Approximation and interpolation of divergence free flows.

Abstract

In many applications like meteorology, atmospheric pollution studies, eolic energy prospection, estimation of instantaneous velocity fields etc., one faces the problem of estimating a velocity field that is assumed to be incompressible. Very often the available data contains just a few and sparse velocity measurements and may be some boundary conditions imposed by solid boundaries. This inverse problem is studied here, and a new method to provide a numerical solution is presented. It is based on the Fourier transform, and allows to include the incompressibility constraint in a simple way, leading to an unconstrained least squares formulation, usually ill-posed. The Tikhonov regularization is applied to stabilize the solution, as well as to provide some smoothness in the estimated fow. As a consequence, the numerical solution will generally approximate the measurements up to a threshold given by the size of the regularization parameter. Moreover, if the available velocity measurements come from a smooth velocity field then the numerical solution can be usually constructed using just a small number of Fourier terms. The choice of the regularization parameter is done using the L curve method, balancing the perturbation and regularization contributions to the error. Perturbation bounds (i.e.), bounds for the condition number of the matrix from the Least Squares formulation are included. Numerical experiments with test problems and real data from the southern part of Uruguay are carried out. In addition, the results are compared with related work and the results are satisfactory.