Adaptive sampling and modal expansions in pattern-forming systems Articles uri icon

publication date

  • August 2021

start page

  • 1

end page

  • 31


  • 4(48)


  • 47

International Standard Serial Number (ISSN)

  • 1019-7168


  • 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.


  • Mathematics


  • pattern-forming systems · reduced order models ·; sampling techniques · collocation methods · proper orthogonal decomposition ·; kuramoto-sivashinsky equation · complex ginzburg-landau equation