Multiscale Stochastic Representation in High-Dimensional Data Using Gaussian Processes with Implicit Diffusion Metrics

Abstract

We develop a stochastic representation of a scalar function defined on a high-dimensional space conditional on marginal statistics of the function at a finite set of localities and a high-dimensional correlation structure. The representation leverages a particular structure of the functional dependence that exhibits scale separation. In the process, we construct a polynomial chaos representation for the scalar quantity of interest (QoI) whose coefficients are themselves random. The intrinsic randomness of the polynomial chaos expansion (PCE) reflects local uncertainty and captures dependence on a subset (say $S_1$) of the parameters, while randomness in the PCE coefficients captures a global structure of the uncertainty and dependence on the remaining parameters (say $S_2$) in the high-dimensional space (let $S=S_1\cup S_2$). This construction is demonstrated by predicting wellbore signatures in the Gulf of Mexico (GoM) where 100 tabulated data values are available at several thousand wellbore locations throughout the GoM. Reservoir simulators describing the physics of multiphase flow in porous media are used to calculate the PCE representations at the sites where data is available. In this context, random parameters describing the subsurface define the parameter set $S_1$. A Gaussian process model is then developed for each coefficient in these representations, construed as a function on $S_2$ over which an intrinsic diffusion metric is defined.