Data-driven stochastic representations of unresolved features in multiscale models

Abstract

Abstract. In this study we investigate how to use sample data, generated by a fully resolved multiscale model, to construct stochastic representations of unresolved scales in reduced models. We explore three methods to model these stochastic representations. They employ empirical distributions, conditional Markov chains and conditioned Ornstein-Uhlenbeck processes, respectively. The Kac-Zwanzig heat bath model is used as a prototype model to illustrate the methods. We demonstrate that all tested strategies reproduce the dynamics of the resolved model variables accurately. Furthermore, we show that the computational cost of the reduced model is several orders of magnitude lower than that of the fully resolved model.

Covariate-based stochastic parameterization of baroclinic ocean eddies

“…We model R as a stochastic process, following the approach discussed in Verheul and Crommelin [47]. This approach is a form of resampling, in which R is sampled uniformly from conditioned observed values of R. However, whereas in [47] we considered a situation in which R was a scalar quantity, here we are dealing with a spatially extended system in which R is a two-dimensional eld.…”

“…This approach is a form of resampling, in which R is sampled uniformly from conditioned observed values of R. However, whereas in [47] we considered a situation in which R was a scalar quantity, here we are dealing with a spatially extended system in which R is a two-dimensional eld. Therefore, we extend the method from [47] to a multidimensional setting, and apply it pointwise to sample the eld R. In this extension, we preserve the modular design philosophy behind the stochastic methodology, as well as the ability to represent non-linear and nonGaussian behavior.…”

“…However, the model allows for straightforward implementation of our stochastic modeling approach, as we discuss in the next section. Therefore, we consider it an appropriate test model for the purposes of developing the stochastic methodology from Verheul and Crommelin [47] in the setting of a spatially extended model.…”

“…Following the set-up from Verheul and Crommelin [47], we apply an equidistant binning procedure to approximate the CPDFs in conditioning procedure (2). We refer to uniformly drawing samples from these bins as sampling from the empirical distribution.…”

“…In the current work we explore how to use the novel data-driven stochastic methodology posed in Verheul and Crommelin [47] for eddy parameterization. The main assumption for our parameterization is that sample data from a 'high-resolution' ocean model is available.…”