The version in Eulerian coordinates is fast; the fully-Lagrangian variant is much slower for a given basis size N, but is highly adaptive for advection-dominated flows.
When the Hilbert space is of infinite dimension, state vectors can be represented by functions, the so-called 'wave functions.' In any given basis, there is a unique wave function that corresponds to the state vector in that basis.
In the second part, the authors propose a methodology to enforce that condition in order to perform an optimal reconstruction of the "closest" physical system starting from a given basis complex modes.
A brief review of major approaches that are suggested for the third step was given in Section 2. A signal x ∈ R n may be sparse in a given basis but not sparse in a different basis.
It is known that a multivariate function f can be well approximated by the linear combination of the elements of a given basis: f = ∑ k α k γ k (1).
It is important to mention that the performance of these methods, however, is strongly dependent on the key assumption that the source signals have a sparse representation in some given basis.
