Basis norm rescaling for nonlinear parameter estimation

We consider the ill-posed, nonlinear problem of recovering the diffusion function in a one-dimensional elliptic equation. Previous work has revealed a correlation between high nonlinearity, low sensitivity, and short scale for this problem. It was noted that representing the unknown function by the multiscale Haar basis gave faster convergence than use of a local basis. The superior performance of the Haar basis was found to be related to the norms of the individual basis elements of this basis not being equal. The norm decreases with decreasing scale, enhancing the influence of parameters associated with low nonlinearity. Also, a hierarchical scale-by-scale approach was suggested to utilize the correlation between nonlinearity, scale, and sensitivity to increase the convergence speed of the parameter estimation. Through numerical experiments, we consider the effect of a systematic rescaling of the norms of the basis elements on the efficiency of the Broydon-Fletcher -Goldfarb-Shanno (BFGS) quasi-Newton minimization procedure. Norm rescaling is considered both for estimation of all parameters simultaneously, and for hierarchical scale-by-scale optimization.