Robust methods in inverse theory

Due to ill-conditioning of the linearised forward problem and the presence of noise in most data, inverse problems generally require some kind of 'regularisation' in order to generate physically plausible solutions. The most popular method of regularised inversion is damped least squares. Damping sometimes forces the solution to be smoother than it otherwise would be by raising all of the eigenvalues in an ad hoc fashion. An alternative is described, based upon the method of least-absolute deviation, which has a property known in the statistical literature as robustness. An account of robust inversion methods-their history and computational developments-is given. The key computational technique turns out to be preconditioned conjugate gradient, an algorithm which had as its genesis 'the method of orthogonal vectors' by Fox, Huskey and Wilkinson (1948). Applications are illustrated from seismic tomography and inverse scattering, two of the most computationally intensive tasks in inverse theory.