Parallelization Method for a Continuous Property

• Published in: Foundations of computational mathematics Vol. 10; no. 1; pp. 93 - 114
• Main Author: Pilarczyk, Paweł
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.02.2010; Springer; Springer Nature B.V
• Subjects: Algorithms; Analysis; Applications of Mathematics; Computer Science; Economics; Estimating techniques; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematics; Mathematics and Statistics; Matrix Theory; Methods; Numerical Analysis; Parallel processing; Parallel computing; Rigorous numerics; Heuristics; 65G40; 65D99; 68W10; Iterative method; Algorithm; Interval arithmetic
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-009-9050-8

An automated general purpose method is introduced for computing a rigorous estimate of a bounded region in ℝ n whose points satisfy a given property. The method is based on calculations conducted in interval arithmetic and the constructed approximation is built of rectangular boxes of variable sizes. An efficient strategy is proposed, which makes use of parallel computations on multiple machines and refines the estimate gradually. It is proved that under certain assumptions the result of computations converges to the exact result as the precision of calculations increases. The time complexity of the algorithm is analyzed, and the effectiveness of this approach is illustrated by constructing a lower bound on the set of parameters for which an overcompensatory nonlinear Leslie population model exhibits more than one attractor, which is of interest from the biological point of view. This paper is accompanied by efficient and flexible software written in C++ whose source code is freely available at http://www.pawelpilarczyk.com/parallel/ .