An Efficient Bayesian Calibration Approach Using Dynamically Biorthogonal Field Equations

Present paper proposes new dynamic-biorthogonality based Bayesian formulation for calibration of computer simulators with parametric uncertainty. The formulation uses decomposition of solution field into mean and random field. The random field is represented as a convolution of separable Hilbert spaces in stochastic and spacial dimensions. Both the dimensions are spectrally represented using respective orthogonal bases. In particular, present paper investigates polynomial chaos basis for stochastic dimension and eigenfunction basis for spacial dimension. Dynamic evolution equations are derived such that basis in stochastic dimension is retained while basis in spacial dimension is changed such that dynamic orthogonality is maintained. Resultant evolution equations are used to propagate prior uncertainty in input parameters to the solution output. Whenever new information is available through experimental observations or expert opinion, Bayes theorem is used to update the basis in stochastic dimension. Efficacy of the proposed methodology is demonstrated for calibration of 2D transient diffusion equation with uncertainty in source location. Computational efficiency of the method is demonstrated against Generalized Polynomial Chaos and Monte Carlo method.