Critical Evaluation of Stochastic Modeling Approaches for Subsurface Flow: How to Remove the Curse of Dimensionality?

Prediction of subsurface flow and transport is subject to uncertainties, which can result from the heterogeneity of the media and our incomplete knowledge about their properties. Such uncertainties render the model parameters random and the equations describing flow and transport in the media stochastic. Monte Carlo simulation method (MCS) is the most common and conceptually straightforward approach to solve the stochastic differential equations numerically. However, its main disadvantage is the requirement of large computational efforts owing to the approximation of probability density function with a large number of realizations. This is the so called “curse of dimensionality”. Recently, a number of alternative stochastic approaches have been developed with the attempt to quantify prediction uncertainties more efficiently. This talk evaluates four of such methods: The moment equation method (ME), the polynomial chaos expansion method (PCE), the Karhunen-Loeve based moment equation method (KLME), and the probabilistic collocation method (PCM). The efficiency of these methods depends on how the random (probability) space is approximated and how the random dimensionality is reduced. This talk also discusses the path forward and strategies for removing the curse of dimensionality.

Dongxiao Zhang, University of Oklahoma

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