Keywords: Bayesian prediction, error, posterior, Burgers' equation, high-fidelity simulations, low-fidelity simulations.

The purpose of this work is to establish a framework to quantify the uncertainty associated with the low-fidelity simulations of stochastic phenomena in the light of high-fidelity simulation data, and to propose a predictive methodology based on Bayesian inference to improve the accuracy of the low-fidelity simulations. We demonstrate the effectiveness of the methodology in the case of Burgers' equation with random viscosity.

The uncertainty in low-fidelity simulations is due to the approximation inherent in the theoretical model and the discretization errors coupled with the random nature of the physical parameters and the indeterminacy of the initial/boundary conditions. One may quantify this uncertainty given an independent observation or data. Such observation/data can be provided by experiments and/or high-order accuracy theoretical models, which we call high-fidelity simulations. In the present context, we choose the latter data. The mathematical formulation is general and accommodates any kind of data including the mixed type of data.

The methodology is applied to the stochastic solution of the one-dimensional Burgers' equation with the viscosity as a random variable. We choose the "numerically exact" solution of the of the Burgers' equation as observation/data (high-fidelity simulation). The "numerically exact" periodic solution is well known (see for example [1] or pages 183-185 of [2]). For the low-fidelity solutions of Burgers equation, we use two different numerical methods: a Fourier spectral method and a finite-difference method.

The general methododology developed in this work quantifies the parametric uncertainty in simulations in the light of accurate data, and based on Bayesian inference, improves the results of low fidelity simulations. The methodology is verified in the case of stochastic Burgers' equation with a random parameter. The uncertainty of the stochastic Burgers' solution on a coarse grid is quantified (in terms of confidence intervals), given fine-grid (numerically exact) Burgers' solutions. Then the Bayesian prediction based on the coarse-grid solution is shown to improve accuracy considerably.

The discrete posterior distribution may also be considered as a filter, which can be used to smooth out spurious oscillations in simulations, which arise due to lack of resolution.

1

G.B. Whitham, "Linear and Nonlinear Waves", Wiley-Interscience Series in Pure and Applied Mathematics, 1974.

2

C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, "Spectral methods in fluid dynamics", Springer-Verlag, 1988.

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