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Comput Geosci (2015) 19:311326
DOI 10.1007/s10596-015-9471-1
ORIGINAL PAPER
Constrained probabilistic collocation method for uncertainty quantification of geophysical models
Qinzhuo Liao Dongxiao Zhang
Received: 5 January 2014 / Accepted: 2 February 2015 / Published online: 21 February 2015 Springer International Publishing Switzerland 2015
Abstract The traditional probabilistic collocation method (PCM) uses either polynomial chaos expansion (PCE) or Lagrange polynomials to represent the model output response. Since the PCM relies on the regularity of the response, it may generate nonphysical realizations or inaccurate estimations of the statistical properties under strongly nonlinear/unsmooth conditions. In this study, we develop a new constrained PCM (CPCM) to quantify the uncertainty of geophysical models accurately and efficiently, where the PCE coefficients are solved via inequality constrained optimization considering the physical constraints of model response, different from that in the traditional PCM where the PCE coefficients are solved using spectral projection or least-square regression. Through solute transport and multiphase flow tests in porous media, we show that the CPCM achieves higher accuracy for statistical moments as well as probability density functions, and produces more reasonable realizations than does the PCM, while the computational effort is greatly reduced compared to the Monte Carlo approach.
Keywords Constrained probabilistic collocation method
Uncertainty quantification Physical constraints Strong
nonlinearity
1 Introduction
Uncertainty quantification methods, which estimate the statistics of system outputs based on the probabilistic description of model characteristics and inputs, are of great importance for stochastic analysis in geophysics [17, 34, 38]. Monte Carlo (MC) method is one of the most common approaches, where the model inputs are randomly generated to obtain the corresponding outputs, which can be further analyzed statistically. However, it often requires a large number of realizations to reduce the sampling error, thus it may become prohibitively expensive, especially for large-scale problems [2].
As a reasonably fast and attractive alternative, polynomial chaos expansion (PCE) approaches have found increased use in uncertainty quantification and sensitivity analysis over the past decades. First introduced by Wiener [33], the PCE constructs a model response surface by polynomial of uncertain parameters and offers an efficient way of including nonlinear effects in stochastic analysis [11]. The original PCE is limited to Hermite polynomials, which are optimal for Gaussian distributed random variables. For other parametric distributions (e.g., Beta, Gamma, Uniform), generalized...