-\chapterauthor{Gleb Beliakov}{School of Information Technology, Deakin University, Burwood 3125, Australia}
-\chapterauthor{Shaowu Liu}{School of Information Technology, Deakin University, Burwood 3125, Australia}
+\chapterauthor{Gleb Beliakov and Shaowu Liu}{School of Information Technology, Deakin University, Burwood 3125, Australia}
+%\chapterauthor{Shaowu Liu}{School of Information Technology, Deakin University, Burwood 3125, Australia}
\chapter{Parallel Monotone Spline Interpolation and Approximation on GPUs}
The failure of splines to preserve monotonicity has prompted fundamental research in this area since 1960s. One of the first methods to remedy this problem were splines in tension by Schweikert \cite{Sch}, where a tension parameter controlled the shape of exponential splines \cite{Spath1969}. Later on several monotonicity preserving polynomial spline algorithms were proposed \cite{Schumaker1983,PasRoul1977,AndElf1987,Andersson1991_JAT,McAllister1981_ACM,PasRoul1977}. These algorithms typically rely on introducing additional spline knots between the abscissae of the data. Algorithmic developments are active to this day, see for example \cite{Kvasov2000_book,Abbas2011}.
-When in addition to the pairs $(x_i, y_i)$ the slopes of the function are available, i.e., the data comes in triples $(x_i, y_i, p_i)$, the interpolation problem is called Hermite, and the Hermite splines are used. However, even when the sequence $y_i$ is increasing and the slopes $p_i$ are non-negative, cubic Hermite splines may still fail to be monotone, as illustrated in Figure \ref{ch11:fig2}. Thus monotone Hermite splines are needed \cite{Gregory1982}. \index{Hermite spline}
+When in addition to the pairs $(x_i, y_i)$ the slopes of the function are available, i.e., the data comes in triples $(x_i, y_i, p_i)$, the interpolation problem is called Hermite, and the Hermite splines are used. However, even when the sequence $y_i$ is increasing and the slopes $p_i$ are non-negative, cubic Hermite splines may still fail to be monotone, as illustrated in Figure \ref{ch11:fig2}. Thus monotone Hermite splines are needed \cite{Gregory1982}. \index{Hermite splines}
Another issue with monotone approximation is noisy data. In this case, inaccuracies in the data make the input sequence $y_i$ itself non-monotone, and hence monotone spline interpolation algorithms will fail. Monotone spline smoothing algorithms are available, e.g. \cite{Andersson1991_JAT,Elfving1989_NM}. Such algorithms are based on solving a quadratic (or another convex) programming problem numerically, and have not been yet adapted to parallel processing.
It is almost straightforward to parallelise this scheme for GPUs, by processing each subinterval $[x_i,x_{i+1}]$ independently in a separate thread. However, it is not known in advance whether an extra knot $t_i$ needs to be inserted or not, and therefore calculation of the position of the knot in the output sequence of knots ${t_i}$ is problematic for parallel implementation (for a sequential algorithm no such issue arises). To avoid serialisation, we decided to insert an additional knot in every interval $[x_i,x_{i+1}]$, but set $t_i=x_i$ when the extra knot is not actually needed. This way we know in advance the position of the output knots and the length of this sequence is $2(n-1)$, and therefore all calculations can now be performed independently. The price we pay is that some of the spline knots can coincide. However, this does not affect spline evaluation, as one of the coinciding knots is simply disregarded, and the spline coefficients are replicated (so for a double knot $t_i=t_{i+1}$, we have $\alpha_i=\alpha_{i+1}$, $\beta_i=\beta_{i+1}$, $\gamma_i=\gamma_{i+1}$). Our implementation is presented in Figures \ref{ch11:algcoef}-\ref{ch11:algcoef1}.
+\lstinputlisting[label=ch11:algcoef1,caption=Calculation of monotone spline knots and coefficients.]{Chapters/chapter11/code2.cu}
+
+
At the spline evaluation stage we need to compute $s(z_k)$ for a sequence of query values ${z_k}, k=1,\ldots,K$. For each $z_k$ we locate the interval $[t_i,t_{i+1}]$ containing $z_k$, using bisection algorithm presented in Figure \ref{ch11:algeval}, and then apply the appropriate coefficients of the quadratic function. This is also done in parallel.
The bisection algorithm could be implemented using texture memory (to cache the array \texttt{z}), but this is not shown in Figure \ref{ch11:algeval}.
+\pagebreak
\lstinputlisting[label=ch11:algcoef,caption=Implementation of the kernel for calculating spline knots and coefficients. Function fmax is used to avoid division by zero for data with coinciding abscissae.]{Chapters/chapter11/code1.cu}
%% \end{figure}
-\lstinputlisting[label=ch11:algcoef1,caption=Calculation of monotone spline knots and coefficients.]{Chapters/chapter11/code2.cu}
%% \begin{figure}[!hp]
%% \renewcommand{\baselinestretch}{1}
%% %\renewcommand{\baselinestretch}{1}
\begin{table}[!h]
\begin{center}
-\caption{The average CPU time (sec) of the serial PAVA, MLS and parallel MLS algorithms. } \label{ch11:table1}
\begin{tabular}{|r|r|r|r|}
-
+\hline
Data & PAVA & MLS & GPU MLS \\ \hline
monotone increasing $f$ & & & \\
$n=10 \times 10^6$ &1.9& 1.9& 3500 \\
$n=20 \times 10^6$ &3.5& 4.0&-- \\
$n=50 \times 10^6$ &11& 11& -- \\
-
+\hline
\end{tabular}
\end{center}
+\caption{The average CPU time (sec) of the serial PAVA, MLS and parallel MLS algorithms. }
+\label{ch11:table1}
\end{table}
%% %\renewcommand{\baselinestretch}{2}
