Distributional Sensitivity for Uncertainty Quantification (Communications in Computational Physics, 2011)
In this work we consider a general notion of distributional sensitivity, whichmeasures the variation in solutions of a given physical/mathematical system with respectto the variation of probability distribution of the inputs. This is distinctivelydifferent from the classical sensitivity analysis, which studies the changes of solutionswith respect to the values of the inputs. The general idea is measurement of sensitivityof outputs with respect to probability distributions, which is a well-studied conceptin related disciplines. We adapt these ideas to present a quantitative framework inthe context of uncertainty quantification for measuring such a kind of sensitivity anda set of efficient algorithms to approximate the distributional sensitivity numerically.A remarkable feature of the algorithms is that they do not incur additional computationaleffort in addition to a one-time stochastic solver. Therefore, an accurate stochasticcomputation with respect to a prior input distribution is needed only once, andthe ensuing distributional sensitivity computation for different input distributions is apost-processing step. We prove that an accurate numerical model leads to accurate calculationsof this sensitivity, which applies not just to slowly-converging Monte-Carloestimates, but also to exponentially convergent spectral approximations. We providecomputational examples to demonstrate the ease of applicability and verify the convergenceclaims.