Figure \ref{ch5:fig:stencilperformance} confirms, that larger stencils improve the kernel performance. Notice that even though these performance results are favorable compared to single core systems ($\sim 10$ GFlops double precision on a $2.5$-GHz processor), they are still far from their peak performance, e.g., $\sim 2.4$ TFlops single precision for the GeForce GTX590. The reason is that the kernel is bandwidth bound, i.e., performance is limited by the time it takes to move memory between the global GPU memory and the chip. The Tesla K20 performs better than the GeForce GTX590 because it obtains the highest bandwidth. Being bandwidth bound is a general limitation for matrix-vector-like operations that arise from the discretization of PDE problems. Only matrix-matrix multiplications, which have a high ratio of computations versus memory transactions, are able to reach near-optimal performance results~\cite{ch5:Kirk2010}. These kinds of operators are, however, rarely used to solve PDE problems.
+Solution time for the heat conduction problem is in itself not very interesting, as it is only a simple model problem. What is interesting for GPU kernels, such as the finite differences kernel, is that increased computational work often comes with a very small price, because the fast computations can be hidden by the relatively slower memory fetches. Therefore, we are able to improve the accuracy of the numerical solution via more accurate finite differences (larger stencil sizes), while improving the computational performance in terms of floating point operations per second (flops). Figure \ref{ch5:fig:stencilperformance} confirms, that larger stencils improve the kernel performance. Notice that even though these performance results are favorable compared to single core systems ($\sim 10$ GFlops double-precision on a $2.5$-GHz processor), they are still far from their peak performance, e.g., $\sim 2.4$ TFlops single-precision for the GeForce GTX590. The reason is that the kernel is bandwidth bound, i.e., performance is limited by the time it takes to move memory between the global GPU memory and the chip. The Tesla K20 performs better than the GeForce GTX590 because it obtains the highest bandwidth. Being bandwidth bound is a general limitation for matrix-vector-like operations that arise from the discretization of PDE problems. Only matrix-matrix multiplications, which have a high ratio of computations versus memory transactions, are able to reach near-optimal performance results~\cite{ch5:Kirk2010}. These kinds of operators are, however, rarely used to solve PDE problems.
 
 \begin{figure}[!htb]
 \setlength\figureheight{0.35\textwidth}
@@ -273,14 +274,14 @@ Solution time for the heat conduction problem is in itself not very interesting,
 }%
 %{\includegraphics[width=0.5\textwidth]{Chapters/chapter5/figures/AlphaPerformanceGTX590_N16777216_conv.pdf}}
 }
-\caption[Single- and double-precision floating point operations per second for a two-dimensional stencil operator on a numerical grid of size $4096^2$.]{Single and double precision floating point operations per second for a two dimensional stencil operator on a numerical grid of size $4096^2$. Various stencil sizes are used $\alpha=1,2,3,4$, equivalent to $5$pt, $9$pt, $13$pt, and $17$pt stencils.}\label{ch5:fig:stencilperformance}
+\caption[Single- and double-precision floating point operations per second for a two-dimensional stencil operator on a numerical grid of size $4096^2$.]{Single- and double-precision floating point operations per second for a two-dimensional stencil operator on a numerical grid of size $4096^2$. Various stencil sizes are used $\alpha=1,2,3,4$, equivalent to $5$pt, $9$pt, $13$pt, and $17$pt stencils.}\label{ch5:fig:stencilperformance}
 \end{figure}
 
 
 
 
 \subsection{Poisson equation}\index{Poisson equation}
-The Poisson equation is a second-order elliptic differential equation, often encountered in applications within scientific fields such as electrostatics and mechanics. We consider the two- dimensional BVP \index{boundary volume problem} defined in terms of Poisson's equation with homogeneous Dirichlet boundary conditions on the form
+The Poisson equation is a second-order elliptic differential equation, often encountered in applications within scientific fields such as electrostatics and mechanics. We consider the two-dimensional BVP \index{boundary volume problem} defined in terms of Poisson's equation with homogeneous Dirichlet boundary conditions on the form
 \begin{subequations}\begin{align}
 \nabla^2 u = f(x,y),& \qquad (x,y) \in \Omega([0,1]\times[0,1]), \\
 u = 0,& \qquad (x,y) \in \partial\Omega.
@@ -295,7 +296,7 @@ The spatial derivative in \eqref{ch5:eq:poissoneq} is again approximated with fi
 \begin{align}
 \mymat{A}\myvec{u}=\myvec{f}, \qquad \myvec{u},\myvec{f} \in \mathbb{R}^{N}, \quad \mymat{A} \in \mathbb{R}^{N\times N}, \label{ch5:eq:poissonsystem}
 \end{align}
-where $\mymat{A}$ is the sparse matrix formed by finite difference coefficients, $N$ is the number of unknowns, and $\myvec{f}$ is given by \eqref{ch5:eq:poissonrhs}. Equation \eqref{ch5:eq:poissonsystem} can be solved in numerous ways, but a few observations may help do it more efficiently. Direct solvers based on Gaussian elimination are accurate and use a finite number of operations for a constant problem size. However, the arithmetic complexity grows with the problem size by as much as $\mathcal{O}(N^3)$ and does not exploit the sparsity of $\mymat{A}$. Direct solvers are therefore mostly feasible for dense systems of limited sizes. Sparse direct solvers exist, but they are often difficult to parallelize, or applicable for only certain types of matrices. Regardless of the discretization technique, the discretization of an elliptic PDE into a linear system as in \eqref{ch5:eq:poissonsystem} yields a very sparse matrix $\mymat{A}$ when $N$ is large. Iterative methods\index{iterative methods} for solving large sparse linear systems find broad use in scientific applications, because they require only an implementation of the matrix-vector product, and they often use a limited amount of additional memory. Comprehensive introductions to iterative methods may be found in any of~\cite{ch5:Saad2003,ch5:Kelley1995,ch5:Barrett1994}.
+where $\mymat{A}$ is the sparse matrix formed by finite difference coefficients, $N$ is the number of unknowns, and $\myvec{f}$ is given by \eqref{ch5:eq:poissonrhs}. Equation \eqref{ch5:eq:poissonsystem} can be solved in numerous ways, but a few observations may help do it more efficiently. Direct solvers based on Gaussian elimination are accurate and use a finite number of operations for a constant problem size. However, the arithmetic complexity grows with the problem size by as much as $\mathcal{O}(N^3)$ and does not exploit the sparsity of $\mymat{A}$. Direct solvers are therefore mostly feasible for dense systems of limited sizes. Sparse direct solvers exist, but they are often difficult to parallelize, or applicable for only certain types of matrices. Regardless of the discretization technique, the discretization of an elliptic PDE into a linear system as in \eqref{ch5:eq:poissonsystem} yields a very sparse matrix $\mymat{A}$ when $N$ is large. Iterative methods\index{iterative method} for solving large sparse linear systems find broad use in scientific applications, because they require only an implementation of the matrix-vector product, and they often use a limited amount of additional memory. Comprehensive introductions to iterative methods may be found in any of~\cite{ch5:Saad2003,ch5:Kelley1995,ch5:Barrett1994}.
 
 One benefit of the high abstraction level and the template-based library design is to allow developers to implement their own components, such as iterative methods for solving sparse linear systems. The library includes three popular iterative methods: conjugate gradient, defect correction\index{defect correction}, and geometric multigrid. The conjugate gradient method is applicable only to systems with symmetric positive definite matrices. This is true for the two-dimensional Poisson problem, when it is discretized with a five-point finite difference stencil, because then there will be no off-centered approximations near the boundary. For high-order approximations ($\alpha>1$), we use the defect correction method with multigrid preconditioning. See e.g., \cite{ch5:Trottenberg2001} for details on multigrid methods.
 
@@ -408,7 +409,7 @@ Defect correction in combination with multigrid preconditioning enables efficien
 
 \section{Optimization strategies for multi-GPU systems}\label{ch5:sec:multigpu}\index{multi-GPU}
 
-CUDA enabled GPUs are optimized for high memory bandwidth and fast on-chip performance. However, the role as a separate coprocessor to the CPU can be a limiting factor for large scale scientific applications, because the GPU memory capacity is fixed and is only in the range of a few gigabytes. In comparison, it is not unusual for a high-end workstation to be equipped with $\sim32$GB of main memory, plus a terabyte hard disk capacity for secondary storage. Therefore, large scale scientific applications that process gigabytes of data, require distributed computations on multiple GPU devices. Multi-GPU desktop computers and clusters can have a very attractive peak performance, but the addition of multiple devices introduces the potential performance bottleneck of slow data transfers across PCIe busses and network interconnections, as illustrated in Figure \ref{ch5:fig:gpu2gputransfer}. The ratio between data transfers and computational work has a significant impact on the possibility for latency hiding and thereby overall application performance.
+CUDA-enabled GPUs are optimized for high memory bandwidth and fast on-chip performance. However, the role as a separate coprocessor to the CPU can be a limiting factor for large scale scientific applications, because the GPU memory capacity is fixed and is only in the range of a few gigabytes. In comparison, it is not unusual for a high-end workstation to be equipped with $\sim32$GB of main memory, plus a terabyte hard disk capacity for secondary storage. Therefore, large scale scientific applications that process gigabytes of data, require distributed computations on multiple GPU devices. Multi-GPU desktop computers and clusters can have a very attractive peak performance, but the addition of multiple devices introduces the potential performance bottleneck of slow data transfers across PCIe busses and network interconnections, as illustrated in Figure \ref{ch5:fig:gpu2gputransfer}. The ratio between data transfers and computational work has a significant impact on the possibility for latency hiding and thereby overall application performance.
 
 \begin{figure}[!htb]
 \begin{center}
@@ -419,7 +420,7 @@ CUDA enabled GPUs are optimized for high memory bandwidth and fast on-chip perfo
 \end{figure}
 
 
-Developing applications that exploit the full computational capabilities of modern clusters--GPU-based or not--is no trivial matter. Developers are faced with the complexity of distributing and coordinating computations on nodes consisting of many-core CPUs, GPUs and potentially other types of accelerators as well. These complexities give rise to challenges in finding numerical algorithms, that are well suited for such systems, forcing developers to search for novel methods that utilize concurrency.
+Developing applications that exploit the full computational capabilities of modern clusters--GPU-based or not--is no trivial matter. Developers are faced with the complexity of distributing and coordinating computations on nodes consisting of many-core CPUs, GPUs, and potentially other types of accelerators as well. These complexities give rise to challenges in finding numerical algorithms, that are well suited for such systems, forcing developers to search for novel methods that utilize concurrency.
 
 To ease software development, we use MPI-2 for message passing and ensure a safe and private communication space by creation of a communicator private to the library during initialization, as recommended by Hoefler and Snir~\cite{ch5:Hoefler2011}. With the addition of remote direct memory access (RDMA) for GPUDirect it is possible to make direct memory transfers between recent generation of GPUs (Kepler), eliminating CPU overhead. Unfortunately there are some strict system and driver requirements to enable these features. Therefore, in the following examples, device memory is first transferred to the CPU main memory before invoking any MPI calls. The library provides device-to-device transfers via template-based routines that work directly with GPU vector objects. This hides the complexity of message passing from the developer and helps developers design new components for multi-GPU execution. 
 
@@ -443,11 +444,13 @@ An alternative to the preconditioning strategy is to have each subdomain query i
 \caption[Domain distribution of a two-dimensional grid into three subdomains.]{Domain distribution of a two-dimensional grid into three subdomains. {\large$\bullet$} and {\scriptsize$\textcolor[rgb]{0.5,0.5,0.5}{\blacksquare}$} represent internal grid points and ghost points, respectively.}\label{ch5:fig:dd2d}
 \end{figure}
 
-Topologies are introduced via an extra template argument to the grid class. A grid is by default not decomposed, because the default template argument is based on a non distribution topology implementation. The grid class is extended with a new member function \texttt{update()}, which makes sure that all ghost points are updated according to the grid topology. The library contains topologies based on one-dimensional and two-dimensional distributions of the grid. The number of grid subdomains will be equal to the number of MPI processes executing the program.
+Topologies are introduced via an extra template argument to the grid class. A grid is by default not decomposed, because the default template argument is based on a nondistribution topology implementation. The grid class is extended with a new member function \texttt{update()}, which makes sure that all ghost points are updated according to the grid topology. The library contains topologies based on one-dimensional and two-dimensional distributions of the grid. The number of grid subdomains will be equal to the number of MPI processes executing the program.
 
-If grid ghost layers are updated whenever information from adjacent subdomains is needed, e.g., before a stencil operation, all interior points will be exactly the same as they would be for the non distributed setup. Therefore, one advantage of this approach is that the algorithmic efficiency of an application can be preserved, if grid updates are consistently invoked at the proper times.
+If grid ghost layers are updated whenever information from adjacent subdomains is needed, e.g., before a stencil operation, all interior points will be exactly the same as they would be for the nondistributed setup. Therefore, one advantage of this approach is that the algorithmic efficiency of an application can be preserved, if grid updates are consistently invoked at the proper times.
 
-Distributed performance for the finite difference stencil operation is illustrated in Figure \ref{ch5:fig:multigpu}. The timings include the compute time for the finite difference approximation and the time for updating ghost layers via message passing. It is obvious from Figure \ref{ch5:fig:multigpu:a} that communication overhead dominates for the smallest problem sizes, where the non distributed grid (1 GPU) is fastest. However, communication overhead does not grow as rapidly as computation times, due to the surface-to-volume ratio. Therefore message passing becomes less influential for large problems, where reasonable performance speedups are obtained. Figure \ref{ch5:fig:multigpu:b} demonstrates how the computational performance on multi-GPU systems can be significantly improved for various stencil sizes. With this simple domain decomposition technique, developers are able to implement applications based on heterogeneous distributed computing, without explicitly dealing with message passing and it is still possible to provide user specific implementations of the topology class for customized grid updates.
+Distributed performance for the finite difference stencil operation is illustrated in Figure \ref{ch5:fig:multigpu}. The timings include the compute time for the finite difference approximation and the time for updating ghost layers via message passing. It is obvious from Figure \ref{ch5:fig:multigpu:a} that communication overhead dominates for the smallest problem sizes, where the nondistributed grid (1 GPU) is fastest. However, communication overhead does not grow as rapidly as computation times, due to the surface-to-volume ratio. Therefore message passing becomes less influential for large problems, where reasonable performance speedups are obtained. Figure \ref{ch5:fig:multigpu:b} demonstrates how the computational performance on multi-GPU systems can be significantly improved for various stencil sizes. With this simple domain decomposition technique, developers are able to implement applications based on heterogeneous distributed computing, without explicitly dealing with message passing and it is still possible to provide user specific implementations of the topology class for customized grid updates.
+
+\clearpage
 
 % TODO: Should we put in the DD algebra?
 
@@ -460,7 +463,7 @@ Distributed performance for the finite difference stencil operation is illustrat
     %{\includegraphics[width=0.5\textwidth]{Chapters/chapter5/figures/MultiGPUAlpha3TimingsTeslaM2050_conv.pdf}}
     \label{ch5:fig:multigpu:a}
     }%
-    \subfigure[Performance at $N=4069^2$, single precision.]{
+    \subfigure[Performance at $N=4069^2$, single-precision.]{
     {\scriptsize\input{Chapters/chapter5/figures/MultiGPUAlphaPerformanceTeslaM2050_N16777216.tikz}}
 	%{\includegraphics[width=0.5\textwidth]{Chapters/chapter5/figures/MultiGPUAlphaPerformanceTeslaM2050_N16777216_conv.pdf}}
     \label{ch5:fig:multigpu:b}
@@ -483,7 +486,7 @@ The parareal algorithm was first presented in 2001, in a paper by Lions et al.~\
 \begin{center}
 \input{Chapters/chapter5/figures/ParallelInTime.tikz}
 \end{center}
-\caption[Time domain decomposition.]{Time domain decomposition. A compute node is assigned to each individual time subdomain to compute the initial value problem. Consistency at the time subdomain boundaries is obtained with the application of a computationally cheap integrator in conjunction with the parareal iterative predictor-corrector algorithm}\label{ch5:fig:ParallelInTime}
+\caption[Time domain decomposition.]{Time domain decomposition. A compute node is assigned to each individual time subdomain to compute the initial value problem. Consistency at the time subdomain boundaries is obtained with the application of a computationally cheap integrator in conjunction with the parareal iterative predictor-corrector algorithm.}\label{ch5:fig:ParallelInTime}
 \end{figure}%
 Initial states for these problems are needed and supplied by a simple, less accurate, but computationally cheap sequential integrator. The smaller independent evolution problems can then be solved in parallel. The information, generated during the concurrent solution of the independent evolution problems with accurate propagators and inaccurate initial states, is used in a predictor-corrector fashion in conjunction with the coarse integrator to propagate the solution faster, now using the information generated in parallel. We define the decomposition into $N$ intervals, that is,
 \begin{align}
@@ -506,7 +509,7 @@ Using the defined $\mathcal{F}_{\Delta T}$ and $\mathcal{G}_{\Delta T}$ operator
 \begin{equation}\label{ch5:eq:PARAREAL}
 U_{n}^{k+1}=\mathcal{\mathcal{G}}_{\Delta T}\left(U_{n-1}^{k+1}\right)+\mathcal{\mathcal{F}}_{\Delta T}\left(U_{n-1}^{k}\right)-\mathcal{\mathcal{G}}_{\Delta T}\left(U_{n-1}^{k}\right),\quad U_{0}^{k}=u^{0},
 \end{equation}
-with the initial prediction $U^{0}_{n} = \mathcal{G}_{\Delta T}^{n} u^{0}$ for $n=1\ldots N$ and $k=1\ldots K$. $N$ being the number of time subdomains, while $K\geq1$ is the number of predictor-corrector iterations applied. The parareal algorithm is implemented in the library as a separate time-integration component, using a fully distributed work scheduling model, as proposed by Aubanel~\cite{ch5:EA10}. The model is schematically presented in Figure \ref{ch5:fig:FullyDistributedCores}. The parareal component hides all communication and work distribution from the application developer.  It is defined such that a user only has to decide what coarse and fine propagators to use. Setting up the type definitions for parareal time-integration using forward Euler for coarse propagation and fourth order Runge-Kutta for fine propagation could then be defined as in Listings \ref{ch5:lst:parareal}. The number of GPUs used for parallelization depends on the number of MPI processes executing the application.
+with the initial prediction $U^{0}_{n} = \mathcal{G}_{\Delta T}^{n} u^{0}$ for $n=1\ldots N$ and $k=1\ldots K$. $N$ being the number of time subdomains, while $K\geq1$ is the number of predictor-corrector iterations applied. The parareal algorithm is implemented in the library as a separate time-integration component, using a fully distributed work scheduling model, as proposed by Aubanel~\cite{ch5:EA10}. The model is schematically presented in Figure \ref{ch5:fig:FullyDistributedCores}. The parareal component hides all communication and work distribution from the application developer.  It is defined such that a user only has to decide what coarse and fine propagators to use. Setting up the type definitions for parareal time-integration using forward Euler for coarse propagation and fourth order Runge-Kutta for fine propagation could then be defined as in Listing~\ref{ch5:lst:parareal}. The number of GPUs used for parallelization depends on the number of MPI processes executing the application.
 \lstset{label=ch5:lst:parareal,caption={assembling a parareal time integrator using forward Euler for coarse propagation and a Runge-Kutta method for fine propagation}}
 \begin{lstlisting}
 typedef gpulab::integration::forward_euler          coarse;
@@ -543,9 +546,10 @@ We can now estimate the speedup, here denoted $\psi$, as the ratio between the c
 \begin{align}\label{ch5:eq:EstiSpeedBasicPara}
 \psi=\frac{N\frac{\Delta T}{\delta t}\mathcal{C}_\mathcal{F}}{\left(k+1\right)N\mathcal{C}_\mathcal{G}\frac{\Delta T}{\delta T}+k\mathcal{C}_\mathcal{F}\frac{\Delta T}{\delta t}}=\frac{N}{\left(k+1\right)N\frac{\mathcal{C}_\mathcal{G}}{\mathcal{C}_\mathcal{F}}\frac{\delta t}{\delta T}+k}.
 \end{align}
-If we additionally assume that the time spent on coarse propagation is negligible compared to the time spent on the fine propagation, i.e., the limit $\frac{\mathcal{C}_\mathcal{G}}{\mathcal{C}_\mathcal{F}}\frac{\delta t}{\delta T}\rightarrow0$, the estimate reduces to $\psi=\frac{N}{k}$. It is thus clear that the number of iterations $k$ for the algorithm to converge poses an upper bound on obtainable parallel efficiency. The number of iterations needed for convergence is intimately coupled with the ratio $R$ between the speed of the fine and the coarse integrators $\frac{\mathcal{C}_\mathcal{F}}{\mathcal{C}_\mathcal{G}}\frac{\delta T}{\delta t}$. Using a slow, but more accurate coarse integrator will lead to convergence in fewer iterations $k$, but at the same time it also makes $R$ smaller. Ultimately, this will degrade the obtained speedup as can be deduced from \eqref{ch5:eq:EstiSpeedBasicPara}, and by Amdahl's law it will also lower the upper bound on possible attainable speedup. Thus, $R$ \emph{cannot} be made arbitrarily large since the ratio is inversely proportional to the number of iterations $k$ needed for convergence. This poses a challenge in obtaining speedup and is a trade-off between time spent on the fundamentally sequential part of the algorithm and the number of iterations needed for convergence. It is particularly important to consider this trade-off in the choice of stopping strategy; a more thorough discussion on this topic is available in \cite{ch5:ASNP12} for the interested reader. Measurements on parallel efficiency are typically observed in the literature to be in the range of 20--50\%, depending on the problem and the number of time subdomains, which is also confirmed by our measurements using GPUs. Here we include a demonstration of the obtained speedup of parareal applied to the two-dimensional heat problem \eqref{ch5:eq:heateq}. In Figure \ref{ch5:fig:pararealRvsGPUs} the iterations needed for convergence using the forward Euler method for both fine and coarse integration are presented. $R$ is regulated by changing the time step size for the coarse integrator. In Figure \ref{ch5:fig:pararealGPUs} speedup and parallel efficiency measurements are presented. Notice, when using many GPUs it is advantageous to use a faster, less accurate coarse propagator, despite it requires an extra parareal iteration that increases the total computational complexity.
+If we additionally assume that the time spent on coarse propagation is negligible compared to the time spent on the fine propagation, i.e., the limit $\frac{\mathcal{C}_\mathcal{G}}{\mathcal{C}_\mathcal{F}}\frac{\delta t}{\delta T}\rightarrow0$, the estimate reduces to $\psi=\frac{N}{k}$. It is thus clear that the number of iterations $k$ for the algorithm to converge poses an upper bound on obtainable parallel efficiency. The number of iterations needed for convergence is intimately coupled with the ratio $R$ between the speed of the fine and the coarse integrators $\frac{\mathcal{C}_\mathcal{F}}{\mathcal{C}_\mathcal{G}}\frac{\delta T}{\delta t}$. Using a slow, but more accurate coarse integrator will lead to convergence in fewer iterations $k$, but at the same time it also makes $R$ smaller. Ultimately, this will degrade the obtained speedup as can be deduced from \eqref{ch5:eq:EstiSpeedBasicPara}, and by Amdahl's law it will also lower the upper bound on possible attainable speedup. Thus, $R$ \emph{cannot} be made arbitrarily large since the ratio is inversely proportional to the number of iterations $k$ needed for convergence. This poses a challenge in obtaining speedup and is a trade-off between time spent on the fundamentally sequential part of the algorithm and the number of iterations needed for convergence. It is particularly important to consider this trade-off in the choice of stopping strategy; a more thorough discussion on this topic is available in \cite{ch5:ASNP12} for the interested reader. Measurements on parallel efficiency are typically observed in the literature to be in the range of 20--50\%, depending on the problem and the number of time subdomains, which is also confirmed by our measurements using GPUs. Here we include a demonstration of the obtained speedup of parareal applied to the two-dimensional heat problem \eqref{ch5:eq:heateq}. In Figure \ref{ch5:fig:pararealRvsGPUs} the iterations needed for convergence using the forward Euler method for both fine and coarse integration are presented. $R$ is regulated by changing the time step size for the coarse integrator. In Figure \ref{ch5:fig:pararealGPUs} speedup and parallel efficiency measurements are presented. Notice, when using many GPUs it is advantageous to use a faster, less accurate coarse propagator, despite it requires an extra parareal iteration that increases the total computational complexity.\eject
 
-\begin{figure}[!htb]
+
+\begin{figure}[t!]
     \setlength\figureheight{0.32\textwidth}
     \setlength\figurewidth{0.35\textwidth}
     \begin{center}
@@ -556,9 +560,9 @@ If we additionally assume that the time spent on coarse propagation is negligibl
     \subfigure[Iterations $K$ needed to obtain a relative error less than $10^{-5}$.]{
     {\small\input{Chapters/chapter5/figures/pararealKvsRvsGPUs.tikz}}
     \label{ch5:fig:pararealRvsGPUs:b}
-    }
+    }\vspace*{-8pt}
     \end{center}
-    \caption[Parareal convergence properties as a function of $R$ and number of GPUs used.]{Parareal convergence properties as a function of $R$ and number GPUs used. The error is measured as the relative difference between the purely sequential solution and the parareal solution.}\label{ch5:fig:pararealRvsGPUs}
+    \caption[Parareal convergence properties as a function of $R$ and number of GPUs used.]{Parareal convergence properties as a function of $R$ and number of GPUs used. The error is measured as the relative difference between the purely sequential solution and the parareal solution.}\label{ch5:fig:pararealRvsGPUs}
 \end{figure}
 
 \begin{figure}[!htb]
@@ -572,11 +576,10 @@ If we additionally assume that the time spent on coarse propagation is negligibl
     \subfigure[Measured parallel efficiency]{
     {\small\input{Chapters/chapter5/figures/pararealEfficiencyvsRvsGPUs.tikz}}
     \label{ch5:fig:pararealGPUs:b}
-    }
+    }\vspace*{-8pt}
     \end{center}
-    \caption[Parareal performance properties as a function of $R$ and number of GPUs used.]{Parareal performance properties as a function of $R$ and number GPUs used. Notice how the obtained performance depends greatly on the choice of $R$ as a function of the number of GPUs. Executed on test environment 3.}\label{ch5:fig:pararealGPUs}
+    \caption[Parareal performance properties as a function of $R$ and number of GPUs used.]{Parareal performance properties as a function of $R$ and number GPUs used. Notice how the obtained performance depends greatly on the choice of $R$ as a function of the number of GPUs. Executed on test environment 3.}\label{ch5:fig:pararealGPUs}\vspace*{-12pt}
 \end{figure}
-
 %TODO: Do we make this into a subsubsection:
 %\subsubsection{Library Implementation}\label{ch5:subsec:libimpl}
 %Describe library features. Stopping criteria. Usage of different numerical integrators. Examples including c++ code. Speed-up measurements on test cases.
@@ -596,7 +599,6 @@ If we additionally assume that the time spent on coarse propagation is negligibl
 %    \end{center}
 %    \caption{Parareal performance and convergence properties as a function of $R$ and number GPUs used. $R$ is regulated by the time discretization in the coarse propagator. }\label{ch5:fig:pararealGPUs}
 %\end{figure}
-
 \section{Conclusion and outlook}
 
 Massively parallel heterogeneous systems continue to enter the consumer market, and there has been no identification that this trend will stop for years to come. However, these parallel architectures require software vendors to adjust to new programming models and optimization strategies. Good software libraries are important tools for reducing the time and complexity of adjusting to new architectures, and they provide the user with an intuitive programming interface.
@@ -632,7 +634,7 @@ from the Danish Research Council for Technology and Production Sciences. A speci
 %\cite{ch5:Vandevoorde2002}
 %\cite{ch5:Bell2011}
 %\cite{ch5:mooreslaw1965}
-
+\clearpage
 \putbib[Chapters/chapter5/biblio5]
 
 % Reset lst label and caption
