Domain decomposition boosted by the mathematician RUDN


image: The mathematician from RUDN University and his colleagues from France and Hungary have developed an algorithm for parallel computation, which makes it possible to solve applied problems, such as electrodynamics or hydrodynamics. The time savings can be up to 50%.
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Credit: RUDN University

The mathematician from RUDN University and his colleagues from France and Hungary have developed an algorithm for parallel computation, which allows to solve applied problems, such as electrodynamics or hydrodynamics. The time savings can be up to 50%. The results are published in the Journal of Computational and Applied Mathematics.

Parallel computing methods are often used to deal with practical problems in physics, engineering, biology, and other fields. It involves multiple processors coming together in a network to simultaneously solve a single problem – each has its own small part. The way to distribute the work between the processors and to make them “communicate” with each other is a choice based on the specifics of a particular problem. One possible method is domain decomposition. The field of study is divided into distinct parts – subdomains – according to the number of processors. When that number is very high, especially in heterogeneous High Performance Computing (HPC) environments, asynchronous processes are a valuable ingredient. Usually, Schwarz methods are used, in which the subdomains overlap. This provides precise results but is not suitable where overlap is not straightforward. The mathematician from RUDN University and his colleagues from France and Hungary proposed a new algorithm that facilitates asynchronous decomposition in many structural cases – the subdomains do not overlap; the result remains precise with less time required for the calculation.

“Until now, almost all investigations of asynchronous iterations in domain decomposition frameworks have targeted parallel Schwarz-type methods. A first and only attempt to deal with non-overlapping primal decomposition resulted in simultaneous iteration over sub -domains and on the interface between them. This means that the calculation scheme is defined on the whole of the global domain “, Guillaume Gbikpi-Benissan, Engineering Academy of RUDN University.

Mathematicians have proposed an algorithm based on the Gauss-Seidel method. The essence of the innovation is that the computational algorithm is not executed simultaneously on the whole domain, but alternately on the subdomains and the boundaries between them. Consequently, the values ​​obtained during each iteration within the subdomain can be immediately used for calculations on the boundary at no additional cost.

Mathematicians tested the new algorithm on the Poisson equation and the linear elasticity problem. The first is used, for example, to describe the electrostatic field, the second is used in hydrodynamics, to describe the movement of liquids. The new method was faster than the original for both equations. A gain of up to 50% was indeed achieved – with 720 subdomains, the computation of the Poisson equation took 84 seconds while the original algorithm took 170 seconds. In addition, the number of synchronous alternating iterations decreases with an increase in the number of subdomains.

“This is quite an interesting behavior which can be explained by the fact that the alternation ratio increases as the sizes of the subdomains are reduced and more interfaces appear. This work therefore encourages for new possibilities and promising new investigations of the asynchronous computing paradigm “, Guillaume Gbikpi-Benissan, Academy of Engineering, RUDN University.

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