LAUSR

Significance Understanding the statistics of extreme events in dynamical systems of high complexity is of vital importance for reliability assessment and design. We formulate a method to pick samples optimally so that we have rapid convergence of the full statistics of a quantity of interest, including the tails that describe extreme events. This is important for large-scale problems in science and engineering, where we desire to predict the statistics of relevant quantities but can only afford a limited number of simulations or experiments due to their very expensive cost. We demonstrate our approach in a hydromechanical system with millions of degrees of freedom, where only 10–20 carefully selected samples can lead to accurate approximation of the extreme event statistics. We develop a method for the evaluation of extreme event statistics associated with nonlinear dynamical systems from a small number of samples. From an initial dataset of design points, we formulate a sequential strategy that provides the “next-best” data point (set of parameters) that when evaluated results in improved estimates of the probability density function (pdf) for a scalar quantity of interest. The approach uses Gaussian process regression to perform Bayesian inference on the parameter-to-observation map describing the quantity of interest. We then approximate the desired pdf along with uncertainty bounds using the posterior distribution of the inferred map. The next-best design point is sequentially determined through an optimization procedure that selects the point in parameter space that maximally reduces uncertainty between the estimated bounds of the pdf prediction. Since the optimization process uses only information from the inferred map, it has minimal computational cost. Moreover, the special form of the metric emphasizes the tails of the pdf. The method is practical for systems where the dimensionality of the parameter space is of moderate size and for problems where each sample is very expensive to obtain. We apply the method to estimate the extreme event statistics for a very high-dimensional system with millions of degrees of freedom: an offshore platform subjected to 3D irregular waves. It is demonstrated that the developed approach can accurately determine the extreme event statistics using a limited number of samples.