Project 3

An adaptive High Dimensional Stochastic Model Representation technique for uncertainty quantification

The aim of this project is to utilize a new computational tool which decomposes the high-dimensional problem into several lower-dimensional sub-problems that are easy to solve and thus alleviates to some extent the curse of dimensionality. This method utilizes High Dimensional Model Representation (HDMR) technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower-order to higher-order component functions:

HDMR is efficient at capturing the high-dimensional input-output relationship such that the behavior for many physical systems can be modeled only by the first few lower-order terms. An adaptive version of HDMR is also developed to automatically detect the important dimensions and construct higher-order terms only as a function of the important dimensions. In this project, we also incorporate the newly developed adaptive sparse grid collocation (ASGC) method into HDMR to solve the resulting sub-problems. By integrating HDMR and ASGC, it is computationally possible to construct a low-dimensional stochastic reduced-order model of the high-dimensional stochastic problem and easily perform various statistic analysis on the output.

Flow through random heterogeneous media is considered as an example to investigate the effect of the input uncertainty on the efficiency of HDMR:

The permeability K(x,Y) is taken as a random field. The domain of interest is a quarter-five spot problem in a unit square. The log-permeability is taken as zero mean random field with a separable exponential covariance function:

The Karhunen-Loeve (K-L) expansion is used to parameterize the field. In the exmaple, we fix the correlation length at L = 0.25 such that the weight of each dimension from the K-L expansion is nearly the same. Then we explore the effects of the spatial variability, from very small variability to very high variability. The number of stochastic dimensions is N = 500. The cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large input variability. The efficiency of the proposed method is examined by comparing with Monte Carlo (MC) simulation.

The following figure compares the standard deviation of the v velocity-component along the cross section y = 0.5.

The following figure compares the PDF of the v velocity-component at point (0,05).

The following figure plots the convergence result of the figure.

From the above plots, it is seen that this method indeed gave us a very accurate result compared with the MC method. In addition, it also exhibits fast convergence rate than that of MC method.

The paper and PowerPoint presentations can be downloaded from here.