Multiscale estimation with spline wavelets, with application to two-phase porous-media flow
Geir Nævdal, Trond Mannseth, Kari Brusdal, Jan‐Erik Nordtvedt · Inverse Problems · 2000
We consider the inverse problem of recovery of unknown coefficient functions in differential equations. The set of PDEs constituting the current forward 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 estimation efficiency. The investigation into these issues is facilitated by applying a multiscale spline wavelet parametrization of the unknown function. Earlier investigations with an ODE forward model found that use of the multiscale Haar parametrization had a positive effect on the estimation efficiency of a quasi-Newton algorithm. Recently, a way to systematically enhance these effects has been suggested. In this paper, we further this approach with the Levenberg-Marquardt algorithm. This results in three variants of the Levenberg-Marquardt algorithm, each incorporating a possibility to enhance multiscale effects. Through numerical experiments with the PDE forward model, we assess the estimation efficiency of the variants when varying the enhancement of multiscale effects.