Publications 2016

Coordinates transformation and polynomial chaos for the Bayesian Inference of a Gaussian process with parameterized prior covariance function
I. Sraj, O. Le Maitre, O. Knio, and I. Hoteit
Computer Methods in Applied Mathematics and Engineering, 298, 205-228, 2016
I. Sraj, O. Le Maitre, O. Knio, and I. Hoteit
Bayesian inference; Dimensionality reduction; Karhunen-Loève expansion; Markov Chain Monte Carlo
​This paper addresses model dimensionality reduction for Bayesian inference based on prior Gaussian fields with uncertainty in the covariance function hyper-parameters. The dimensionality reduction is traditionally achieved using the Karhunen-Loève expansion of a prior Gaussian process assuming covariance function with fixed hyper-parameters, despite the fact that these are uncertain in nature. The posterior distribution of the Karhunen-Loève coordinates is then inferred using available observations. The resulting inferred field is therefore dependent on the assumed hyper-parameters. Here, we seek to efficiently estimate both the field and covariance hyper-parameters using Bayesian inference. To this end, a generalized Karhunen-Loève expansion is derived using a coordinate transformation to account for the dependence with respect to the covariance hyper-parameters. Polynomial Chaos expansions are employed for the acceleration of the Bayesian inference using similar coordinate transformations, enabling us to avoid expanding explicitly the solution dependence on the uncertain hyper-parameters. We demonstrate the feasibility of the proposed method on a transient diffusion equation by inferring spatially-varying log-diffusivity fields from noisy data. The inferred profiles were found closer to the true profiles when including the hyper-parameters' uncertainty in the inference formulation.

DOI: 10.1016/j.cma.2015.10.002