3 research outputs found
A spectral surrogate model for stochastic simulators computed from trajectory samples
Stochastic simulators are non-deterministic computer models which provide a
different response each time they are run, even when the input parameters are
held at fixed values. They arise when additional sources of uncertainty are
affecting the computer model, which are not explicitly modeled as input
parameters. The uncertainty analysis of stochastic simulators requires their
repeated evaluation for different values of the input variables, as well as for
different realizations of the underlying latent stochasticity. The
computational cost of such analyses can be considerable, which motivates the
construction of surrogate models that can approximate the original model and
its stochastic response, but can be evaluated at much lower cost.
We propose a surrogate model for stochastic simulators based on spectral
expansions. Considering a certain class of stochastic simulators that can be
repeatedly evaluated for the same underlying random event, we view the
simulator as a random field indexed by the input parameter space. For a fixed
realization of the latent stochasticity, the response of the simulator is a
deterministic function, called trajectory. Based on samples from several such
trajectories, we approximate the latter by sparse polynomial chaos expansion
and compute analytically an extended Karhunen-Lo\`eve expansion (KLE) to reduce
its dimensionality. The uncorrelated but dependent random variables of the KLE
are modeled by advanced statistical techniques such as parametric inference,
vine copula modeling, and kernel density estimation. The resulting surrogate
model approximates the marginals and the covariance function, and allows to
obtain new realizations at low computational cost. We observe that in our
numerical examples, the first mode of the KLE is by far the most important, and
investigate this phenomenon and its implications
Spectral representation of stochastic field data using sparse polynomial chaos expansions
Uncertainty quantification is an emerging research area aiming at quantifying the variation in engineering system outputs due to uncertain inputs. One approach to study problems in uncertainty quantification is using polynomial chaos expansions. Though, a well-known limitation of polynomial chaos approaches is that their computational cost becomes prohibitive when the dimension of the stochastic space is large. In this paper, we propose a procedure to solve high dimensional stochastic problems with a limited computational budget. The methodology is based on an existing non-intrusive model reduction scheme for polynomial chaos representation, introduced by Raisee et al. [1], that is further extended by introducing sparse polynomial chaos expansions. Specifically, an optimal stochastic basis is calculated from a coarse scale analysis, using proper orthogonal decomposition and sparse polynomial chaos and is then utilized in the fine scale analysis. This way, the computational expense on both the coarse and fine discretization levels is drastically reduced. Two application examples are considered to validate the proposed method and demonstrate its potential in solving high dimensional uncertainty quantification problems. One analytical stochastic problems is first studied, where up to 20 uncertainties were introduced in order to challenge the proposed method. A more realistic CFD type application is then discussed. It consists of a two dimensional NACA 0012 symmetric profile operating at subsonic flight conditions. It is shown that the proposed reduced order method based on sparse polynomial chaos expansions is able to predict statistical quantities with little loss of information, at a cheaper cost than other state-of-the-art techniques. © 2018 Elsevier Inc