Randomized low-rank approximation methods for projection-based model order reduction of large nonlinear dynamical problems

Projection-based nonlinear model order reduction (MOR) methods typically make use of a reduced basis V is an element of R-mxk to approximate high-dimensional quantities. However, the most popular methods for computing V, eg, through a singular value decomposition of an m x n snapshot matrix, have asymptotic time complexities of (O(min(mn(2), m(2)n)) and do not scale well as m and n increase. This is problematic for large dynamical problems with many snapshots, eg, in case of explicit integration. In this work, we propose the use of randomized methods for reduced basis computation and nonlinear MOR, which have an asymptotic complexity of only (O(mnk) or (9(mn log(k)). We evaluate the suitability of randomized algorithms for nonlinear MOR and compare them to other strategies that have been proposed to mitigate the demanding computing times incurred by large nonlinear models. We analyze the computational complexities of traditional, iterative, incremental, and randomized algorithms and compare the computing times and accuracies for numerical examples. The results indicate that randomized methods exhibit an extremely high level of accuracy in practice, while generally being faster than any other analyzed approach. We conclude that randomized methods are highly suitable for the reduction of large nonlinear problems.