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Center for Nonlinear Studies |
Nonlinear parabolic equations with uncertain (random) coefficients play an important role in science and engineering, including geosciences where they are used to describe multiphase flow in heterogeneous porous media. We review several alternative probabilistic approaches for uncertainty quantification (UQ) in such problems, including stochastic collocation (SC) strategies---the current method of choice in the UQ community---and other techniques based on a spectral decomposition of state variables in the probability space. We demonstrate that the performance of SC is strongly tied to the way the stochastic properties of the random input parameters affect the regularity of the systems' stochastic response in the probability space. If random input fields have low variance and large correlation lengths, SC strategies are competitive against alternative uncertainty quantification methods, such as Monte Carlo simulations (MCS). Increasing variance affects the regularity of the stochastic response, requiring higher-order quadrature rules to accurately approximate the moments of interest, thus increasing the overall computational cost beyond that of MCS. We develop reduced complexity models, which yield closed-form semi-analytical expressions for the single- and multi-point probability density functions (PDFs) of the state variables. These solutions enable us to investigate the relative importance of uncertainty in various input parameters (e.g., hydraulic soil properties) and the effects of their cross-correlation. Our simulation results demonstrate the reduced complexity models provide conservative estimates of predictive uncertainty.
Host: Gowri Srinivasan