Adaptive sampling and modal expansions in pattern-forming systems

Abstract

A new sampling technique for the application of proper orthogonal decomposition to a set of snapshots has been recently developed by the authors to facilitate a variety of data processing tasks (J. Comput. Phys. 335, 2017). According to it, robust modal expansions result from performing the decomposition on a limited number of relevant snapshots and a limited number of discretization mesh points, which are selected via Gauss elimination with double pivoting on the original snapshot matrix containing the given data. In the present work, the sampling method is adapted and combined with low-dimensional modeling. This combination yields a novel adaptive algorithm for the simulation of time-dependent non-linear dynamics in pattern-forming systems. Convenient snapshot sets, computed on demand over the evolution, are stored to record local temporal events whose underlying mechanisms are essential for the approximations. Also, a collection of sparse grid points, which are used to construct the mode basis and the reduced system of equations, is adaptively sampled according to unlinked spatial structures. The outcome is a reduced order model of the problem that (i) yields reliable approximations of the dynamical transitions, (ii) is well-suited to describe localized spatio-temporal complexity, and (iii) provides fast computations. Robustness, accuracy, and computational efficiency of the proposed algorithm are illustrated for some relevant pattern-forming systems, in both one and two spatial dimensions, exhibiting solutions with a rich spatio-temporal structure.