On the Symmetry Importance in a Relative Entropy Analysis for Some Engineering Problems

Abstract

This paper aims at certain theoretical studies and additional computational analysis on symmetry and its lack in Kullback-Leibler and Jeffreys probabilistic divergences related to some engineering applications. As it is known, the Kullback-Leibler distance in between two different uncertainty sources exhibits a lack of symmetry, while the Jeffreys model represents its symmetrization. The basic probabilistic computational implementation has been delivered in the computer algebra system MAPLE 2019®, whereas engineering illustrations have been prepared with the use of the Finite Element Method systems Autodesk ROBOT® & ABAQUS®. Determination of the first two probabilistic moments fundamental in the calculation of both relative entropies has been made (i) analytically, using a semi-analytical approach (based upon the series of the FEM experiments), and (ii) the iterative generalized stochastic perturbation technique, where some reference solutions have been delivered using (iii) Monte-Carlo simulation. Numerical analysis proves the fundamental role of computer algebra systems in probabilistic entropy determination and shows remarkable differences obtained with the two aforementioned relative entropy models, which, in some specific cases, may be neglected. As it is demonstrated in this work, a lack of symmetry in probabilistic divergence may have a decisive role in engineering reliability, where extreme and admissible responses cannot be simply replaced with each other in any case.

Keywords

• Kullback-Leibler divergence
• Jeffreys divergence
• relative entropy
• Finite Element Method
• probabilistic divergence of homogenized parameters
• reliability assessment