Optimal Upscaling with Transport Maps

Abstract

Partial differential equation models often involve space-dependent parameters, such as diffusion coefficients and advection fields, that cannot be measured explicitly and are therefore uncertain. Midpoint (MP), spatial averaging (SA), shape function (SF) and series expansion (SE) methods have long existed in the stochastic simulation community and are well known for their computational hurdles. In this work, we compute efficient, spatially adaptive, lower-dimensional approximations of these fields, using transport maps. Such parsimonious representations of the parameter space would greatly improve the efficiency of the resulting stochastic simulations, allow for more targeted use of reduced order models, and aid in the related design of interventions. Numerical examples demonstrate our theoretical results