Probabilistic collocation method for strongly nonlinear problems: 2. Transform by displacement

Abstract

The probabilistic collocation method (PCM) is widely used for uncertainty quantification and sensitivity analysis. In paper 1 of this series, we demonstrated that the PCM may provide inaccurate results when the relation between the random input parameter and the model response is strongly nonlinear, and presented a location-based transformed PCM (xTPCM) to address this issue, relying on the transform between response and location. However, the xTPCM is only applicable for one-dimensional problems, and two or three-dimensional problems in homogeneous media. In this paper, we propose a displacementbased transformed PCM (dTPCM), which is valid in two or three-dimensional problems in heterogeneous media. In the PCM, we first select collocation points and run model/simulator to obtain response, and then approximate the response by polynomial construction. Whereas, in the dTPCM, we apply motion analysis to transform the response to displacement. That is, the response field is now represented by the displacement field. Next, we approximate the displacement instead of the response by polynomial, since the displacement is more linear to the input parameter than the response. Finally, we randomly generate a sufficient number of displacement samples and transform them back to obtain response samples to estimate statistical properties. Through multiphase flow and solute transport examples, we demonstrate that the dTPCM provides much more accurate statistics than does the PCM, and requires considerably less computer time than does the Monte Carlo (MC) method.

Deep Convolutional Encoder‐Decoder Networks for Uncertainty Quantification of Dynamic Multiphase Flow in Heterogeneous Media

“…It can then be used as a full replacement of the actual model when performing UQ tasks. Many surrogate methods based on the polynomial chaos expansion (Xiu & Karniadakis, ), Gaussian processes (Rasmussen & Williams, ), and neural networks (Hornik et al, ) have been applied widely to address UQ tasks in groundwater models with random inputs and have shown an impressive approximation accuracy and computational efficiency in comparison to MC methods (Chan & Elsheikh, ; Crevillén‐García et al, ; Li et al, ; Liao & Zhang, ; ; , Liao et al, ; Meng & Li, ; Müller et al, , ; Tian et al, , ).…”

“…Second, for multiphase flow problems, the saturation profile is discontinuous in the spatial domain due to capillarity effects. This discontinuity leads to one in the stochastic space as well making it extremely challenging to accurately approximate due to the fact that most surrogate models are continuous and often differentiable (Asher et al, ; Liao & Zhang, , , ; Liao et al, ; Lin & Tartakovsky, ; Mo et al, ; Xiu & Hesthaven, ). A variety of approaches are proposed to handle discontinuities.…”

Constrained probabilistic collocation method for uncertainty quantification of geophysical models

“…And P 1 may be less than the number of unknowns Q, which leads to an underdetermined system. To avoid this possible ill-posed problem, we use the regularization technique via choosing a prior solution [22]. Therefore, the optimization problem can be modified as…”

“…However, the sampling results from approximated polynomials in the traditional PCM may reveal nonphysical values out of this range. This is because the saturation profile has a discontinuous shock front in space domain, which is then translated to a discontinuous shock front in parametric domain [21,22]. Thus, the Gibbs phenomenon with overshoot occurs when we use globally smooth polynomials to approximate the strongly nonlinear functions.…”

“…Another instance is that in solute transport, the concentration values should always be nonnegative. However, it was observed that the PCM may produce realizations with negative concentrations as well as inaccurate moments under certain circumstances [21][22][23]39]. The reason is that with small porescale dispersion, the point concentration has a bimodal PDF, with one of the modes being zero and the other close to the concentration at the source at early times [4,8], which brings about difficulty in predicting the statistics by the PCM In summary, the strong nonlinearity and non-Gaussianity are accompanied by physical constraints especially in some advection dominated cases, where the traditional PCM is no longer suitable.…”