Nonlinear least squares and regularization

摘要

A problem frequently encountered in the earth sciences requires deducing physical parameters of the system of interest from measurements of some other (hopefully) closely related physical quantity. The author presents and discusses some general ideas about iterative nonlinear output least-squares methods. The main result is that, if it is possible to do forward modeling on a physical problem in a way that permits the output (i.e., the predicted values of some physical parameter that could be measured) and the first derivative of the same output with respect to the model parameters (whatever they may be) to be calculated numerically, then it is possible (at least in principle) to solve the inverse problem using the method described. The main trick learned in this analysis is that the steps in the model updates may have to be quite small in some cases for the implied guarantees of convergence to be realized.