The Wasserstein Distances Between Pushed-Forward Measures with Applications to Uncertainty Quantification

In the study of dynamical and physical systems, the input parameters are often uncertain or randomly distributed according to a measure $\varrho$. The system’s response $f$ pushes forward $\varrho$ to a new measure $f\circ \varrho$whichwewouldliketostudy\mathrm{.}However,wemightnothaveaccessto$f$butonlytoitsapproximation$g$. We thus arrive at a fundamental question – if $f$ and $g$ are close in $L^q$, does $g\circ \varrho$ approximate $f\circ \varrho$ well, and in what sense? Previously, we demonstrated that the answer to this question might be negative in terms of the $L^p$ distance between probability density functions (PDF). Here we show that the Wasserstein metric is the proper framework for this question. For any $p\ge 1$, we bound the Wasserstein distance $W_p(f\circ \varrho ,g\circ \varrho )$ from above by $\mathrm{\parallel }f-g{\mathrm{\parallel }}_q$. Furthermore, we provide lower bounds for the cases of $p=1,2$. Finally, we apply our theory to the analysis of common numerical methods in the field of computational uncertainty quantification.