Scalability Evaluation of Iterative Algorithms Used for Supercomputer Simulation of Physical processes

The paper is devoted to the development of a methodology for evaluating the scalability of compute-intensive iterative algorithms used in simulating complex physical processes on supercomputer systems. The proposed methodology is based on the BSF (Bulk Synchronous Farm) parallel computation model, which makes it possible to predict the upper scalability bound of an iterative algorithm in early phases of its design. The BSF model assumes the representation of the algorithm in the form of operations on lists using high-order functions. Two classes of representations are considered: BSF-M (Map BSF) and BSF-MR (Map-Reduce BSF). The proposed methodology is described by the example of the solution of the system of linear equations by the Jacobi method. For the Jacobi method, two iterative algorithms are constructed: Jacobi-M based on the BSF-M representation and Jacobi-MR based on the BSF-MR representation. Analytical estimations of the speedup, parallel efficiency and upper scalability bound are constructed for these algorithms using the BSF cost metrics on multiprocessor computing systems with distributed memory. An information about the implementation of these algorithms in C++ language using the BSF program skeleton and MPI parallel programming library are given. The results of large-scale computational experiments performed on a cluster computing system are demonstrated. Based on the experimental results, an analysis of the adequacy of estimations obtained analytically by using the cost metrics of the BSF model is made.

💡 Research Summary

The paper presents a methodology for evaluating the scalability of compute‑intensive iterative algorithms used in large‑scale physical‑process simulations on supercomputers. The approach is built on the Bulk Synchronous Farm (BSF) parallel computation model, which extends the classic BSP model with a master‑worker (master‑slave) paradigm and explicit macro‑step synchronization. The BSF model defines six fundamental cost parameters: the number of worker nodes K, the time st for the master to send a command to a single worker (excluding latency), the time wt for a worker to execute one command, the total time Rt for the master to receive results from all workers, the total time pt for the master to evaluate received results and check the stopping criterion, and the network latency L for a 1‑byte message.

A key contribution is the representation of iterative algorithms as operations on lists using higher‑order functions from the Bird–Meertens formalism: Map and Reduce. In the BSF‑M (Map) representation, the algorithm is expressed solely with Map; in BSF‑MR (Map‑Reduce), both Map and Reduce are used. This functional abstraction allows the analytical cost model to be applied directly, yielding closed‑form expressions for speedup, parallel efficiency, and the upper bound on scalability.

For BSF‑M, speedup S(K) is given by
S(K)=K·(wt+st) /