Transport reversal for model reduction of hyperbolic partial differential equations
For researchers in model reduction of hyperbolic PDEs, this work offers a method to improve projection-based techniques, though it is an incremental extension of prior symmetry reduction work.
The paper addresses the slow singular value decay in snapshot matrices from hyperbolic PDEs, which hinders projection-based model reduction. It proposes an iterative algorithm decomposing snapshots into shifting profiles with speeds, demonstrating applicability via numerical examples.
Snapshot matrices built from solutions to hyperbolic partial differential equations exhibit slow decay in singular values, whereas fast decay is crucial for the success of projection- based model reduction methods. To overcome this problem, we build on previous work in symmetry reduction [Rowley and Marsden, Physica D (2000), pp. 1-19] and propose an iterative algorithm that decomposes the snapshot matrix into multiple shifting profiles, each with a corresponding speed. Its applicability to typical hyperbolic problems is demonstrated through numerical examples, and other natural extensions that modify the shift operator are considered. Finally, we give a geometric interpretation of the algorithm.