Effective filtering and interpolation of 2D discrete velocity fields with Navier–Stokes equations. (21st September 2016) - Saumier, Louis-Philippe; Khouider, Boualem; Agueh, Martial
Abstract: We introduce a new variational technique to interpolate and filter a two-dimensional velocity vector field which is discretely sampled in a region of R 2 and sampled only once at a time, on a small time-interval [ 0, Δ t ] . The main idea is to find a solution of the Navier–Stokes equations that is closest to a prescribed field in the sense that it minimizes the l 2 norm of the difference between this solution and the target field. The minimization is performed on the initial vorticity by expanding it into radial basis functions of Gaussian type, with a fixed size expressed by a parameter ϵ . In addition, a penalty term with parameter k e is added to the minimizing functional in order to select a solution with a small kinetic energy. This additional term makes the minimizing functional strongly convex, and therefore ensures that the minimization problem is well-posed. The interplay between the parameters k e and ϵ effectively contributes to smoothing the discrete velocity field, as demonstrated by the numerical experiments on synthetic and real data.