Multiscale estimation with spline wavelets, with application to two-phase porous-media flow

We consider the inverse problem of recovery of unknown coefficient function s in differential equations. The set of PDEs constituting the current forwa rd model describes a special case of two-phase porous-media flow. The focus of the paper is on the influence of different length scales on parameter e stimation efficiency. The investigation into these issues is facilitated by applying a multiscale spline wavelet parametrization of the unknown functi on. Earlier investigations with an ODE forward model found that use of the multiscale Haar parametrization had a positive effect on the estimation eff iciency of a quasi-Newton algorithm. Recently, a way to systematically enha nce these effects has been suggested. In this paper, we further this approa ch with the Levenberg-Marquardt algorithm. This results in three variants o f the Levenberg-Marquardl algorithm, each incorporating a possibility to en hance multiscale effects. Through numerical experiments with the PDE forwar d model, we assess the estimation efficiency of the variants when varying t he enhancement of murtiscale effects.