Probabilistic Robustness Analysis: Examples

The purpose of this document is to demonstrate a methodology to bound the uncertainty of an output function(s), given some level of uncertainty in the input variables to the model. In a companion paper, the theory and formulation of the method have been discussed. Several different types of data are available for use in the prescription of the uncertainty in the input variables. These include random variables, intervals, sets of intervals, and combinations of the three, e.g. random variables with parameters that are intervals. The propagation of the uncertainty is accomplished by the formulation and solution of two global optimization problems with bounded constraints. The solutions of these two problems yield an interval for the output function in the scalar case and a hyper-box in the vector output case. Along with the bounds on the outputs, a probability estimate that the output lies in the pertinent region, e.g. the interval or hyper-box, is calculated using Frechet bounds. It can be shown that these estimates are always conservative in nature. Several examples will be given demonstrating the efficacy of the method and its increased potential efficiency over traditional sampling based approaches. Its use in the vector output case will be also be discussed. Introduction Current trends in engineering analysis require a significantly increased reliance on computational models of systems, some of which can be described as possessing uncertainty both in their physical description and in their service environments. The propagation of uncertainty through terascale computational models is a daunting computational task. Uncertainty occurs in various forms. There are two main classes of uncertainty. The first is known as reducible or epistemic uncertainty; this uncertainty generally results from a lack of information about some aspect of the problem being considered. For instance, the form of the constitutive model of a joint connecting two members of a complex structural system is not known; this uncertainty results from a lack of information about the joint’s behavior. Another example of epistemic uncertainty is the related situation where the constitutive model for the joint is known, but sufficient data to fully characterize the uncertainty in the models’s parameters is not available. Most current uncertainty propagation techniques employ a statistical approach where probability distributions are required to be specified for the uncertain parameters contained within the computational models, and the induced probability distributions of desired output quantities computed. AIAA-2002-1645 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con 22-25 April 2002, Denver, Colorado AIAA 2002-1645 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. 2 American Institute of Aeronautics and Astronautics However, oftentimes, insufficient information is known to completely specify probability distributions for all of the uncertain parameters within the model. Reiterating, this lack of complete knowledge about the proper choice of probability distribution is one instance of epistemic uncertainty. Consequently, a flexible, uncertainty propagation methodology which can accommodate specification of uncertain variables not only as random variables but also as intervals, sets of intervals, and combinations of these is desirable. The approach to be discussed here is an extension of classical interval analysis, i.e. “Given bounds on the input parameters, what range of output function is possible?” Since the computational models of interest are blackboxes, only those techniques that are not intrusive to the analysis code are germane.The analysis is especially difficult in the case of blackboxes as the minimum and maximum of the output(s) often occur at interior points of the ranges of the input variables, e.g. resonance phenomena. The propagation of the uncertainty is accomplished by the formulation and solution of two global optimization problems with bounded constraints. It can be shown that this formulation is also applicable if there is enough information to specify some input parameters as random variables and others as intervals or a combination of these. In this mixed situation, the bounded constraints for each variable are chosen differently depending on the nature of that input variable.The solutions of these two problems yield an interval for the output function in the scalar case and a hyper-box in the vector output case. Along with the bounds on the outputs, a probability estimate that the output lies in the pertinent region, e.g. the interval or hyper-box, is calculated using Frechet bounds. It can be shown that these probability estimates are lower bounds, thus always conservative in nature. Several examples will be given. The global optimization problems are formulated in the next section. Problem Formulation Consider a model (simulation code), , which maps a vector of uncertain inputs given, , to a vector of output(s), . This inputoutput relationship is displayed in Figure 1. In the probabilistic robustness approach, the uncertain variables, , are modeled as random variables or deterministic intervals or combinations of the two. One seeks bounds on along with a bound on the probability of being contained in the enveloping region. The approach differs from a traditional statistical approach as it does not require full distribution specification but permits lower grades of information such as intervals to be specified when appropriate. As a result, the proposed methodology is more robust to changes to the uncertainty specification as it deals only with the support, i.e. areas of non-negligible probability of the uncertain variables and the induced outputs, rather than the entire distributions. However, a penalty is incurred as the methodology addresses only the extremes of the output and not intermediate values; this is usually acceptable in the uncertainty quantification process used in certification, i.e. a go-no go situation; however, it is not in cases such as optimization under uncertainty where design trade-offs need to be performed. M