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 $f_\ast \varrho$ which we would like to study. However, we might not have access to $f$, but to its approximation $g$. This problem is common in the use of surrogate models for numerical uncertainty quantification (UQ). We thus arrive at a fundamental question – if $f$ and $g$ are close in an $L^q$ space, does the measure $g_\ast \varrho$ approximate $f_\ast \varrho$ well, and in what sense? Previously, it was demonstrated that the answer to this question might be negative when posed in terms of the $L^p$ distance between probability density functions (PDF). Instead, we show in this paper that the Wasserstein metric is the proper framework for this question. For domains in $\mathbb{R}^d$, we bound the Wasserstein distance $W_p (f_\ast \varrho , g_\ast \varrho)$ from above by ${\lVert f - g \rVert}_q \:$. Furthermore, we prove lower bounds for the cases where $p=1$ and $p=2$ (for $d=1$) in terms of moments approximation. From a numerical analysis standpoint, since theWasserstein distance is related to the cumulative distribution function (CDF), we show that the latter is well approximated by methods such as spline interpolation and generalized polynomial chaos (gPC).

Keywords

Wasserstein, uncertainty quantification, density estimation, optimal transport, approximation

2010 Mathematics Subject Classification

28A10, 60A10, 65D99

Full Text (PDF format)

Received 23 March 2019

Accepted 18 November 2019

Published 30 June 2020