  Many inverse problems are phrased as optimization problems in which the
objective function is the sum of a data-fidelity term and a regularization.
Often, the Hessian of the fidelity term is computationally unavailable while
the Hessian of the regularizer allows for cheap matrix-vector products. In this
paper, we study an LBFGS method that takes advantage of this structure. We show
that the method converges globally without convexity assumptions and that the
convergence is linear under a Kurdyka--{\L}ojasiewicz-type inequality. In
addition, we prove linear convergence to cluster points near which the
objective function is strongly convex. To the best of our knowledge, this is
the first time that linear convergence of an LBFGS method is established in a
non-convex setting. The convergence analysis is carried out in infinite
dimensional Hilbert space, which is appropriate for inverse problems but has
not been done before. Numerical results show that the new method outperforms
other structured LBFGS methods and classical LBFGS on non-convex real-life
problems from medical image registration. It also compares favorably with
classical LBFGS on ill-conditioned quadratic model problems. An implementation
of the method is freely available.
