A Dynamic Bi-orthogonal Field Equation Approach for Efficient Bayesian Calibration of Large-Scale Systems

Abstract: This paper proposes a novel computationally efficient dynamic bi-orthogonality based approach for calibration of a computer simulator with high dimensional parametric and model structure uncertainty. The proposed method is based on a decomposition of the solution into mean and a random field using a generic Karhunnen-Loeve expansion. The random field is represented as a convolution of separable Hilbert spaces in stochastic and spacial dimensions that are spectrally represented using respective orthogonal bases. In particular, the present paper investigates generalized polynomial chaos bases for stochastic dimension and eigenfunction bases for spacial dimension. Dynamic orthogonality is used to derive closed form equations for the time evolution of mean, spacial and the stochastic fields. The resultant system of equations consists of a partial differential equation (PDE) that define dynamic evolution of the mean, a set of PDEs to define the time evolution of eigenfunction bases, while a set of ordinary differential equations (ODEs) define dynamics of the stochastic field. This system of dynamic evolution equations efficiently propagates the prior parametric uncertainty to the system response. The resulting bi-orthogonal expansion of the system response is used to reformulate the Bayesian inference for efficient exploration of the posterior distribution. Efficacy of the proposed method is investigated for calibration of a 2D transient diffusion simulator with uncertain source location and diffusivity. Computational efficiency of the method is demonstrated against a Monte Carlo method and a generalized polynomial chaos approach.