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

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.