Title: Solution of uncertain linear systems of equations with probability-box parameters

Authors: Naijia Xiao; Robert L. Mullen; Rafi L. Muhanna

Addresses: Department of Microbiology and Plant Biology, University of Oklahoma, Norman, OK 73019, USA ' Department of Civil and Environmental Engineering, University of South Carolina, Columbia, SC 29208, USA ' School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

Abstract: The solution of linear systems of equations is often a component of engineering simulation and modelling. Often, the system parameters are uncertain. One representation of this uncertainty is the use of probability-boxes (or p-boxes), which do not require complete information about the probability distribution underlying the random variables. P-boxes are the bounds on allowable continuous distribution function for the random variables. Arithmetic operations on p-boxes yield guaranteed bounds on the probability distribution of the solution, regardless the nature of dependency. The solutions of p-box linear systems of equations are presented in the context of FEA of structural systems. Loading and material uncertainties are described by p-boxes. Earlier Monte-Carlo p-box approach was limited to independent uncertainties. The governing p-box linear equations are solved by an iterative approach using a fixed-point formulation. The resulting formulation gives guaranteed bounds of the probability distribution of the structural responses, at a high computational efficiency and a low overestimation level.

Keywords: uncertainty; probability-box; matrix decomposition; iterative enclosure method.

DOI: 10.1504/IJRS.2018.092515

International Journal of Reliability and Safety, 2018 Vol.12 No.1/2, pp.147 - 165

Received: 16 May 2017
Accepted: 19 Jan 2018
Published online: 23 Jun 2018 *

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