Incomplete statistical information limits the utility of high-order polynomial chaos expansions

Polynomial chaos expansion (PCE) is a well-established massive stochastic model reduction technique that approximates the dependence of model output on uncertain input parameters. In many practical situations, only incomplete and inaccurate statistical knowledge on uncertain input parameters are available. Fortunately, to construct a finite-order expansion, only some partial information on the probability measure is required that can be simply represented by a finite number of statistical moments. Such situations, however, trigger the question to what degree higher-order statistical moments of input data are increasingly uncertain. On the one hand, increasing uncertainty in higher moments will lead to increasing inaccuracy in the corresponding chaos expansion and its result. On the other hand, the degree of expansion should adequately reflect the non-linearity of the analyzed model to minimize the approximation error of the expansion. Observation of the PCE convergence when statistical input information is incomplete demonstrates that higher-order PCE expansions without adequate data support are useless. Moreover, it makes apparent that PCE of a certain order is adequate just for a corresponding amount of available input data. The key idea of the current work is to align the order of expansion with a compromise between the degree of non-linearity of the model and the reliability of statistical information on the input parameters. To assure an optimal choice of the expansion order, we offer a simple relation that helps to align available input statistical data with an adequate expansion order. As fundamental steps into this direction, we propose overall error estimates for the statistical type of error that results from inaccurate statistical information plus the error that results from truncating the expansion of a non-linear model. Our key message is that any order of expansion is only justified if accompanied by reliable statistical information on input moments of a certain higher order.