Li, X., Hulshoff, S., Hickel, S. (2022)
Computers & Mathematics with Applications 126: 43-59. doi: 10.1016/j.camwa.2022.09.007

We present an enhanced online algorithm based on incremental Singular Value Decomposition (SVD), which can be used to efficiently perform a Proper Orthogonal Decomposition (POD) analysis on the fly. The proposed enhanced algorithm for modal analysis has significantly better computational efficiency than the standard incremental SVD and good parallel scalability, such that the strong reduction of computational cost is maintained in parallel computations.

POD plays an important role in the analysis of complex nonlinear systems governed by partial differential equations (PDEs), since it can describe the full-order system in a simplified but representative way using a handful of dominant dynamic modes. However, determining a POD from the results of complex unsteady simulations is often impractical using traditional approaches due to the need to store a large number of high-dimensional solutions. As an alternative, incremental SVD can be used to avoid the storage problem by performing the analysis on the fly using a single-pass updating algorithm. Nevertheless, the total computing cost of incremental SVD is more than traditional approaches.

In order to reduce this total cost, we incorporate POD mode truncation into the incremental procedure, leading to an enhanced algorithm for incremental SVD. The accuracy of the method depends on the truncation number (M) of the enhanced process. Results obtained with the enhanced method converge to the results of a standard SVD for large M. Two error estimators are formulated for this enhanced incremental SVD based on an aggregated expression of the snapshot solutions, equipping the proposed algorithm with criteria for choosing the truncation number. The effectiveness of these estimators and the parallel efficiency of the enhanced algorithm are demonstrated using transient solutions from representative model problems. Numerical results show that the enhanced algorithm can significantly improve the computing efficiency for different kinds of datasets, and that the proposed algorithm is scalable in both the strong and weak sense.