Efficient Task Scheduling in the Parallel Result-Verifying Solution of Nonlinear Systems

Abstract

Nonlinear systems occur in diverse applications, e.g., in the steady state analysis of chemical processes. If safety concerns require the results to be provably correct then result-verifying algorithms relying on interval arithmetic should be used for solving these systems. Since such algorithms are very computationally intensive, the coarse-grained inter-box parallelism should be exploited to make them feasible in practice. In this paper we briefly describe our framework SONIC for the verified solution of nonlinear systems and give detailed information about its parallelization with OpenMP and MPI. Our numerical results show that the implemented parallelization schemes are indeed successful. The more sophisticated MPI implementation seems to be superior to the easy-to-implement OpenMP version and shows almost linear speedup up to a large number of processors.

A class of problems that can be solved using interval algorithms

“…Several tools can be used to parallelize interval computations. In particular, sharedmemory parallelization has already been done using OpenMP [2], POSIX threads [14], Boost threads [16], Intel Threading Building Blocks (TBB) [17].…”

Módulo Python para Matemática Intervalar

“…Mas também tem sido extensiva a pesquisa de novos algoritmos intervalares para a solução de problemas relacionados as mais diversasáreas onde o tratamento da incertezaé fundamental [11], como, por exemplo, Inteligência Artificial e Sistemas Multiagentes [5], Soft Computing, Geoinformática, Engenharia Elétrica [7] etc., oferecendo métodos para sistemas de equações lineares e não lineares, otimização, equações diferenciais etc. Em particular, os métodos para sistemas lineares intervalares têm sido um constante foco de pesquisa; foram introduzidos por E. Hansen [8] e continuados por um grande número de pesquisadores [9,10,16], inclusive em abordagens de programação paralela [2].…”

Hand Gesture Recognition in an Interval Fuzzy Approach

“…Interval Mathematics was introduced in [17] for the automatic and rigorous controlling of the errors that arise in numerical computations, providing techniques to deal with the uncertainty and to obtain verified results in several different contexts (see, e.g., [3,9,14]). Any real number x ∈ R that is uncertain for some reason (e.g., if it is obtained by a measuring instrument with limited resolution) is represented by a real interval X = [x 1 ; x 2 ], with x 1 , x 2 ∈ R and x 1 ≤ x ≤ x 2 .…”