From 2282c9980dcb4a35455c48753d543af16d0fe2b6 Mon Sep 17 00:00:00 2001 From: Marco De Lucia Date: Wed, 10 Apr 2024 23:10:39 +0200 Subject: [PATCH] added doc/ with latex documentation --- doc/Description.pdf | Bin 0 -> 533494 bytes doc/Description.tex | 372 ++++++++++++++++++++ doc/images/barite_200_field_Ba_crop.png | Bin 0 -> 9560 bytes doc/images/barite_200_field_alphax_crop.png | Bin 0 -> 33641 bytes doc/images/barite_large_field_Ba_crop.png | Bin 0 -> 282882 bytes doc/images/barite_large_init_locs.pdf | Bin 0 -> 22122 bytes doc/images/surfex_field_Na.png | Bin 0 -> 9189 bytes doc/images/surfex_field_U6.png | Bin 0 -> 7403 bytes 8 files changed, 372 insertions(+) create mode 100644 doc/Description.pdf create mode 100644 doc/Description.tex create mode 100644 doc/images/barite_200_field_Ba_crop.png create mode 100644 doc/images/barite_200_field_alphax_crop.png create mode 100644 doc/images/barite_large_field_Ba_crop.png create mode 100644 doc/images/barite_large_init_locs.pdf create 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b/doc/Description.tex @@ -0,0 +1,372 @@ +%% Time-stamp: "Last modified 2024-04-10 23:08:54 delucia" +\documentclass[a4paper,10pt]{article} + +\usepackage{listings} +\usepackage{amsmath} +\usepackage{xcolor} +\lstset{basicstyle=\ttfamily\footnotesize} + + +\usepackage{graphicx} +\usepackage{subcaption} +\usepackage[normalem]{ulem} +\usepackage{fancyvrb} +\usepackage{fullpage} +\usepackage{hyperref} + +\DeclareRobustCommand*\chem[1] +{\ensuremath{% + {\mathcode`\-="0200\mathcode`\=="003D% no space around "-" and "=" + \ifx\f@series\testbx\mathbf{#1}\else\mathrm{#1}\fi}}} + +\title{\texttt{TUG} ``standalone'' benchmarks \\ + (and some hopefully useful metrics)} + +% \author{\Large Marco \\ % +% \vspace{0.25cm} \url{delucia@gfz-potsdam.de}} + +% \date{\today} +\date{} +\sloppy +\begin{document} +\maketitle + +\section{Benchmark description} + +All benchmarks are specified and can be modified (e.g., number of +iterations) in the \url{eval/bench_defs.hpp.in} file. + +\subsection{\texttt{barite\_200}} + +At a glance: +\begin{table}[!h] + \centering + \begin{tabular}{|c|c|} + \hline + Grid & 200x200 \\ \hline + Size & 1x1~m$^2$ \\ \hline + Timestep & 1000~s \\ \hline + Iterations & 50 \\ \hline + $\alpha_x, \alpha_y$ & heter., aniso. \\\hline + Species \# & 7 \\ \hline + Init & homog. \\ \hline + \end{tabular} + \caption{Summary of parameters for the barite\_200 benchmark} + \label{tab:b200} +\end{table} + +\noindent \textbf{Initial Conditions (IC):} all concentrations are +initially homogeneous, refer to the \textbf{IC} values in table +\ref{tab:b200val}. The actual numerical values read by the benchmark +from file \url{eval/barite_200/barite_200_init.csv} have higher +significant digits. + +\noindent \textbf{Boundary conditions (BC):} the top left corner +(first 5 element boundaries in both N and W sides) are set to constant +value of 0.1 molal \chem{BaCl_2}. All other boundaries are closed. + +\begin{table*}[!h] + \centering + \begin{tabular}{|r|r|r|r|r|r|r|r|}\hline + & H & O & Charge & Ba & Cl & S\_6\_ & Sr \\\hline + \textbf{IC} & 110.0124 & 55.5086 & -1.2163e-09 & 4.4553e-07 & 2.0e-12 & 6.1516e-5 & 6.1472e-5 \\\hline + \textbf{BC} & 111.0124 & 55.5062 & -3.3370e-08 & 0.1 & 0.2 & 0 & 0 \\\hline + \end{tabular} + \caption{Initial and boundary values of all transported variables in + the \texttt{barite\_200} benchmark.} + \label{tab:b200val} +\end{table*} + +Spatially heterogeneous values for $\alpha_x$ and $\alpha_y$ are read +from \url{eval/barite_200/alpha_[xy].csv}. They result from a single +geostatistical simulation $\mathcal{F}$ of a $\mathcal{N}$(0, 1) +variable (anisotropic spherical variogram of correlation length 5 at +-30\textdegree and 20 at 60\textdegree, and sill 1). This +$\mathcal{F}$ field was scaled with an approximate order of magnitude +for $\alpha_x$ and $\alpha_y$ respectively: + +\begin{equation*} + \begin{cases} + \displaystyle \alpha_x & \displaystyle = 10^{-7} + 10^{-6} \frac{\mathcal{F}-\min{(\mathcal{F})}}{\max{(\mathcal{F})}}\\ + \alpha_y & \displaystyle = 10^{-7} + 10^{-7} \frac{\mathcal{F}-\min{(\mathcal{F})}}{\max{(\mathcal{F})}} + \end{cases} +\end{equation*} + +Figure~\ref{fig:b200a} displays the picture for $\alpha_x$ and +\ref{fig:b200b} the results for Ba after the 50 iterations in the +benchmark. + +\begin{figure}[!htb] + \centering + \begin{subfigure}{0.6\textwidth} + \includegraphics[width=\textwidth]{images/barite_200_field_alphax_crop.png} + \caption{$\alpha_x$ field\label{fig:b200a}} + \end{subfigure} + \begin{subfigure}{0.6\textwidth} + \includegraphics[width=\textwidth]{images/barite_200_field_Ba_crop.png} + \caption{\chem{log_{10}Ba} after 50 iterations\label{fig:b200b}} + \end{subfigure} + \caption{Diffusivity field and endresult for Ba in the + \texttt{barite\_200} benchmark\label{fig:b200}} +\end{figure} + + +This benchmarks runs in $\sim$11~s on 8 CPUs on my desktop. + +\clearpage + +\subsection{\texttt{barite\_large}} + +Larger grid version of the \texttt{barite\_200} benchmark, this time +with heterogeneous initial conditions, closed boundaries everywhere +and homogeneous diffusion coefficients. The sense of this benchmark is +to check for mass conservation. + +At a glance: + +\begin{table}[!h] + \centering + \begin{tabular}{|c|c|} + \hline + Grid & 1000x1000 \\ \hline + Size & 10x10~m \\ \hline + Timestep & 100~s \\ \hline + Iterations & 50 \\ \hline + $\alpha$ & homog. 1E-6 \\\hline + Species \# & 7 \\ \hline + Init & heter. \\ \hline + \end{tabular} + \caption{Summary of parameters for the \texttt{barite\_large} benchmark} + \label{tab:blarge} +\end{table} + +\noindent \textbf{Boundary conditions (BC):} all boundaries are +closed. As for initial conditions, background concentrations are set +in the whole grid (\textbf{All} record in table~\ref{tab:blargeval}). +1000 randomly selected grid cells (cfr figure~\ref{fig:blargea} for +their position) are assigned initial values of the \textbf{Locations} +record. The non-rounded values are read from file +\url{eval/barite_large/barite_large_init.csv}. + + +\begin{table*}[!h] + \centering + \begin{tabular}{|r|r|r|r|r|r|r|r|}\hline + & H & O & Charge & Ba & Cl & S\_6\_ & Sr \\\hline + \textbf{All} & 110.0124 & 55.50868 & -1.2163e-09 & 4.4553e-07 & 0 & 0.0006152 & 0.00061 \\\hline + \textbf{Locations} & 111.0124 & 55.50622 & -3.0000e-07 & 1 & 2 & 0.01 & 0.001 \\\hline + \end{tabular} + \caption{\texttt{Barite\_large} benchmark, initial conditions: the + whole grid has the values in the \textbf{All} record, while 1000 + cells as displayed in figure~\ref{fig:blargea} are assigned the + values of \textbf{Locations} record. } + \label{tab:blargeval} +\end{table*} + + +\begin{figure}[!htb] + \centering + \begin{subfigure}[T]{0.6\textwidth} + \vskip 0pt + \includegraphics[width=\textwidth]{images/barite_large_init_locs.pdf} + \caption{Locations of ``heterogeneities'' in initial + conditions\label{fig:blargea}} + \end{subfigure} + \begin{subfigure}[T]{0.6\textwidth} + \vskip 0pt + \includegraphics[width=\textwidth]{images/barite_large_field_Ba_crop.png} + \caption{\chem{log_{10}Ba} after 5 iterations\label{fig:blargeb}} + \end{subfigure} + \caption{Diffusivity field and endresult for Ba in the + \texttt{barite\_200} benchmark\label{fig:blarge}} +\end{figure} + +This benchmark runs in $\sim$30~s on my desktop using 8 CPUs. + +\clearpage + +\subsection{\texttt{surfex}} + +Homogeneous benchmark with values inspired from POET's +\texttt{surfex}, transporting 21 species. Here we use actual +physically true values for $\alpha$ (isotropic and homogeneous set to +1.1E-12) and simulate a rectangular domain of 2x1 \chem{cm^2}. At a +glance: + +\begin{table}[!h] + \centering + \begin{tabular}{|c|c|} + \hline + Grid & 200x100 \\ \hline + Size & 0.02x0.01~m \\ \hline + Timestep & 3600~s (1~h) \\ \hline + Iterations & 20 \\ \hline + $\alpha$ & homog. 1.1E-12 \\ \hline + Species \# & 21 \\ \hline + Init & homog. \\ \hline + \end{tabular} + \caption{Summary of parameters for the \texttt{surfex} benchmark} + \label{tab:bsurf} +\end{table} + +\noindent \textbf{Boundary conditions (BC):} \textbf{all} domain +boundaries are set to constant \textbf{BC} values. \textbf{Initial + conditions (IC):} homogeneous, cfr table~\ref{tab:bsurfval}. + +\begin{table*}[!h] + \centering + \begin{tabular}{|l|r|r|}\hline + & \textbf{IC} & \textbf{BC} \\ \hline + H & 1.11e+02 & 120.0 \\ \hline + O & 5.55e+01 & 55.1 \\ \hline + Charge & -2.0e-13 & 8.0e-17 \\ \hline + C & 2.0e-16 & 2.0e-15 \\ \hline + CH4 & 2.0e-03 & 0.2 \\ \hline + Ca & 2.0e-01 & 0.03 \\ \hline + Cl & 3.0e-01 & 0.5 \\ \hline + Fe2 & 1.4e-04 & 0.0002 \\ \hline + Fe3 & 1.3e-09 & 2.0e-08 \\ \hline + H0 & 6.0e-12 & 2.0e-11 \\ \hline + K & 2.0e-03 & 1.0e-05 \\ \hline + Mg & 1.0e-02 & 0.2 \\ \hline + Na & 2.0e-01 & 0.3 \\ \hline + HS2 & 5.9e-10 & 0 \\ \hline + S2 & 8.3e-15 & 8.3e-12 \\ \hline + S4 & 2.1e-14 & 5.1e-14 \\ \hline + S6 & 1.6e-02 & 0.026 \\ \hline + Sr & 4.5e-04 & 0.045 \\ \hline + U4 & 2.5e-09 & 2.5e-08 \\ \hline + U5 & 1.6e-10 & 1.6e-10 \\ \hline + U6 & 2.3e-07 & 1.0e-05 \\ \hline + \end{tabular} + \caption{\texttt{surfex} benchmark, homogeneous initial conditions + \textbf{IC} and boundary values \textbf{BC}} + \label{tab:bsurfval} +\end{table*} + + +\begin{figure}[!htb] + \centering + \begin{subfigure}[T]{0.6\textwidth} + \includegraphics[width=\textwidth]{images/surfex_field_U6.png} + \caption{\chem{\log_{10}(U6)} after 20 + iterations\label{fig:bsurfa}} + \end{subfigure} + \begin{subfigure}[T]{0.6\textwidth} + \includegraphics[width=\textwidth]{images/surfex_field_Na.png} + \caption{Na (linear scale) after 5 iterations\label{fig:bsurfb}} + \end{subfigure} + \caption{Results for U6 and Na in the \texttt{surfex} + benchmark\label{fig:bsurf}} +\end{figure} + +This benchmark runs in $\sim$7~s on my desktop using 8 CPUs. + +\clearpage + +\section{Some hopefully useful metrics} + +The problem with the \emph{classical}, \emph{central} measures such as +MAE (Mean Absolute Error) or RMSE (Root Mean Square Error), even when +scaled using a constant (e.g., the range of true variable, its mean, +or standard deviation) is that the measure does underestimate +discrepancies for very small values and conversely is only sensible to +large values. In practice these measures only work if the distribution +of the variable of interest is uniform on a small range or symmetric +distributed, e.g., gaussian. However, many variables we deal with in +geochemical models are however rather \emph{lognormally} distributed +or \emph{uniformly distributed on a logarithmic scale}. In any case, +it is common that variables we try to regress or match with models +span many orders of magnitude, and the error measures defined above +are biased towards large values. + +This problem can be partially solved operating on logarithms of the +variables. Plugging the logarithms of the true and predicted variables +$y$ and $\hat{y}$ in the usual MAE and RMSE formula we get, +\textbf{assuming both variables strictly larger than 0}: + +\begin{equation} + \label{eq:MAElog} + \text{MAE}_{\text{log}} = \frac{1}{N}\sum \left| \log{y_i} - \log{\hat{y_i}}\right| = \frac{1}{N}\sum \left| \log{\frac{y_i}{\hat{y_i}}}\right| +\end{equation} + +\begin{equation} + \label{eq:RMSElog} + \text{RMSE}_{\text{log}} = \sqrt{\frac{1}{N}\sum \left( \log{y_i} - \log{\hat{y_i}}\right)^2} = \sqrt{\frac{1}{N}\sum \left( \log{\frac{y_i}{\hat{y_i}}}\right)^2} +\end{equation} + +It is usual to define a slightly different variant of +\chem{RMSE_{log}}, called RMSLE (Root Mean Square Logarithmic Error) +by adding 1 to both the predicted and the true value to avoid dividing +by 0: + +\begin{equation} + \label{eq:RMSLE} + \text{RMSLE} = \sqrt{\frac{1}{N}\sum \left[ \log{(y_i+1)} - \log{(\hat{y_i}+1)}\right]^2} = \sqrt{\frac{1}{N}\sum \left( \log{\frac{y_i+1}{\hat{y_i}+1}}\right)^2} +\end{equation} + +All these measures yield 0 if $y$ and $\hat{y}$ are identical. Note +that the \chem{MAE_{log}} ressembles a \emph{geometric mean of the + absolute values of the quotients} of $y$ and $\hat{y}$ per +observation (putting either $y$ or $\hat{y}$ at denominator is +equivalent): + +\begin{equation} + \label{eq:GMAQ} + \text{Geometric Mean of Absolute Quotients} = \left(\prod + \left|\frac{\hat{y}_{i}}{y_i}\right|\right)^{\frac {1}{N}} +\end{equation} + +The geometric mean of the quotients would be 1 if the two variables +are identical. The connection with the \chem{MAE_{log}} is easy to see +since by definition a geometric mean is the $N$-th root of the product +of the terms: + +\begin{equation} + \label{eq:5} + \exp \left[{\frac {1}{N}}\sum\log a_{i}\right]= \left(\prod + a_{i}\right)^{\frac {1}{N}} +\end{equation} + +So the \chem{MAE_{log}} is the logarithm of the actual geometric mean +of the (absolute) logarithms of the quotients between the variables. +Instead of the simple quotient of the true and predicted values $y$ +and $\hat{y}$ we can define a relative (\textbf{per observation}) +error $\alpha_i$ as: +\begin{equation} + \label{eq:relalpha} + \alpha_i = + \begin{cases} + \displaystyle \frac{ y_i-\hat{y_i}}{y_i} & \text{if~} \hspace{0.1cm} y_i,\hat{y}_i \neq 0 \\ + 1 & \text{if~} \hspace{0.1cm} y_i=0 \text{\hspace{0.1cm} and \hspace{0.1cm}} \hat{y}_i \neq 0 \\ + 0 & \text{if~} \hspace{0.1cm} y_i=0 \text{\hspace{0.1cm} and \hspace{0.1cm}} \hat{y}_i = 0 \\ + \end{cases} +\end{equation} + +The same treatment of the case when a variable is 0 of course also +applies to the above introduced relative measures. Without using +logarithms we can then define some relative measures such as the Mean +Absolute Percentage Error (\textbf{MAPE}) and Relative RMSE +(\textbf{RRMSE}): + +\begin{equation} + \label{eq:MAPE} + \text{MAPE} = \frac{100\%}{N}\sum \left| \alpha_i \right| +\end{equation} + +\begin{equation} + \label{eq:RRMSE} + \text{RRMSE} = \sqrt{\frac{1}{N}\sum \left( \alpha_i\right)^2} +\end{equation} + +These relative measures account for discrepancies across all +magnitudes of the $y$ and $\hat{y}$ variables and preserve the +physical meaning of 0. + +\end{document} + + +%%% Local Variables: +%%% mode: xelatex +%%% TeX-master: t +%%% End: diff --git a/doc/images/barite_200_field_Ba_crop.png b/doc/images/barite_200_field_Ba_crop.png new file mode 100644 index 0000000000000000000000000000000000000000..30a00ce75ed7c1fbb524934c9a1181c2cd6e74aa GIT binary patch literal 9560 zcmbVy2{e>#|Nm`CXe@&mOO~;gh}78ku}f0vv2TSKvSrV{jF3Gh38784vM*&{LPSD{ 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