A Statistical Method for Regularizing Nonlinear Inverse Problems

Inverse problems are typically ill-posed or ill-conditioned and require regularization. Tikhonov regularization is a popular approach and it requires an additional parameter called the regularization parameter that has to be estimated. The χ method introduced by Mead in [8] uses the χ distribution of the Tikhonov functional for linear inverse problems to estimate the regularization parameter. However, for nonlinear inverse problems the distribution of the Tikhonov functional is not known. In this thesis, we extend the χ method to nonlinear problems through the use of Gauss Newton iterations and also with Levenberg Marquardt iterations. We derive approximate χ distributions for the quadratic functionals that arise in Gauss Newton and Levenberg Marquardt iterations. The approach is illustrated on two ill-posed nonlinear inverse problems: a nonlinear cross-well tomography problem and a subsurface electrical conductivity estimation problem. We numerically test the validity of assumptions in this approach by demonstrating that the theoretical χ distributions agree closely with actual distributions. The nonlinear χ method is implemented in two algorithms, based on Gauss Newton and the Levenberg Marquardt methods, that dynamically estimate the regularization parameter using χ tests. We compare parameter estimates from the nonlinear χ method with estimates found using Occams inversion and the discrepancy principle on the cross-well tomography problem and on the subsurface electrical conductivity estimation problem. The χ method is shown to provide similar parameter estimates to estimates found using the discrepancy principle and is computationally less expensive. In addition, the χ method provided much better parameter estimates than Occams Inversion. iii