Towards Gradient-Enhanced Surrogate Modeling for Groundwater Flow Problems
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In this contribution, we present our ongoing research into developing surrogate models for Uncertainty Quantification (UQ) in groundwater flow simulations. The physical problem is governed by Darcy's law, which describes fluid flow in a porous medium. More specifically, we consider the free-boundary formulation of the Darcy problem, in which the shape of the wet domain (saturated zone) is unknown a priori and is determined through a shape optimization process. To account for the inherent uncertainty in the soil composition we introduce a parametrized family of permeability fields, and consider the values of the parameters as uncertain. To construct an efficient surrogate model for quantities of interest (QoI) of the problem, such as the total flux across a surface, we employ a novel gradient-enhanced sparse grid technique. This approach integrates the information about the derivatives of the QoI with respect to the uncertain parameters into the construction of surrogate models. By exploiting both QoI values and their gradients, which can be obtained at a lower computational cost than QoI evaluations, we aim to reduce the computational costs needed to achieve an accurate representation of the input-output relationship.