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An aspect of parallel computing that is closely linked to the choice of an adequate parallel architecture concerns the parallelisation of sequential computations. When a computational solution to a given problem, being it, for example, a simulation algorithm or a numerical method for the solution of a system of differential equations is introduced, it is usually devised in a sequential fashion, which is: However, in order to exploit the computational efficiency provided by parallel architectures, standard centralised algorithms have to be turned into parallel ones.

As a consequence, the parallelisation step helps in determining the kind of parallelism that characterises the problem, thus in choosing the best architecture.

INTRODUCTION

Algorithms for determining a partition of a given computational problem are usually referred to as clustering algorithms or graph partitioning algorithms. In a WG, nodes and arcs denote computational units and communications, respectively. Partitioning of the WG is based on the following simple criteria: A whole plethora of solutions exists in the literature about clustering methods.

Here, we limit ourselves to sketching the basic principles behind two most important families of such algorithms, namely geometric algorithms and structural algorithms. Solutions in both families are based on bisection according to which a partition is determined by recursive application of a division procedure that splits the original WG into two disjoint sub-graphs.

Geometric algorithms require nodes of the input graph to be equipped with geometric coordinates, which are used to calculate the bisection i. Furthermore, geometric algorithms rely on the assumption that nodes connectivity is equivalent to geometric proximity, an assumption which is not always reasonable. Structural algorithms, on the other hand, determine a bisection of the WG based exclusively on the graph's connectivity information. With level-structure bisection, the simplest form of structural bisection, two nodes of near-maximal distance is found, and a bisection is obtained through a breadth-first traversing that starting from one such node reaches as many as half the vertices of the graph: Several variants of this simple structural approach exist.

A detailed overview on the subject can be found in [ 7 ]. Numerical analysis is a field of mathematics that includes studying algorithms for the approximation of the solution of ODE [ 8 ]. Since the middle of the s, there have been significant efforts in designing efficient numerical techniques for ODE solution exploiting parallel and distributed architectures.

In the following, we sketch the possible techniques together with pointers to some popular software tools that implement them. Performance improvements can be achieved by applying parallel linear algebra techniques [ 10 ] and using available software libraries as in [ 11—14 ]. A more tailored approach to improve the performance of numerical methods for ODEs is to redesign or modify a sequential algorithm in order to exploit a specific target parallel architecture. The type of approach adopted deeply influences the performance improvement that can be achieved.

Parallelism across the method regards the use of parallel architectures to increment the strength and the efficiency of existing sequential algorithm. These kinds of methods are particularly used within the class of Runge—Kutta methods, and have a simple implementation on a parallel machine. A limit is given by the large data exchanged with respect to the workload per processor, and therefore this type of parallelism can capitalise the recent spread of cheap multi-core systems. Clearly, this approach allows only small-scale parallelism, and in general, it is used to obtain the same performance of sequential methods but at stringent tolerances.

Massive parallelism, in which a large number of processing units are available, requires different approaches. Parallelism across the system involves the decomposition of the domain of a problem into simpler sub-domains that can be solved independently. This approach requires sophisticate techniques and it is not always applicable. The general idea is to decompose an IVP into sub-problems that can be solved with different methods and different step-size strategies.

Waveform relaxation is a well-known class of decomposition techniques, where a continuous problem is split and the corresponding interactions a la Picard is defined. These methodologies require a stringent synchronisation of the computations in order to assure the consistency of the results. A different method exploits parallelism by performing concurrently several integration steps with a given interaction method, leading to the class of techniques of parallelism across the steps.

These techniques may be theoretically employed with a large number of processors, but intrinsic poor convergence behaviour could lead to robustness problems. However, this parallelisation method is receiving great attention because of its potential in scaling up the size of the problem that can be managed e. These are the main approaches to the parallelisation of numerical methods for ODE.

For a deeper introduction, we refer to the good monograph [ 8 ] and to the special issue [ 16 ]. As in the case of parallel linear algebra, many libraries that can be included in general purpose softwares have been developed. These libraries are then used within complex tools such as, for example, the Systems Biology Workbench [ 20 ], a tool which offers design, simulation and analysis instruments. Behind the use of libraries within specific simulation and analysis tools, new research lines specifically tailored on biological pathways deserve a separate discussion.

The on-chip solver computes concentration of substances for each time step by integrating rate law functions. Often biologists use cluster computers to launch many simulations of the same model with different parameters at the same time. ReCSiP is particularly suited for this kind of job offering a speed about to fold compared with modern microprocessors, and it is cheaper than the cluster solution. Both linear algebra applications and ODE solvers are actively studied, but actually specific works on pathway-related problems are not available.

A new and fruitful research line can be opened. Another interesting proposal is to parallelise algorithms that are specific to the analysis of biological pathways, as opposed to general ODE methods.

Bioinformatics: High Performance Parallel Computer Architectures (Embedded Multi Core Systems)

For instance, extreme pathways [ 23 ] are an algorithm for the analysis of metabolic networks. A solution of the IVP describes a particular metabolic phenotype, while extreme pathways analysis aims at finding the cone of solutions corresponding to the theoretical capabilities of a metabolic genotype.

Extreme pathways algorithm shows combinatorial complexity, but its parallel version [ 24 ] exhibits super-linear scalability, which means that the execution time decreases faster than the rate of increase in the number of processors. Stochastic simulation algorithms are computer programs that generate a trajectory i.

The SSA is referred to biochemical systems consisting of a well-stirred mix of molecular species that chemically interact, through so-called reaction channels, inside some fixed volume and at a constant temperature. Based on the CME, a propensity function is defined for each reaction j , giving the probability that a reaction j will occur in the next infinitesimal interval.

Bioinformatics: High Performance Parallel Computer Architectures (Embedded Multi Core Systems)

Then, relying on standard Monte Carlo method reactions are stochastically selected and executed forming in that a way a simulated trajectory in the discrete state-space corresponding to the CME. Several variants of the SSA exist, but all of them are based on the following common template: The different instances of the SSA vary in how the next reaction is selected and in the data structures used to store chemical species and reactions. In particular, the Next Reaction Method [ 28 ] is based on the so-called dependency graph, a direct graph whose nodes represent reactions, and whose arcs denote dependencies between reactions i.

Another research direction aims at integrating SSA with spatial information [ 29 ].

The Next-Subvolume Method NSM [ 30 ] simulates both reaction events and diffusion of molecules within a given volume; the algorithm partitions the volume into cubical sub-volumes that represent well-stirred systems. Stochastic simulation approaches, such as the one implemented by the SSA, are not new in systems biology; however only in relatively recent times they have received much attention, as an increasing number of studies revealed a fundamental characteristic of many biological systems.

It has been observed, in fact, that most key reactant molecules are present only in a small amount in living systems e. This renewed attention also showed the main limit of SSA: The resource requirements of SSA could be reduced either by using approximate algorithms or through parallelisation.

The latter research line is a really recent one, but some interesting proposals are emerging. A first improvement can be achieved by considering that many independent running of SSA are needed to compute statistics about a discrete and stochastic model.


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It is straightforward to run different simulations on different processes, but much attention has to be paid to the generation of random numbers [ 33 ]. This kind of parallelism is called parallelism across the simulation. The use of GRID architectures to run many independent simulations is promising because of inherent scalability [ 34 ]. Parallelism across the simulation is an effective technique when many simulations are needed, but there are instances where a single simulation of a large system think for example to a colony of cells is required.

In this case, the researches are only at the very beginning. Basically, there are two approaches to distribute computation to the processing units: The Distributed-based Stochastic Simulation Algorithm or DSSA [ 35 ] is developed on the intuition that the main computational requirement of any SSA variants comes from Steps 2 and 3, namely the random selection and the execution of the next reaction.

The DSSA relies on cluster architecture to tackle the complexity of these steps. In particular, a cluster processing unit, termed server, coordinates the activities of the other processing units or clients in the cluster, resulting in the following workflow: The partitioning algorithm employed in Step 1 uses the dependency graph as a weighted graph to minimise communications between the server and the clients; in particular, not all the clients need to be updated after a reaction is selected by the server.

The authors outline some experimental and performance analysis showing that the performance improvement with respect to SSA is linearly dependent on the number of client nodes. Another approach that is receiving great attention is based on geometric clustering.

A pioneer work is [ 36 ], but the algorithm reached a good maturation only with the recent efforts in integrating SSA with space information. In particular, in [ 37 ] the NSM is parallelised by using a geometric clustering algorithm to map set of sub-volumes to processing units. The algorithm scales well on a cluster architecture, where the main limit is the linear relation between the diffusion coefficient and the number of messages exchanged among the processing units.

The authors also test a promising GRID version of the algorithm, but the overhead due to the synchronisation among processing units requires more investigations. Finally, we refer a couple of applications of non-standard parallel hardware to speed up stochastic simulation [ 38 ].

1st Edition

Another work [ 40 ] exploits the high parallel structure of modern GPUs to obtain parallelism across the simulation without the costs of a computer cluster. He was the graphic of his American foundation to generate the business members. He not established the to find it. He pointed 0 by what he designed and later was it submitted as a.


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