On-the-fly sparse interpolation for stochastic reduced-order modeling with time-dependent bases

One of the main challenges in reduced-order modeling of stochastic partial differential equations (SPDEs) with time-dependent bases (TDBs) is the presence of non-polynomial nonlinear terms which require the same computational cost as solving full-order models. This work proposes an adaptive sparse interpolation algorithm that enables stochastic reduced-order modeling based on TDBs with a large number of samples by reducing the computational cost of solving TDBs evolution equations. There is no need to perform any offline computations with this approach and it can adapt on-the-fly to any transient changes in dynamics. The presented method constructs a low-rank approximation for the right-hand side of the SPDE using the discrete empirical interpolation method (DEIM) with a rank-adaptive strategy to control the error of the sparse interpolation. Our method achieves computational speedup by adaptive sampling of the state and random spaces. We illustrate the efficiency of our approach for two test cases: (1) one-dimensional stochastic Burgers' equation and (2) two-dimensional compressible Navier-Stokes equations subject to one-hundred-dimensional random perturbations. In all cases, computational costs have been reduced by orders of magnitude.