Dynamics of data-driven ambiguity sets for hyperbolic conservation laws with uncertain inputs

Francesca Boso, Dimitris Boskos, Sonia Martínez, Jorge Cortés, Daniel M. Tartakovsky
SIAM Journal of Scientific Computing, 43 (3) (2021) A2102-A2129

Abstract:

Ambiguity sets of probability distributions are used to hedge against uncertainty about the true probabilities of uncertain inputs and random quantities of interest (QoIs). When available, these ambiguity sets are constructed from both data (collected at the initial time and along the boundaries of the physical domain) and concentration-of-measure results on the Wasserstein metric. To propagate the ambiguity sets into the future, we use a physics-dependent equation governing the evolution of cumulative distribution functions (CDFs) obtained through the method of distributions. We investigate the spatio-temporal evolution of data-driven ambiguity sets and their associated guarantees when the random QoIs they describe obey hyperbolic partial differential equations with random inputs. For general nonlinear hyperbolic equations with smooth solutions, the CDF equation is used to propagate the upper and lower envelopes of pointwise ambiguity bands. For linear dynamics, the CDF equation allows us to construct an evolution equation for tighter ambiguity balls. We demonstrate that, in both cases, the ambiguity sets are guaranteed to contain the true (unknown) distribution within a prescribed confidence.

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Bib-tex entry:

@article{FB-DB-JC-SM-DT:21-siamsc,
author = {F. Boso and D. Boskos and J. Cort\'es and and S. Mart{\'\i}nez and D. Tartakovsky},
title = {Dynamics of data-driven ambiguity sets for hyperbolic conservation laws with uncertain inputs},
journal= {SIAM Journal of Scientific Computing},
pages = {A2102-A2129 },
volume = {43},
number = {3},
year = {2021}
}