Subspace methods for large-scale nonlinear inversion

Abstract

Many nonlinear inverse problems can be cast into the form of determining the minimum of a misfit function between observations and theoretical predictions, subject to a regularisation condition on the form of the model. For large scale problems, linearised techniques involving inversion of a Hessian matrix rapidly become difficult to handle as the size of the problem increases. It can therefore be computationally advantageous to use techniques which can achieve convergence without the inversion of large matrices. In this class are descent algorithms or modifications thereto, the simplest approach is to use a single direction of search at each step (usually related to the gradient of the misfit function). The efficiency of the search for a minimum can be improved by the introduction of a second search direction at each stage, e.g. the rate of change of the gradient, or the gradient of the regularisation term.