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\chapter{Parallel Architectures and  Iterative Applications}
\label{ch1}
%% Introduction
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\section{Introduction}
\label{ch1:1}
Almost of the software applications  are traditionally programmed as a sequential programs according to the Von Neumann report in 1993 \cite{ref50}. The structure of the 
program code is understandable by the human brain as a series of instructions that execute one after the other. From many years until a short time, the users of the sequential applications are moving their thinking   towards that these applications must run faster with each new generation of microprocessors. This idea is no longer valid nowadays, because the recent release of the microprocessors have many computing units embedded in one chip and these programs are only run over one computing unit sequentially.
Consequently, the traditional applications not have improved their performance a lot over the new architectures, whereas the new applications run faster over them in a parallel. The parallel application is executed over  all the available  computing  units at the same time to improve its performance. Furthermore, the concurrency revolution has been referred to the drastically improvement in the performance of the new applications side by side to  the new parallel architectures \cite{ref51}. Therefore, parallel applications and parallel architectures are closely tied together. It is hard to think about any of a parallel applications without thinking of the parallel hardware that executing them.
 
In this work, the iterative parallel applications, which is the most popular type of the parallel applications, are interested and running them over different parallel architectures to optimize their energy consumptions is the goal. 
As a result, this chapter is aimed to give a brief overview for a parallel hardware architectures,  parallel iterative applications and the energy model from the other authors used to measure the energy consumption of these applications. 
The reminder of this chapter is organized as follows: section \ref{ch1:2}  is  devoted 
to describe the types of  parallelism and the types of the parallel platforms. It is also gives some information about the parallel programming models.  Section \ref{ch1:3} explains both the synchronous and asynchronous parallel iterative methods and comparing them. Section \ref{ch1:4}, presents the well accepted energy model from the state of the art that can be used to measure the energy consumption of the parallel iterative applications when changing the frequency of the processor. Finally,  section \ref{ch1:5} summaries this chapter.


\section{Parallel Computing  Architectures} 
\label{ch1:2}
The process of the simultaneous application of the calculations is called the parallel computing.
It has main principle refer to the ability of dividing the large problem into smaller sub-problems that can be solved  at the same time \cite{ref2}. 
Mainly, solving the sub-problems of the main problem in a parallel computing are carried out on multiple parallel processors.
Indeed, the parallel processors architecture is a computer system composed from many processing elements connected via network model in addition to the software tools required to make the processing units work together \cite{ref1}.
Consequently, parallel computing architecture consist of software and hardware resources. 
The hardware resources are the processing units and the memory model in addition to the network system connecting them. The software resources include the specific operating system, the programming language and the compiler, or the runtime libraries. Furthermore, parallel computing can have different levels of parallelism, which can perform in software or hardware. There are five types of parallelism as follows:
\begin{itemize}

\item \textbf{Bit-level parallelism (BLP)}: The appearance of very-large-scale integration (VLSI) in 1970s has been considered the first approach towards the parallel computing. It is used to increase the number of bits  in word size  being processed by a processor as in the figure~\ref{fig:ch1:1}. Year after year, the number of bits is increased starting from 4-bit microprocessors reaching until 64 bit microprocessors . For example, the recent  x86-64 architecture becomes the most familiar architecture nowadays. Therefore, the biggest word size is given more parallelism level and  thus less instructions to be executed by the processor at the same time.
 
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{fig/ch1/bits-para.pdf}
\caption{Bit-level parallelism }
\label{fig:ch1:1}
\end{figure}

\item \textbf{Data-level parallelism (DLP)}:Data parallelism is the process of distributing the data vector between different parallel processors and each one performs the same operation on its data sub-vector. Therefore, many arithmetic operations can be performed on the same data vector in a simultaneous manner.  This type of parallelism can be used in many programs, especially from the area of scientific computing. Usually, data-parallel operations are only provided for arrays operations, see figure \ref{fig:ch1:2}. As an example about the applications  of this type of parallelism are the vectors multiplication, image and signal processing. 

\begin{figure}[h!]
\centering
\includegraphics[scale=1]{fig/ch1/data-para.pdf}
\caption{Data-level parallelism }
\label{fig:ch1:2}
\end{figure}

\item \textbf{Instruction-level parallelism (ILP)}: Generally, the sequential program composed of many instructions. These instructions can be executed in a parallel at the same time, if each of them is independent from the others. In particular, parallelism can be achieved in the instruction level by using  pipeline. It means all the independent instructions of the program are overlapped the execution of each  others. For example, if we have two instruction $I_1$ and $I_2$, they are independent if there is no control and data dependency between them.
 In pipeline stages, the execution of each instruction is divided into multiple steps that can be overlapped with the steps of other instructions by the pipeline hardware unit.
Figure~\ref{fig:ch1:3} demonstrates four instructions each one has four steps denoted as  fetch, decode, execute and write which are implemented in a hardware units by pipeline. 

\begin{figure}[h!]
\centering
\includegraphics[scale=1]{fig/ch1/pipelines.pdf}
\caption{Instruction-level parallelism by pipelines}
\label{fig:ch1:3}
\end{figure}



\item \textbf{Thread-level parallelism (TLP)}: It is also known as a task-level parallelism.
According to the Moore’s law \cite{ref9}, the processor can have a number of transistors by a double 
each two years to increase the frequency of the processor and thus its performance.  Beside that, cache and main memories sizes are must increased together to satisfy this increased. 
This leads to some limits come from two main reasons, the first one is when the cache size is drastically increased leading to a larger access time. The second  is related to the big increase in the number of the transistors per CPU can be increased significantly the heat dissipation. As a result, the programmers sub divided their programs into multiple tasks which can be executed in parallel over distributed processors or shared multi-cores processors to improve the performance of the program, see figure~\ref{fig:ch1:4}. Each processor can have a multiple or individual thread dedicated for each task. A thread can be defined as a part of a parallel program which shares processor resources with other threads. 

\begin{figure}[h!]
\centering
\includegraphics[scale=1]{fig/ch1/thread-para.pdf}
\caption{Thread-level parallelism}
\label{fig:ch1:4}
\end{figure}

Therefore, we can consider the execution time of a sequential program composed of 
$N$ tasks as the sum of the execution times of all tasks as follows:

\begin{equation}
\label{ch1:eq1}
    Sequential~execution~time = \sum_{i=1}^{N} T_i
\end{equation}

Whereas, if the tasks are executed synchronously over multiple processing units in a parallel, the execution time of the program is the execution time of the task that has the maximum execution time (the slowest task) as follows:

 
\begin{equation}
\label{ch1:eq2}
    Parallel~execution~time = \max_{i=1,\dots,N} T_i
\end{equation}


\item \textbf{Loop-level parallelism (LLP)}:
The numerical algorithms and many other algorithms are executed iteratively the same program portion, the computations, using different forms of the loop statements allowed in the programming languages. At each iteration,  the program need to scan a large data structure such as an array structure to make the arithmetic calculations. Inside the loop structure, there are many instructions, which are independent or dependent. In a sequential loop execution the $i$ iteration must be executed after the completion of $(i-1)$ iteration. 
While, if each iteration is independent from the others, then all the iterations are distributed over many  processors to be executed in a parallel,
for example see figure\ref{fig:ch1:5}. Thus, this type of a loop is  called $parallel~loop$.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{fig/ch1/loop-para.pdf}
\caption{Loop-level parallelism}
\label{fig:ch1:5}
\end{figure}

The execution time of the parallel loop portion can be computed as 
the execution time of a sequential loop portion has $N_{iter}$ iterations divided by the number of the processing units $N_{processors}$ as follows:

\begin{equation}
\label{ch1:eq3}
 Parallel~loop~time = \frac{Sequential~loop~time}{N_{processors}}
                  =\frac{\sum_{i=1}^{N_{iter}} Time~of~iter_i}               
                   {N_{processors}}
\end{equation}

For more detail about the levels of parallelism see \cite{ref3,ref4,ref6,ref7}.
\end{itemize}

\subsection{Types of Parallel platforms} 
\label{ch1:2:1}
The main goal behind using a parallel computers is to solve bigger problem faster.
A collection of processing elements composing them must to work together to perform the final solution of the main problem. However, many different architectures have been proposed  
and classified according to the parallelism in the instruction and data
streams. In 1966, Michel Flynn has been proposed a simple model of categorizing all computers that still useful until know \cite{ref10}. His taxonomy considered the data and the operations performed on this data to produce four types of computer systems as follows:

\begin{itemize}
 
\item \textbf{Single instruction, single data (SISD) stream}: A single processor executes a single instruction stream executing one data stream stored in an individual memory model, see figure \ref{fig:ch1:6}. As an example of this type is the  conventional sequential computer according to the Von Neumann model, it is also called the Uniprocessors.
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{fig/ch1/sisd.pdf}
\caption{SISD machine architecture}
\label{fig:ch1:6}
\end{figure}
 
\item \textbf{Single instruction, multiple data (SIMD) stream}: All the processors execute the same instructions on different data. 
Each processor stores the data in its local memory, the processor communicates with each others typically via simple communication model, see figure \ref{fig:ch1:7}. Many scientific and engineering
applications are suitable to this type of parallel scheme.
Vector and array processors are a well know  examples of this type. 
As an example about the applications executed over this architecture are the graphics processing, video compression and medical image analysis applications.

\begin{figure}[h!]
\centering
\includegraphics[scale=1]{fig/ch1/simd.pdf}
\caption{SIMD machine architecture}
\label{fig:ch1:7}
\end{figure}

\item \textbf{Multiple instruction, single data (MISD) stream}: Many operations from multiple processing elements are executed over the same data stream. Each processing element has its local memory to store the private multiple program instructions  applied  to unique global memory data stream as in figure \ref{fig:ch1:8}. While the MISD machine is not commonly used,  there are some interesting uses such as the systolic arrays and dataflow machines.

\begin{figure}[h!]
\centering
\includegraphics[scale=1]{fig/ch1/misd.pdf}
\caption{MISD machine architecture}
\label{fig:ch1:8}
\end{figure}


\item \textbf{Multiple instruction, Multiple data (MIMD) stream}: There are multiple processing elements each of which has a separate instruction  and data local memories.
At any time, different processing elements may be executing different instructions on different data fragment, see figure \ref{fig:ch1:9}. There are two types of the MIMD machines: the share memory and massage passing MIMD machines. 
In the share memory architectures, a processors are communicated via a share memory model, while in the message passing architecture each processor has its own local memory and all processors communicate via communication network model. The  multi-core processors, local
clusters and grid systems are an examples for the MIMD machine.
Many of applications have conducted over this architecture 
such as computer-aided design, computer-aided manufacturing, simulation, modeling, iterative applications and so on.

 \begin{figure}[h!]
\centering
\includegraphics[scale=1]{fig/ch1/mimd.pdf}
\caption{MIMD machine architecture}
\label{fig:ch1:9}
\end{figure}
\end{itemize}
 For more details about this architectural taxonomy see \cite{ref11,ref5,ref13,ref14}.

The work of this thesis is dedicated to MIMD machines architecture. Therefore, we discuss in
this chapter some of the commonly used parallel architectures that belong to MIMD machines.
As explained before, the MIMD architectures can be classified into two types, the shared memory and the distributed message passing ones. Furthermore, these classifications are based on 
how MIMD processors access the memory model. The shared MIMD machines communication topology can be bus-based, extended or hierarchical type. Whereas, the distributed memory MIMD machines may have hypercube or mesh inter connected networks. In the following are some well known MIMD parallel computing platforms:

\begin{itemize}
\item \textbf{Multi-core processors}:
The multi-core processor is a single chip component with two or more processing units.
These processing units are called cores, which connected with each other via main memory model as in the figure \ref{fig:ch1:10}. Each individual core has its cache memory to store its data and execute different data or instructions stream in a parallel. Moreover, each core can have  one or more threads to execute a specific programming task as shown in the thread-level parallelism. Historically, the multi-cores of the CPU began as two-core processors, with the number of cores approximately doubling with each semiconductor process generation \cite{ref12}. The very quick improvements in the performance and thus the increase in the number of cores  is devoted in a graphical processing unit (GPU). A current exemplar of GPU  is the NVIDIA  GeForce TITAN Z with 5700 cores in 2015 \cite{ref17}. While the general-purpose microprocessors (CPU) has less number of the cores, for example the TILE-MX processor from Tilera   has 100 cores in the same year \cite{ref16}.
For more details about the multi-core processors see \cite{ref15}.

\begin{figure}[h!]
\centering
\includegraphics[scale=1]{fig/ch1/multicores.pdf}
\caption{Multi-core processor architecture}
\label{fig:ch1:10}
\end{figure}


\item \textbf{Local Cluster}:
 is generally collection of independent  computers that are connected
to each other  via standard network switches and cables, which is a high speed  
local area networks (LAN) with low latency and big bandwidth. Moreover, each node is distributed from each other and it is communicated with other nodes  using distributed massage passing model. All the nodes in the cluster must be controlled by one node called the master node, which is a specific node handles the scheduling and management of the other nodes as shown in the figure \ref{fig:ch1:11}. Usually, the hardware specifications of all nodes are homogeneous in term of the computing power and memory and it is also called tightly-coupled fashion. Also, each computing node in the cluster has the same copy of the operating system. See \cite{ref18, ref19} for more information about the cluster and its applications.

\begin{figure}[h!]
\centering
\includegraphics[scale=1]{fig/ch1/cluster.pdf}
\caption{Local cluster architecture}
\label{fig:ch1:11}
\end{figure}


\item \textbf{Grid (Distributed clusters)}:
Grid is a collection of local computing clusters from different sites connected via wide area network (WAN), which can be appeared virtually to the benefit users as a complete computing system  \cite{ref20}. 
In particular, different local clusters composing the grid are geographically far away from each others. Usually, each local cluster composed from homogeneous nodes, which are different from the nodes of the others cluster located in different sites. These nodes can be different in the hardware and software  specifications such as the computing power, memory, operating system, local network latency and bandwidth. Figure \ref{fig:ch1:12} presents an example of the grid composed from three heterogeneous local clusters located in a different sites which are connected throw wide area network.  Furthermore, the grid can be referred to an infrastructure  applies the integration and the collaboration by using a collection  of different computers, networks, databases servers , and scientific devices belong to  many companies and universities. Therefore, wide heterogeneous computing resources are allowed to many users simultaneously. While the only bottleneck of the grid is the high latency communications between the nodes from different sites. The grid is also called the loosely-coupled fashion platform. However, the fault tolerance  is required to guarantee the process of sending and receiving  the messages between the computing nodes and thus keeps all the messages    from the lost.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.85]{fig/ch1/grid.pdf}
\caption{Grid architecture}
\label{fig:ch1:12}
\end{figure}


\begin{figure}[h!]
\centering
\includegraphics[scale=1]{fig/ch1/grid5000.pdf}
\caption{Grid5000's sites distribution in France and Luxembourg}
\label{fig:ch1:13}
\end{figure}
\end{itemize}


Grid'5000 \cite{ref21} can be considered as a good example for this distributed platform. 
It is a large-scale testbed that consists of ten sites distributed
all over metropolitan France and Luxembourg. These sites are: Grenoble, Lille, Luxembourg, Lyon, Nancy, Reims, Rennes , Sophia, Toulouse, Bordeaux. Figure \ref{fig:ch1:13} shows the geographical distribution of grid'5000 sites over France and Luxembourg. All the sites are connected together via a  special long distance network called RENATER, which is the French
National Telecommunication Network for Technology. Each site in the grid is
composed of a few heterogeneous computing clusters and each cluster contains
many homogeneous nodes. In total, Grid'5000 has about one thousand heterogeneous nodes and eight thousand cores. In each site, the clusters and their nodes
are connected via high speed local area networks. Two types of local networks
are used, Ethernet or Infiniband networks, which have different characteristics
in terms of bandwidth and latency.
Grid'5000 is dedicated as a test-bed for grid computing and thus users can book the required nodes from different sites. It  allows the user to deploy his configured image of the operating system over the reserved nodes. Therefore, many software tools are available to the user to control and manage the reservation and deployment processes from his local machine. For example, OAR \cite{ref22} is a  batch scheduler used to manage the heterogeneous resources of the grid'5000.



\subsection{Parallel programming Models} 
\label{ch1:2:2}.
There are many parallel programming languages and libraries have been developed 
to explore the computing power of the parallel architectures. In this section,
the parallel computing programming languages are divided into two main types, 
which is the shared and the distributed models. Moreover, these two types are divided  into two subcategories according to the support level to the number of computing units composing them.
Figure \ref{fig:ch1:14} presents this classification hierarchy of the parallel programming 
paradigm. It is also show three parallel languages examples for each sub-category.


\begin{figure}[h!]
\centering
\includegraphics[scale=0.75]{fig/ch1/classification.pdf}
\caption{The classification of the parallel Programming Models}
\label{fig:ch1:14}
\end{figure}


Many programming interfaces and libraries have been developed to compile and run the 
parallel applications over the parallel architectures. In the following are 
some examples for each type of the parallel programming models:

\begin{itemize}
\item \textbf{Local cluster programming models}
  \begin{itemize}
    \item \textbf{MPI} \cite{ref23} is the Message Passing Interface, is a standardization 
                      dedicated for message passing in distributed memory environment.
                      The first version of MPI designated by a group of researchers in
                      1991. It is a library,  not a language and its subroutines 
                      can be called from many programming languages such as C, Fortran and 
                      Java. The programmes that users write in these languages are 
                      compiled with ordinary compilers and linked with the MPI library.  
                      Its library functions are not only for peer to peer operations throw 
                      send and receive messages, but it allowed many others collective 
                      operations such as gathering and reduction operations. MPI user feel
                      free form the network topology, synchronization, and communication
                      functionality between  group of processes. Furthermore, it has 
                      asynchronous point to point operations, which make the computations 
                      to overlap with communications. While MPI is not devoted to a grid,
                       \textbf{MPICH}  is one of the most 
                      popular implementations of MPI dedicated for grid computing. It is used               
                      as an extended version for MPI, which implements a fault tolerance  
                      \cite{ref52}. In this work, both of MPI and MPICH programming libraries 
                      are used for programming our algorithms and applications which called 
                      inside both Fortran and C programming languages.  
                      
   \item \textbf{PVM} \cite{ref25} is for Parallel Virtual Machine,  which is a collection 
                      of software tools and libraries to allows users working over a   
                      heterogeneous set of machines to operate as a single high performance 
                      parallel platform. It is dedicated for a group of machine that are 
                      distributed and heterogeneous in the operating system environments. 
                      The PVM system is elementarily for parallel programming to be used with 
                      C, C++, and Fortran  languages. 
                      It is considered more robust in fault tolerance 
                      than MPI, easier to add or delete the crashed nodes in the host pool 
                      \cite{ref26}. While MPI has more communication messages support and asynchronous 
                      operations which are not allowed in PVM. 
                      
                      
   \item \textbf{BLACS} \cite{ref27} is for Basic Linear Algebra Communication Subprograms.                 
                 It has a collection of libraries that used to built linear algebra messages 
                 communication model which is applied effectively over distributed memory architectures.
                 The primary goal of using 
                 BLACS is mapping a liner set or processors or any distributed machines into
                 a two dimensional array or grid, which is offer an easy tool for building a
                 linear algebra applications. 
                 
  \end{itemize} 
  
\item \textbf{Grid programming models}
  \begin{itemize}
   \item \textbf{Gridsolve} \cite{ref28}  is the first middleware for grid  and 
                high performance computing that offers a good tool to solve a complex
                scientific applications using distinct distributed machines. It applied the
                fault tolerance and load balancing features to ensure the reliability of the
                applications  when running over a geographically distributed resources. 
                It works with different programming languages such as C,C++, Java and Fortran. 
                
   \item \textbf{GLOBAS} \cite{ref29,ref30} is the most widely standardization tool kit
                for grid computing. It permits the users to share their computing resources securely. 
                While the GLOBAS toolkit is allowed to work with grid, it offers a fault 
                detection mechanism to ensure the delivery of the messages. 
                The first version of Globus toolkit appeared
                in 1998 and now the sixth version is available \cite{ref31}. 
                       

  \item \textbf{Legion} \cite{ref32,ref33} is an object-based and meta-systems software project
                at the University of Virginia on November 1997. 
                It implements many features such as security, portability and fault tolerance.
                Moreover, it is created to support a wide degree of parallelism throw an easy 
                programming tools to built the parallel applications.
              
        
  \end{itemize}
 
\item \textbf{Multi-core CPU programming models} 
  \begin{itemize}
   \item \textbf{OpenMP}  \cite{ref34} is parallel programming tool for shared memory 
                architectures. The main goal of using this programming model is to provide 
                a standard and portable API (application programming interface) for writing 
                shared memory parallel programs. It can be used with programming languages such 
                as C, C++ and Fortran to support different types of shared memory platforms 
                such as multi-core processors.
                OpenMP implements multi-threading, which is a method of parallel programming 
                that organized by using master thread to control a set of slave threads. Each
                thread can be executed in parallel by allocating it to a processor.  
                Moreover, OpenMP can be  used with MPI to support hybrid platforms that have 
                shared and distributed memory models in the same time.
                
  \item \textbf{Cilk} \cite{ref13,ref35} is a linguistic and runtime technology for algorithmic 
                multi-threaded programming originally developed at MIT. 
                It is allowed the programmer to focus on building the program in a structural way 
                to discover the inherent parallelism. Many specification are used in Cilk 
                such as the load balancing, synchronization and communication protocols. 
                
                
                
            
                
   \item \textbf{TBB} \cite{ref36} is for Threading Building Blocks, is a software library used with 
                C++ programming language for multi-core parallel programming developed by Intel.
                It woks on the principle of dividing the computations into many tasks that can be 
                executed in  
                parallel. It also has a management library to schedule the parallel task execution. 
                The difference between OpenMP and TBB, is the latter uses a task-based scheduling 
                mechanism. Furthermore, TBB is more popular with C++ programming language than 
                others languages. It is designed to work with any compiler environments, and thus 
                it easily ported to a new platform. Consequently, TBB has been ported to a 
                different types of operating systems and processors. While, it has limited
                support to vector processing architecture and then it connected with OpenMP  
                and Cilk to support this  platform.
  \end{itemize}
  
  
  
\item \textbf{GPU programming models} 
  \begin{itemize}
   \item \textbf{CUDA} \cite{ref37} Modern graphical processing units (GPUs) have been increasing  chip-level  
                       parallelism.  Current  NVIDIA  GPUs  are  many-core processor having thousands
                       of core.  According to this massively cores parallelism, the NVIDIA in 2007 developed
                       a parallel programming  language called CUDA , which is for Compute Unified Device  
                       Architecture.  A CUDA program has two parts, the first one is called a host which is a
                       set of threads that executed sequentially over the CPU. The second part is called the
                       kernels, which are a set of a threads that can be executed in a parallel over the GPU.
 
   \item \textbf{OpenCL}\cite{ref38} is for Open Computing Language. It is a parallel 
                       programming language dedicated for heterogeneous platform composed 
                       of CPUs and GPUs. The first release is initially developed by Apple  
                       in 2008. Functions executed on an OpenCL device is called kernel, 
                       which can be portably executes on any computing hardware such as CPU or GPU cores. 
                       This parallel programming language can support the homogeneous shared memory 
                       platforms and the multi-core processors, by using one core for control 
                       and the others for computing.
                         
   \item \textbf{HLSL} \cite{ref39} is for High Level Shading Language, is the shader 
                       programming language for Direct3D, which is a part of
                       Microsoft’s DirectX API. It supports the shader construction with 
                       C-like syntax, types, expressions, statements, and functions. It 
                       provides a graphical pipeline parallelism.
                       The last version of HLSL is v5.0 for DirectX 11, which adds a new general-purpose GPU
                       functions like CUDA. Recently, the new OpenCL version starts to replace CUDA 
                       as a multi-platform GPU language.
                                   
  \end{itemize}
 
\end{itemize}


\section{Iterative Methods} 
\label{ch1:3}
Numerical methods are a scientific computations for solving linear and non-linear problems.
Almost of the numerical problems can be represented by mathematical equation form with relations between its components. For example, solving linear equations which is well known in the scientific  area is generally expressed in the following form:

\begin{equation}
  \label{eq:linear}
 A x = b
\end{equation}

Where $A$ is a two dimensional matrix of size $N \times N$, $x$ is the unknown vector,
and $b$ is a vector of constant, each of size $N$ . There are two types of solution methods for solving this linear system.
The first method is called \textbf{Direct methods}, which is a finite number of steps depending on the 
size  of the linear system to give the exact solution. If the problem size is very big this methods are expensive or their
solutions are impossible in some cases.  The second type is called \textbf{Iterative methods}, which is computed 
many times the same block of the operations starting from the initial vector until reaching to the acceptable 
approximation  of the exact solution. However, the iterative methods are faster than direct methods and can be 
applied in parallel. Moreover, iterative methods can be used to solve both of the linear and non-linear equations.
In our work, we are interested in parallelizing the iterative methods because they are more popular and effective than  direct ones.

The sequential iterative algorithm is typically organized as a series of steps essentially  of the form:

\begin{equation}
  \label{eq:iter}
   X^{(k+1)} \longleftarrow F(X^k) 
\end{equation}

Where $F$ is one or set of operations applied to the data vector $X^k$ to produce the new data vector $X^{(k+1)}$. This operation $F$ is applied sequentially many times until convergence condition is satisfy as in the   algorithm \ref{sia}.



\begin{algorithm}[h!]
\begin{algorithmic}[1]
  
    \State Initialize the vector $X^0$ randomly  
    \For {$k:=1$  to \textit{convergence}}
    
      \State $X^{(k+1)} = F(X^k)$ 
   \EndFor
   \end{algorithmic}
   \caption{The iterative sequential algorithm}
  \label{sia}
   
\end{algorithm}

The sequential iterative algorithm at each iteration computes the value of the relative error, which is called the residual that denoted as $R$. This error value is the maximum difference  between the  data components of vectors of the last two successive iterations as follows:

 \begin{equation}
  \label{eq:res}
   R = \max_{i=1, \dots, N}  \abs{X_i^{(k+1)} - X_i^k}
\end{equation}  
Where $N$ is the size of the vector $X$. Then, the iterative sequential algorithm stops its iterations if the maximum error between the last two successive solutions vectors, as in \ref{eq:res}, is less than or equal to the some threshold value. Otherwise, it replaces the new vector $X^{(k+1)}$ with the old vector $X^k$ and computes the new iteration.

\subsection{Synchronous Parallel Iterative method} 
\label{ch1:3:1}
The sequential iterative algorithm \ref{sia}, can be parallelized by executing it on many computing units. To solve this algorithm on $M$ computing units, first the elements of the problem vector $X$ must be subdivided into $M$ sub-vectors, $X^k=(X_1^k,\dots,X_M^k)$. 
Each sub-vector can be solved  independently one computing units as follows:

\begin{equation}
  \label{eq:subvector}
   X_i^{k+1}= F_i(X_1^k,\dots,X_M^k)  \hspace{1cm} where \hspace{0.2cm} i=1,\dots, M
\end{equation}

Where $X_i^k$ is the sub-vector executed over the $i^{th}$ computing unit at the iteration $k$.

\begin{algorithm}[h!]
\begin{algorithmic}[1]
  
  \State Initialize the sub-vectors $(X_1^0,\dots,X_M^0)$ randomly  
    \For {$k:=1$  step $1$ to \textit{convergence}}
       \For {$i:=1$   to \textit{M}}
        \State $X^{(k+1)} = F(X^k)$
      \EndFor
    \EndFor  

   
   \end{algorithmic}
   \caption{The synchronous parallel iterative algorithm}
  \label{spia}
\end{algorithm}





The algorithm \ref{spia}, represents the synchronous parallel iterative algorithm. In contrast to 
sequential iterative algorithm, algorithm \ref{spia}, stops its iterations when the convergence condition  is satisfied. It computes the residual value $R$ as follows: 

\begin{equation}
 \label{eq:res_syn}
   R = \max_{i=1, \dots, M}  (\max_{j=1, \dots, m}\abs{X_{ij}^{(k+1)} - X_{ij}^k})
\end{equation}
This algorithm need to satisfy some convergence condition which  is called the  global convergence condition. In order to detect the global convergence overall computing units, first we need to compute
at each iteration the local residual and store it in the local variable at the computing unit $i$. Then at the end of each iteration, all the local residuals  from $M$ computing units must be reduce to one maximum  value represented by the  global residual, which is represent the global maximum errors overall maximum local errors from $M$ computing units. Where $m$ is the size of the $i$  sub-vector.
For example, in MPI this operation is directly applied using a high level communication procedure  called \textit{AllReduce}. The goal of this communication procedure is to apply the reduction operation on all local variables computed by the computing units.


\begin{figure}[h!]
\centering
\includegraphics[scale=0.75]{fig/ch1/sisc.pdf}
\caption{The SICS Model}
\label{fig:ch1:15}
\end{figure}


In synchronous iterative algorithm, computing processors needs to communicate with each other to 
exchange data at each iteration. Algorithm \ref{spia} can be used synchronous iterations and synchronous communications and denoted as \textbf{SISC} model.  At each iteration, the computing processor waits until 
it has receive all the data computed at the previous iteration from the other processors to perform the next iteration. This type of communication model used if there are a dependencies between the parallel tasks. Figure \ref{fig:ch1:15}, shows that using SICS model in a heterogeneous platform may results in a big periods  of the idle times represented by the white dashed spaces between two successive iterations. Indeed, this happens when the fast computing processor waits for the slow ones to finish their iterations to be able to synchronously send its data to them. This operation is wasted a big amount of the computing power of the faster processors and thus their energy consumptions. The increased in the level of the heterogeneity between the computing powers of the computing processors may increased propositionally these idle times.
For this reason, this algorithm is effectively implemented over a local cluster where a high speed local network is exist to reduce these idle times.  


\begin{figure}[h!]
\centering
\includegraphics[scale=0.75]{fig/ch1/siac.pdf}
\caption{The SIAS Model}
\label{fig:ch1:16}
\end{figure}

The communications of the synchronous iterative algorithm can be implemented asynchronously. Therefore, this algorithm is called the synchronous iteration and asynchronous 
communication algorithm and denoted as \textbf{SIAC} algorithm. The main principle of this algorithm is to use a synchronized iterations while exchanging the data between the computing units asynchronously.
Moreover, each computing unit not has to wait for its neighbours to receive the data messages 
that its has sent, while it only waits for receiving the  data from them. This can be implemented in SISC algorithm by replacing the synchronous send of the messages by asynchronous ones and keeps 
the synchronous receive of the data messages. The only advantage of this technique is to reduce the idle times between the iterations by making the communications to overlap partially
with computations, see figure \ref{fig:ch1:16}. The idle times are not totally eliminated  because the
fast computing nodes have to wait for slow ones to send their data messages. 
Both of the SISC and SIAC algorithms are not tolerate to the loss of data messages. Consequently, if one node is crashed,  all the other computing nodes are blocked together and all the system is crashed.


 
\subsection{Asynchronous Parallel Iterative method} 
\label{ch1:3:2}
The asynchronous iterations mean that all processors perform their iterations without considering the works of the other processors. Each processor not has to wait for receiving
the data messages from the others processors and continue computing the next iteration depending on its own data received at a specific time. While all processors not have to wait 
for data delivery from each other, there are not existence for the idle times at all between the iterations, see figure \ref{fig:ch1:17}. As shown in this figure, the fast processors can perform more iterations than the others at the same time. 
The asynchronous iterative algorithm uses asynchronous communications and called  \textbf{AIAC} algorithm. As same as in SISC algorithm, the AIAC algorithm subdivides the global Vectors $X$ into $M$ sub-vectors between the computing units. The main different between the two algorithm is that these $M$ sub-vectors are not updated at each iteration in the AIAC algorithm because both of the iterations and communications are asynchronous. 
However, there are two mechanisms to update the data vectors in AIAC algorithm:
 \begin{itemize}
 \item  The local vectors can be updated randomly on the order of $M$ computing units.
        This leads to some of these local vectors to not update at a certain time. 
 \item  According to the time period $t$, each computing unit checks if one of the its 
        dependencies components have been updated. If the computing node detects any update 
        case, it updates its own local vector data using the last received data messages.
        Otherwise, it does nothing at the time $t$.
 \end{itemize}


\begin{figure}[h!]
\centering
\includegraphics[scale=0.75]{fig/ch1/aiac.pdf}
\caption{The AIAC Model}
\label{fig:ch1:17}
\end{figure}

The global convergence of the parallel iterative method depend on the scientific application.
For more information about the convergence detection techniques of the asynchronous iterative methods,
we refer to \cite{ref40,ref41,ref42,ref43} for more details. 


The implementation of the AIAC method is not easy, but it gives many advantages over the traditional synchronous iterative method:

\begin{itemize}
\item It prevents the existence of the  idle times because each processor not has to wait 
      for the others to receive the data messages. Then, no idle times between each two  
      successive iterations.
      
\item Less sensitive for the heterogeneous communications and nodes' computing powers. In a 
      heterogeneous platform, the fast nodes not have to wait for the slow ones and so it 
      performs more iterations than them. While in the traditional synchronous iterative 
      methods, the fast computing nodes perform the same number of iterations as the slow ones
      because they are blocked together. 

\item The loss of the data messages is totally tolerant because each computing unit is not 
      blocked with the others. If the message is lost, the destination node not has to wait  
      for this data message and it uses the last received data to perform its iteration  
      independently.
      
\item In the distributed grid architecture, the local clusters from different sites are 
      connected via slow network with a high latency. The use of the AIAC model reduces the
      delay of sending the data message over such slow network link and thus the performance 
      of the applications is improved.
\end{itemize}


In addition to the difficulty of applying the asynchronous iterative method, it has some 
disadvantages as follows:

\begin{itemize}
\item It is not compatible to all types of the iterative applications because some of these 
      applications need to receive the data messages from its neighbours at each iteration.   
      Therefore, they required a fix number of iterations to converge. Otherwise, the 
      application is perform  infinity number of iterations  and then all of the system 
      is crash. 
 
\item The application of the asynchronous iterative method requires more iterations compared 
      to the synchronous ones to converge  when it is executed over the local cluster. 
      The increase in the number of  the iterations may increases proportionally 
      the execution time of the application. 
      Especially, the local computing cluster uses a high speed network, then running the 
      synchronous version over such platform is quicker to converge.
    
\item While the process not receive the new data messages at each iteration, the mechanism of 
       synchronous iterative methods for detecting the global convergence cannot be used for 
      asynchronous ones. Therefore, in AIAC algorithm a process can performs many iterations    
      without receiving any data messages from its neighbours. The absence of receiving new 
      data messages makes the data component not vary at the computing units and thus it detect 
      a false local convergence. This mean that the local residual value is less than the   
      required threshold. This fake convergence is conflicted at the reception of the first data message 
      because the local subsystem will locally diverges after computing the next iteration. 
      Therefore, special mechanisms are required for detecting the global convergence of a parallel 
      iterative algorithm implemented according to the asynchronous iteration model.


\end{itemize}

Generally, the interested readers can find more details about both of synchronous and asynchronous 
iterative methods in \cite{ref44,ref45}.

In our works, we are interested to implement both of a synchronous and asynchronous 
iterative methods for solving different problems over local homogeneous cluster, local heterogeneous cluster and distributed grid. Then, the process of optimizing their energy consumptions and performance is the main objective of this work as shown in the coming chapters. 


\section{The energy consumption model of the parallel applications } 
\label{ch1:4}

Many researchers~\cite{ref46,ref47,ref48,ref49} divide the power consumed by a processor into
two power metrics: the static and the dynamic power.  While the first one is
consumed as long as the computing unit is on, the latter is only consumed during
computation times.  The dynamic power $P_{dyn}$ is related to the switching
activity $\alpha$, load capacitance $C_L$, the supply voltage $V$ and
operational frequency $F$, as shown in EQ~(\ref{eq:pd}).
\begin{equation}
  \label{eq:pd}
  P_\textit{dyn} = \alpha \cdot C_L \cdot V^2 \cdot F
\end{equation}
The static power $P_{static}$ captures the leakage power as follows:
\begin{equation}
  \label{eq:ps}
   P_\textit{static}  = V \cdot N_{trans} \cdot K_{design} \cdot I_{leak}
\end{equation}
Where V is the supply voltage, $N_{trans}$ is the number of transistors,
$K_{design}$ is a design dependent parameter and $I_{leak}$ is a
technology-dependent parameter.  


The dynamic voltage and frequency scaling technique (\textbf{DVFS}) is a process that is allowed in modern processors to reduce the dynamic
power by scaling down the voltage and frequency.  Its main objective is to
reduce the overall energy consumption~\cite{ref77}.  The operational frequency $F$
depends linearly on the supply voltage $V$ as follows:
\begin{equation} 
\label{eq:v}
V = \beta \cdot F
\end{equation}

 Where $\beta$ is some of constant.  This equation is used to study the change of the dynamic
voltage with respect to various frequency values in~\cite{ref47}.  The reduction
process of the frequency can be expressed by the scaling factor $S$ which is the
ratio between the maximum and the new frequency as in EQ~(\ref{eq:s}).
\begin{equation}
  \label{eq:s}
 S = \frac{F_\textit{max}}{F_\textit{new}}
\end{equation}
The value of the scaling factor $S$ is greater than 1 when changing the
frequency of the CPU to any new frequency value~(\emph{P-state}) in the
governor.  The CPU governor is an interface driver supplied by the operating
system's kernel to lower a core's frequency \cite{ref8}.  

Depending on the equation \ref{eq:s}, the new frequency $F_{new}$ can be calculates as follows:

\begin{equation}
  \label{eq:fnew}
  F_\textit{new}= S^{-1} \cdot  F_\textit{max}
\end{equation}


Replacing $V$  in \ref{eq:pd} as in \ref{eq:v} gives the following equation of the dynamic power consumption
as a function of the constant $\beta$ instead of $V$: 

\begin{equation}
  \label{eq:pd-beta}
   P_{dyn}= \alpha \cdot C_L \cdot (\beta \cdot F) ^2 \cdot  F  =\alpha \cdot C_L \cdot \beta^2 \cdot  F^3
\end{equation}

Replacing $F_{new}$ in \ref{eq:pd-beta} as in \ref{eq:fnew} gives the following equation for dynamic power consumption:

\begin{multline}
  \label{eq:pdnew}
   P_{dynNew} = \alpha \cdot C_L \cdot \beta^2 \cdot  F_{new}^3 = 
   \alpha \cdot C_L \cdot \beta^2 \cdot  F_{max}^3 \cdot S^{-3} =
   \alpha \cdot C_L \cdot (\beta \cdot  F_{max})^2 \cdot F_{max} \cdot S^{-3} \\
   {} =\alpha \cdot C_L \cdot V^2 \cdot F_{max} \cdot S^{-3} = P_{dyn} \cdot S^{-3}
\end{multline}

Where $P_{dynNew}$  and $P_{dyn}$ are the  dynamic power consumed with the
new frequency and the maximum frequency respectively.

According to (\ref{eq:pdnew}) the dynamic power is reduced by a factor of
$S^{-3}$ when reducing the frequency of a processor by a factor of $S$.
The energy consumption is measured in Joule, and can be calculated by
multiplying the power consumption, measured in watts, by the  execution time of the program as follows:

\begin{equation}
  \label{eq:E}
  Energy = Power \cdot T
\end{equation}

According to the equation \ref{eq:E}, the dynamic energy consumption of the program executed in the time $T$ over one processor is the dynamic power multiply by the execution time. Moreover, the frequency scaling factor $S$ increases the execution time of the processor linearly, then the new dynamic energy consumption can be computed as follows:

\begin{equation}
  \label{eq:Edyn}
   E_{dynNew} = P_{dyn} \cdot S^{-3} \cdot (T \cdot S)= S^{-2} \cdot P_{dyn} \cdot  T
\end{equation}


According to \cite{ref46,ref47}, the static power consumption $P_{static}$  is not changes when the frequency of the processors is scaled down. Therefore, the static energy consumption can be computed as follows:
\begin{equation}
  \label{eq:Estatic}
   E_{static} = S \cdot  P_{static}  \cdot T
\end{equation}


Therefore, the energy consumption of the individual task running over one processor can be computed as follows:
\begin{equation}
 \label{eq:Eind}
  E_{ind} =  E_{dynNew} + E_{static} = S^{-2} \cdot P_{dyn} \cdot  T + S \cdot  P_{static}  \cdot T
\end{equation}

 
Depending on \ref{eq:Eind}, the total energy consumption of $N$  parallel task  running on $N$ processors is the summation  of the individual energies consumed by all processors. This model is developed and used by  Rauber and Rünger~\cite{ref47}. They modeled 
the total energy consumption for a parallel tasks running on a homogeneous platform by sorting the execution time of the all parallel tasks in a descending order, then their model can be  written as a function of the scaling factor $S$, as in EQ~(\ref{eq:energy}).

\begin{equation}
  \label{eq:energy}
  E_\textit{~all~tasks} = P_\textit{dyn} \cdot S_1^{-2} \cdot
    \left( T_1 + \sum_{i=2}^{N} \frac{T_i^3}{T_1^2} \right) +
    P_\textit{static} \cdot T_1 \cdot S_1 \cdot N
 \hfill
\end{equation}

Where $N$ is the number of parallel nodes, $T_i$  for $i=1,\dots,N$ are
the execution times of the sorted tasks.  Therefore, $T1$ is
the time of the slowest task, and $S_1$ its scaling factor which should be the
highest because they are proportional to the time values $T_i$. 
Finally, model \ref{eq:energy}  can be used to measure the energy consumed by any parallel application such as the iterative parallel applications with respect to the new scaled frequency value. 

There are two drawbacks of this energy model as follows:
\begin{itemize}
\item The message passing iterative program consists of the communication and computation times.
      This energy model is assumed that the dynamic power consumes during both these times.
      While the processor during the communication times involved remain idle and only consumes the static 
      power, for more details see \cite{ref53}. 


\item It is not well adapted to a heterogeneous architectures when there are different 
      types of the processors, which are consumed different dynamic and static powers. Then, this model is 
      not able to measured the energy consumption of all the parallel system  because it depends on 
      one value of the static and dynamic powers.
\end{itemize}

Therefore, one of the more important goals of this work is to develop an energy models that 
take into account the communication times in addition to computation times to modelize and measure the energy consumptions of the parallel iterative methods. These models are dedicated to all parallel architectures such as the homogeneous and heterogeneous platforms, which are local or distributed computing clusters.  

\section{Conclusion}
\label{ch1:5}
In this chapter, we have presented in general different types of parallelism levels that can be implemented in a software and hardware techniques. Furthermore, the types of the parallel architectures are demonstrated and classified according to how the computing units are connected to a memory model. 
The two parallel systems  are described,  which are the shared and distributed platforms. Depending on these two types, we have categorized the parallel programming models. The parallel iterative methods are explained and their two types, the synchronous and asynchronous iterative methods, are described. The synchronous iterative methods are well implemented over local homogeneous cluster with a high speed network link, while the asynchronous iterative methods are more conventional to implement over the distributed heterogeneous clusters. 
Consequently, running these two types of the parallel iterative methods over distributed platforms are interested in this work. The energy consumption model for  measuring the energy consumption of the parallel applications from the related literature is  described. This model  cannot be used for all types of parallel architectures. It is assumed to measure the dynamic power during both communication and computation times, while the processor involved remains idle during the communication times and  only consumes the static power. Moreover, it is not well adapted to the heterogeneous architectures when there are different
types of the processors, which are consumed different dynamic and static powers.

However, in the coming chapters of this thesis a new energy consumption models are developed, use for modeling and measuring the energies consumed by a parallel iterative methods running on both homogeneous and heterogeneous architectures. 
