Domain decomposition parallelization method

The well-known domain decomposition (DD) methods, see e.g. Chan Mathew (1994), use data partition induced by a decomposition of the computational domain. This decomposition can be used for two purposes Firstly for parallel implementation of vector updates, inner products and matrix-vector multiplication, i.e. for parallel implementation of the CG method. Secondly, for a construction of efficient preconditioners. 

The third parallel method discussed here for MD simulations of systems with short-range interactions is domain-decomposition (DD) method [31,38-40]. In this method the physical simulation box is divided into small 3D boxes, one for each processor. The partitioning of a simulation box of length L in a DD algorithm is shown in Fig. 12 (2D projection). Each processor z owns a sub-box labeled with edge length L = L P) and... 

In this paper, we shall touch the development of such numerical methods intended for the solution of the coupled evolution problems as e.g. thermoelasticity, which is described in Section 2. Here we also discuss the discretization of the evolution problems. As the computational demands are concentrated mainly in the solution of the arising linear systems, we shall focus on the application of suitable, efficient and parallelizable iterative solvers for these linear systems. Section 3 deals with some general techniques enhancing the efficiency of the iterative solution of discrete evolution problems. Section 4 is devoted to a short discussion of the numerical results. In Section 5, we shall describe solvers, which exploit the domain decomposition and parallel computations. Here we also mention another division techniques as displacement decomposition or composite grid methods. 

Spectral domain-decomposition methods are suggested as an efficient way of resolving the limitation of conventional spectral methods. The basic idea behind the methods is to partition the whole domain into several subdomains and then to simultaneously solve the differential equations in each subdomain and appropriate matching conditions on each interface. These methods have very attractive features, such as rapid convergence, geometric flexibility, and suitability to parallel implementation. Thus, they are appropriate when particular regions need to be resolved or when flows in complex geometries are to be simulated. 

Because ofthe material non-linearities, the mechanical contacts with friction, the large number of elements, many iteration steps, and the choice of 500 Monte Carlo simulations, four parallel computers (with 26 CPU) were used to handle the large computational requirements for this problem. The FETI Domain Decomposition Method (i.e. apphcation of parallel computers) was used, see Figure 11. 

This article reviews some of the progress made in using parallel processor systems to study macromolecules. After an initial introduction to the key concepts required to understand parallelisation, the main part of the article focuses on molecular dynamics. It is shown that simple replicated data methods can be used to carry out molecular dynamics effectively, without the need for major changes from the approach used in scalar codes. Domain decomposition methods are then introduced as a path toward reducing inter-processor communication costs further to produce truly scalable simulation algorithms. Finally, some of the methods available for carrying out parallel Monte Carlo simulations are discussed. 

MD simulations of polymer systems, in particular, require computation of two kinds of interactions bonded forces (bond length stretching, bond angle bending, torsional) and nonbonded forces (van der Waals and Coulombic). Parallel techniques developed [31-33] include the atom-decomposition (or replicated-datd) method, the force-decomposition method, and the spatial (domain)-decomposition method. The three methods differ only in the way atom coordinates are distributed among the processors to perform the necessary computations. Although all methods scale optimally with respect to computation, their different data layouts incur different interprocessor communication costs which affect the overall performance of each method. 

A DPD simulation may involve as few as 1000 particles for siirple equilibrium simulations to several million particles for simulations of complex fluids. Hence, it is important to run DPD codes in a parallel environment. There are several parallelization strategies, such as domain decomposition and force decomposition methods. One method that is appealing because of the particle-based nature of the method is the atom decomposition algorithm in which the computations for the N particles are split among P processors. Further advantage can be taken from the fact that DPD is a short-range method, i. e. each particle does... 

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