Simulated based optimization problems are hard due to both expensive cost of accurate function evaluations and the non convexity of the cost function, especially when the cost function evaluations are derived by finite time-average approximation of an infinite time-average statistics (Minimizing the drag or maximizing the lift-to-drag are examples of such optimization problems). Using a derivative- based optimization method is not appropriate since the exact value of infinite-time average statistics
is not available. Another possibility is to use the a specific type of derivative-free methods, which needs only an approximate value of the cost function with the known value the error of this approximate. In other words, the infinite-time-average value of a cost function can be replaced with the finite-time-average value; nevertheless, an estimate for the error of this approximation is also needed. In the present work, we develop a framework to quantify precisely the uncertainty of such a
finite-time-average approximation of an infinite-time-average statistic of a stationary ergodic process. In the method used, different statistical models for stationary processes have been examined to model the statistical behavior of the time series derived from the turbulence simulation. It is observed that the statistical behavior of some of these models is sufficiently representative of that of the real time series that they provide an accurate estimate of the uncertainty associated with the finite time average approximation of the statistic of interest.
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