Probabilistic collocation method for strongly nonlinear problems: 1. Transform by location

Abstract

[1] In this work, we propose a new collocation method for uncertainty quantification in strongly nonlinear problems. Based on polynomial construction, the traditional probabilistic collocation method (PCM) approximates the model output response, which is a function of the random input parameter, from the Eulerian point of view in specific locations. In some cases, especially when the advection dominates, the model response has a strongly nonlinear profile with a discontinuous shock or large gradient. This nonlinearity in the space domain is then translated to nonlinearity in the random parametric domain, which causes nonphysical oscillation and inaccurate estimation using the traditional PCM. To address this issue, a new location-based transformed probabilistic collocation method (xTPCM) is developed in this study, inspired by the Lagrangian point of view, in which model response is represented by an alternative variable, i.e., the location of a particular response value, which is relatively linear to the random parameter with a smooth profile. The location is then approximated by polynomial construction, from which a sufficient number of location samples are randomly generated and transformed back to obtain the response samples and to estimate the statistical properties. The advantage of the xTPCM is demonstrated through applications to multiphase flow and solute transport in porous media, which shows that the xTPCM achieves higher statistical accuracy than does the PCM, and produces more reasonable realizations without oscillation, while computational effort is greatly reduced compared to the direct sampling Monte Carlo method.

Characterization of Aquifer Multiscale Properties by Generating Random Fractal Field with Truncated Power Variogram Model Using Karhunen–Loève Expansion

“…For example, it has been demonstrated that the mean and variance of hydraulic head obtained with only the first few leading KL expansion terms (e.g., = 6 for a 1D case) is very close to those obtained from thousands of Monte Carlo simulation with full models [25]. Similar observations are also reported in Li et al [26] and Liao and Zhang [27]. In a quantitative sense, if the polynomial chaos expansion (PCE) is used to construct a surrogate model for the state variables (e.g., hydraulic head) and KL expansion on the model parameter (e.g., log hydraulic conductivity), the number of model evaluation required to obtain the PCE coefficients is ( + )!/ !/ !, where is the degree of PCE.…”

“…The advantage of the proposed method is that KL expansion has the capability to conduct the model dimensionality reduction. If it is further incorporated with polynomial chaos expansion on the state variables, such as hydraulic head in the single-phase flow problem [25], saturation in the multiphase problem [26], and contaminant transport problem [27], the computational cost to perform uncertainty analysis in the hydrological research can be greatly reduced. The proposed KL-based multiscale random fractal field generator provides the foundation to establish a high-efficiency stochastic analysis framework.…”

Towards Improving the Efficiency of Bayesian Model Averaging Analysis for Flow in Porous Media via the Probabilistic Collocation Method

“…The Karhunen-Loeve expansion method can convert a correlated random function into a polynomial with independent and identically distributed Gaussian random variables [45,46]. Since the log hydraulic conductivity, Y, is heterogeneous and spatial correlated.…”

“…To obtain the coefficients, the probabilistic collocation method can be used, which is derived on the basis of weighted residual method. Denoting as the differential operator in Equation (1), the stochastic partial differential equation of groundwater flow model can be written as [44,46]:…”

“…An intrusive method to obtain the PCE coefficient has been developed so that the governing equation can be treated as a black box, and any existing solver can be easily adopted. The probabilistic collocation method (PCM) is a non-intrusive method developed by [44] which uses the points of numerical evaluation of the Galerkin integral as collocation points [45,46]. However, the previous response-surface-based inverse modeling methods can only consider a single best model as mentioned before, and thus the model uncertainty has been ignored.…”