Small-sample Bayesian error estimation for ergodic, chaotic systems of ordinary differential equations

The discretization of chaotic systems introduces statistical errors in addition to discretization errors into the estimation of quantities of interest. In order to efficiently arrive at estimates of quantities of interest, these two forms of error should be balanced; however, simulations are run without knowledge of the true/asymptotic outputs of interest or their error behaviors. In this work, we develop a framework for error modeling and identification using small-sample Bayesian inference that allows approximation of the optimal balance between sampling time and discretization precision without the computation of high-cost libraries of reference solutions. The result enables the possibility of running chaotic and turbulent simulations in a way that minimizes the total error between sampling and discretization without prior knowledge of the error behavior of the system.