An Ensemble-type Approach to Numerical Error Estimation

Abstract:

The estimation of the numerical error in a specific physical quantity of interest (goal) is of key importance in geophysical modelling. Towards this aim, we have formulated an algorithm that combines elements of the classical dual-weighted error estimation with stochastic methods. Our algorithm is based on the Dual-weighted Residual method in which the residual of the model solution is weighed by the adjoint solution, i.e. by the sensitivities of the goal towards the residual. We extend this method by modelling the residual as a stochastic process. Parameterizing the residual by a stochastic process was motivated by the Mori-Zwanzig formalism from statistical mechanics.
Here, we apply our approach to two-dimensional shallow-water flows with lateral boundaries and an eddy viscosity parameterization. We employ different parameters of the stochastic process for different dynamical regimes in different regions. We find that for each region the temporal fluctuations of local truncation errors (discrete residuals) can be interpreted stochastically by a Laplace-distributed random variable. Assuming that these random variables are fully correlated in time leads to a stochastic process that parameterizes a problem-dependent temporal evolution of local truncation errors. The parameters of this stochastic process are estimated from short, near-initial, high-resolution simulations. Under the assumption that the estimated parameters can be extrapolated to the full time window of the error estimation, the estimated stochastic process is proven to be a valid surrogate for the local truncation errors.
Replacing the local truncation errors by a stochastic process puts our method within the class of ensemble methods and makes the resulting error estimator a random variable. The result of our error estimator is thus a confidence interval on the error in the respective goal. We will show error estimates for two 2D ocean-type experiments and provide an outlook for the 3D case.