We consider large-scale least squares problems where the coefficient matrix comes from the discretization of an operator in an ill-posed problem, and the right-hand side contains noise. Special techniques known as regularization methods are needed to treat these problems in order to control the effect of the noise on the solution. We pose the regularization problem as a quadratically constrained least squares problem. This formulation is equivalent to Tikhonov regularization, and we note that it is also a special case of the trust-region subproblem from optimization. We analyze the trust-region subproblem in the regularization case and we consider the nontrivial extensions of a recently developed method for general large-scale subproblems that will allow us to handle this case. The method relies on matrix-vector products only, has low and fixed storage requirements, and can handle the singularities arising in ill-posed problems. We present numerical results on test problems, on an inverse interpolation problem with field data, and on a model seismic inversion problem with field data.

Key words. regularization, constrained quadratic optimization, trust region, Lanczos method, ill-posed problems, inverse problems, seismic inversion

Also available as a Rice University Technical Report TR99-26.