A Data-Driven Low-Rank Approximation of Time-Dependent Stochastic Flows

We present a non-intrusive and data-driven method for constructing a low-rank approximation of time-dependent stochastic flows. This method requires a snapshot sequence of samples of the stochastic field in the form of $\mathbf{A} \in \mathbb{R}^{n\times s \times m}$ where $n$ is the number of observable data points, $s$ is the number of samples and $m$ is the number time steps. These samples may be generated using deterministic solvers or time-resolved PIV experimental measurements. In this methodology, the time-dependent data is approximated by an $r$-dimensional reduction in the form of: $\mathbf{A}^r(t)= \mathbf{U}(t) \mathbf{Y}(t)^T $ where $\mathbf{U}(t) \in \mathbb{R}^{n\times r}$ is a set of deterministic time-dependent orthonormal basis and $\mathbf{Y}(t) \in \mathbb{R}^{s\times r}$ are the stochastic coefficients. We derive explicit evolution equations for $\mathbf{U}(t)$ and $\mathbf{Y}(t)$ and use the data to solve these equations. We demonstrate that this reduction technique captures the strongly transient stochastic flows with high-dimensional random dimensions. We present the capability of this method for two classical fluid dynamics problems: (1) flow over a cylinder, and (2) three-dimensional jet in a crossflow.