On 24 May 2011 14:21, <[email protected]> wrote:
> Merge authors:
> Anders Logg (logg)
> ------------------------------------------------------------
> revno: 684 [merge]
> committer: Anders Logg <[email protected]>
> branch nick: fenics-book
> timestamp: Tue 2011-05-24 14:19:56 +0200
> message:
> Merge proofreading fixes from Marie to [oelgaard-2]
> modified:
> TODO
> chapters/kirby-1/chapter.tex
> chapters/oelgaard-2/chapter.tex
>
>
> --
> lp:fenics-book
> https://code.launchpad.net/~fenics-editors/fenics-book/main
>
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> === modified file 'TODO'
> --- TODO 2011-05-24 11:51:50 +0000
> +++ TODO 2011-05-24 12:17:35 +0000
> @@ -44,6 +44,7 @@
> not using text font
> * Fix missing indentation of paragraphs, was removed in rev 625
> * Add chapter numbers to headings
> +* Fix strange / turning into \ in header for DOLFIN chapter
>
> Things to discuss
> * Fix width of C++ and Python code environments, seem different
>
> === modified file 'chapters/kirby-1/chapter.tex'
> --- chapters/kirby-1/chapter.tex 2011-05-24 11:51:50 +0000
> +++ chapters/kirby-1/chapter.tex 2011-05-24 12:17:35 +0000
> @@ -727,7 +727,6 @@
> has far better properties when defined globally compared to its analogous
> definition in terms of a reference element.
>
> -
> \subsection{Local to global mapping of degrees of freedom}
>
> As shown in Figure~\ref{fig:kirby-1:patch}, finite elements are
>
> === modified file 'chapters/oelgaard-2/chapter.tex'
> --- chapters/oelgaard-2/chapter.tex 2011-05-17 08:32:17 +0000
> +++ chapters/oelgaard-2/chapter.tex 2011-05-24 12:17:35 +0000
> @@ -42,7 +42,7 @@
> Therefore, to ensure a proper performance comparison between the
> representations we assume that all functions in a form, including
> coefficient functions, come from a finite element function space. In
> -the case of equation~\eqref{oelgaard-2:eq:weightedlaplacian}, all
> +the case of~\eqref{oelgaard-2:eq:weightedlaplacian}, all
> functions will come from
>
> \begin{equation}
> @@ -51,12 +51,12 @@
> \label{oelgaard-2:eq:space_H1}
> \end{equation}
> %
> -where $P_{q}\brac{T}$ denotes the space of Lagrange
> -polynomials of degree $q$ on the element $T$ of the standard
> -triangulation of $\Omega$, which is denoted by~$\mathcal{T}$. If we
> -let $\bracc{\phi^{T}_{i}}$ denote the local finite element basis that
> -span the discrete function space $V_{h}$ on $T$, the local element
> -tensor for an element $T$ can be computed as
> +where $P_{q}\brac{T}$ denotes the space of Lagrange polynomials of
> +degree $q$ on the element $T$ of the standard triangulation of
> +$\Omega$, which is denoted by~$\mathcal{T}$. If we let
> +$\bracc{\phi^{T}_{i}}$ denote the local finite element basis that span
> +the discrete function space $V_{h}$ on $T$, the local element tensor
> +for an element $T$ can be computed as
>
> \begin{equation}
> A_{T,i} = \int_{T} w \nabla \phi^{T}_{i_1} \cdot \nabla
> @@ -93,19 +93,18 @@
> \label{oelgaard-2:eq:weightedlaplacian_quadraturerepresentation}
> \end{equation}
> %
> -where a change of variables from the reference coordinates
> -$X$ to the real coordinates $x = F_T(X)$ has been used. In the above
> -equation, $N$ denotes the number of integration points, $d$ is the
> -dimension of $\Omega$, $n$ is the number of degrees of freedom for the
> -local basis of~$w$, $\Phi_{i}$ denotes basis functions on the
> -reference element, $\det F_T'$ is the determinant of the Jacobian, and
> -$W^q$ is the quadrature weight at integration point~$X^q$. By
> -default, \ffc{} applies a quadrature scheme that will integrate the
> -variational form exactly. It calls FIAT (see
> -Chapter~\ref{chap:kirby-2}) to compute the quadrature scheme. FIAT
> -supplies schemes that are based on the Gauss--Legendre--Jacobi rule
> -mapped onto simplices (see~\citet{KarniadakisSherwin2005} for details
> -of such schemes).
> +where a change of variables from the reference coordinates $X$ to the
> +real coordinates $x = F_T(X)$ has been used. In the above equation,
> +$N$ denotes the number of integration points, $d$ is the dimension of
> +$\Omega$, $n$ is the number of degrees of freedom for the local basis
> +of~$w$, $\Phi_{i}$ denotes basis functions on the reference element,
> +$\det F_T'$ is the determinant of the Jacobian, and $W^q$ is the
> +quadrature weight at integration point~$X^q$. By default, \ffc{}
> +applies a quadrature scheme that will integrate the variational form
> +exactly. It calls FIAT (see Chapter~\ref{chap:kirby-2}) to compute
> +the quadrature scheme. FIAT supplies schemes that are based on the
> +Gauss--Legendre--Jacobi rule mapped onto simplices
> +(see~\citet{KarniadakisSherwin2005} for details of such schemes).
>
> From the representation
> in~\eqref{oelgaard-2:eq:weightedlaplacian_quadraturerepresentation},
> @@ -129,12 +128,12 @@
> \emp{Psi\_vu\_D10} in Figure~\ref{oelgaard-2:fig:standard_code}.
> After the tabulation of basis functions values, the loop over
> integration points begins. In the example we are considering linear
> -elements have been used, and only one integration point is necessary
> -for exact integration. The loop over integration points has therefore
> -been omitted. The first task inside a loop over integration points is
> -to compute the values of coefficients at the current integration
> -point. For the considered problem, this involves computing the value
> -of the coefficient $w$. The code for evaluating \emp{F0} in
> +elements, and only one integration point is necessary for exact
> +integration. The loop over integration points has therefore been
> +omitted. The first task inside a loop over integration points is to
> +compute the values of coefficients at the current integration point.
> +For the considered problem, this involves computing the value of the
> +coefficient $w$. The code for evaluating \emp{F0} in
> Figure~\ref{oelgaard-2:fig:standard_code} is an exact translation of
> the representation $\sum_{\alpha_{3}=1}^n \Phi_{\alpha_{3}}(X^q)
> w_{\alpha_{3}}$. The last part of the code in
> @@ -210,7 +209,7 @@
> \item[Loop invariant code motion] In short, this procedure seeks to
> identify terms that are independent of one or more of the summation
> indices and to move them outside the loop over those particular
> -indices. For instance, in equation~\eqref{oelgaard-2:eq:%
> +indices. For instance, in~\eqref{oelgaard-2:eq:%
> weightedlaplacian_quadraturerepresentation} the terms regarding the
> coefficient $w$, the quadrature weight $W^q$ and the determinant $\det
> F_T'$ are all independent of the basis function indices $i_1$ and
> @@ -392,6 +391,8 @@
> \label{oelgaard-2:fig:O_simplify_code}
> \end{figure}
>
> +\editornote{Explain what \emp{FE0} etc. mean in
> Figure~\ref{oelgaard-2:fig:O_simplify_code}!}
There is no FE0 in that code extract.
Furthermore, in the text we write:
... in Figure~\ref{oelgaard-2:fig:O_simplify_code}, where again
only code different from that in
Figure~\ref{oelgaard-2:fig:standard_code} has been included.
Any symbols in the code which has not already been accounted for in
the 'standard_code' is explained in the text following
the 'simplify_code'.
> Due to expansion of the expression, many terms related to the geometry
> have been moved outside of the loops over the basis function indices
> \emp{j} and \emp{k} and stored in the array~\emp{G}. Also, note how
> @@ -471,7 +472,7 @@
>
> This optimization strategy is an extension of the strategy described
> in the previous section. In addition to hoisting terms related to the
> -geometry and the integration point, values that depends on the basis
> +geometry and the integration points, values that depends on the basis
> indices are precomputed inside the loops. This will result in a
> reduction in operations for cases in which some terms appear
> frequently inside the loop such that a given value can be reused once
> @@ -558,10 +559,10 @@
> strategies outlined in the previous section on the runtime
> performance. The point is not to present a rigorous analysis of the
> optimizations, but to provide indications as to when the different
> -strategies will be most effective. We also compare the runtime
> +strategies will be most effective. We also compare the runtime
> performance of quadrature representation to the tensor representation,
> which is described in Chapter~\ref{chap:kirby-8}, to illustrate the
> -strength and weaknesses of the two approaches.
> +strengths and weaknesses of the two approaches.
>
> \subsection{Performance of quadrature optimizations}
> \label{oelgaard-2:sec:quad_performance}
> @@ -599,7 +600,7 @@
> Table~\ref{oelgaard-2:tab:laplace_stats_1}, while
> Figure~\ref{oelgaard-2:fig:laplace_stats_2} shows the runtime
> performance for different compiler options for $N = 1 \times 10^7$.
> -The \ffc{} compiler options can be seen on the x-axis in the figure
> +The \ffc{} compiler options can be seen on the $x$-axis in the figure
> and the four g++ compiler options are shown with different colors.
>
> \begin{table}
> @@ -625,12 +626,14 @@
> \begin{figure}
>
> \center\includegraphics[width=\largefig]{chapters/oelgaard-2/pdf/runtime_laplace.pdf}
> \caption{Runtime performance for the weighted Laplace equation for
> - different compiler options. The x-axis shows the \ffc{}
> + different compiler options. The $x$-axis shows the \ffc{}
> compiler options, and the colors denote the g++ compiler
> options.}
> \label{oelgaard-2:fig:laplace_stats_2}
> \end{figure}
>
> +\editornote{Very hard to read legends and axes in
> Figure~\label{oelgaard-2:fig:laplace_stats_2}, please fix!}
Does this apply to both 'stats' figures?
> The \ffc{} and g++ compile times were less than one second for all
> optimization options. It is clear from
> Figure~\ref{oelgaard-2:fig:laplace_stats_2} that runtime performance
> @@ -690,21 +693,24 @@
> \begin{figure}
>
> \center\includegraphics[width=\largefig]{chapters/oelgaard-2/pdf/runtime_hyperelasticity.pdf}
> \caption{Runtime performance for the hyperelasticity example for
> - different compiler options. The x-axis shows the \ffc{} compiler
> + different compiler options. The $x$-axis shows the \ffc{} compiler
> options, and the colors denote the g++ compiler options.}
> \label{oelgaard-2:fig:hyper_stats_2}
> \end{figure}
>
> +\editornote{Mismatch between Figure~\ref{fig:oelgaard-2:fig:hyper_stats_2}
> and text which claims that \emp{-basis -zeros} is the best option.}
This has been fixed a long time ago!
I hope this doesn't mean that some of the other errors have been
reintroduced in the merge!
Kristian
> Comparing the number of flops involved to compute the element tensor
> to the weighted Laplace example, it is clear that this problem is
> considerably more complex. The \ffc{} compile times in
> Table~\ref{oelgaard-2:tab:hyper_stats_1} show that the \emp{-simplify}
> optimization, as anticipated, is the most expensive to perform. The
> g++ compile times for all test cases were in the range two to six
> -seconds for all optimization options. A point to note is that scope
> -for reducing the flop count is considerably greater for this problem
> -than for the weighted Laplace problem, with a difference in the number
> -of flops spanning several orders of magnitude between the different
> +seconds for all optimization options. A point to note is that the
> +scope for reducing the flop count is considerably greater for this
> +problem than for the weighted Laplace problem, with a difference in
> +the number of flops spanning several orders of magnitude between the
> +different
> \ffc{} optimizations. This compares to a difference in flops of
> roughly a factor two between the non-optimized and the most effective
> optimization strategy for the weighted Laplace problem. In the case
> @@ -714,7 +720,7 @@
> this effect becomes less pronounced. Another point to note, in
> connection with the g++ optimizations, is that switching on additional
> optimizations beyond \emp{-O2} does not seem to provide any further
> -improvements in run time. For the hyperelasticity example, the option
> +improvements in run-time. For the hyperelasticity example, the option
> \emp{-zeros} has a positive effect on the performance, not only when
> used alone but in particular when combined with the other \ffc{}
> optimizations. This is in contrast with the weighted Laplace
> @@ -769,8 +775,8 @@
> The test and trial functions are denoted by $v, u \in V_{h}$, with
>
> \begin{equation}
> - V_{h} = \bracc{v \in \brac{H^{1}\brac{\Omega}}^2: \ v\vert_T \in
> - \brac{P_{q}\brac{T}}^2 \foralls T \in \mathcal{T}}
> + V_{h} = \bracc{v \in [H^{1}\brac{\Omega}]^2: \ v\vert_T \in
> + [P_{q}\brac{T}]^2 \foralls T \in \mathcal{T}}
> \label{oelgaard-2:eq:elastictity_H1_vector_space}
> \end{equation}
> %
>
>
>
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