This study deals with both a decision model for making decisions under epistemic uncertainty and how to use it for selecting optimal materials under the same uncertainty. In particular, the proposed decision model employs a set of possibilistic objective functions defined by fuzzy numbers to handle a set of conflicting criteria. In addition, the model can calculate the compliance of a piece of decision-relevant (imprecise) information with a given objective function. Moreover, the model is capable to aggregate the calculated compliances for the sake of ranking a given set of alternatives against the set of conflicting criteria. The problem of selecting materials for making the body of a vehicle is considered as an example. In this problem, the indices for selecting the materials are unknown because the specifications regarding the vehicle body are not given. In addition, the data relevant to material properties entails a great deal of imprecision. The presented decision model can quantify the above-mentioned epistemic uncertainty in a lucid manner and generate a list of optimal materials.

Much of uncertainty quantification to date has focused on determining the effect of variables modeled probabilistically, and with a known distribution, on some physical or engineering system. We develop methods to obtain information on the system when the distributions of some variables are known exactly, others are known only approximately, and perhaps others are not modeled as random variables at all.The main tool used is the duality between risk-sensitive integrals and relative entropy, and we obtain explicit bounds on standard performance measures (variances, exceedance probabilities) over families of distributions whose distance from a nominal distribution is measured by relative entropy. The evaluation of the risk-sensitive expectations is based on polynomial chaos expansions, which help keep the computational aspects tractable.

In this work we consider a general notion of distributional sensitivity, which measures the variation in solutions of a given physical/mathematical system with respect to the variation of probability distribution of the inputs. This is distinctively different from the classical sensitivity analysis, which studies the changes of solutions with respect to the values of the inputs. The general idea is measurement of sensitivity of outputs with respect to probability distributions, which is a well-studied concept in related disciplines. We adapt these ideas to present a quantitative framework in the context of uncertainty quantification for measuring such a kind of sensitivity and a set of efficient algorithms to approximate the distributional sensitivity numerically. A remarkable feature of the algorithms is that they do not incur additional computational effort in addition to a one-time stochastic solver. Therefore, an accurate stochastic computation with respect to a prior input distribution is needed only once, and the ensuing distributional sensitivity computation for different input distributions is a post-processing step. We prove that an accurate numericalmodel leads to accurate calculations of this sensitivity, which applies not just to slowly-converging Monte-Carlo estimates, but also to exponentially convergent spectral approximations. We provide computational examples to demonstrate the ease of applicability and verify the convergence claims.