##### Abstract

We present a new algorithm for the solution of nonlinear least squares problems arising from parameterized imaging problems with diffuse optical tomographic data [D. Boas et al., IEEE Signal Process. Mag., 18 (2001), pp. 57-75]. The parameterization arises from the use of parametric level sets for regularization [M.E. Kilmer et al., Proc. SPIE, 5559 (2004), pp. 381-391], [A. Aghasi, M.E. Kilmer, and E.L. Miller, SIAM J. Imaging Sci., 4 (2011), pp. 618-650]. Such problems lead to Jacobians that have relatively few columns, a relatively modest number of rows, and are ill-conditioned. Moreover, such problems have function and Jacobian evaluations that are computationally expensive. Our optimization algorithm is appropriate for any inverse or imaging problem with those characteristics. In fact, we expect our algorithm to be effective for more general problems with ill-conditioned Jacobians. The algorithm aims to minimize the total number of function and Jacobian evaluations by analyzing which spectral components of the Gauss-Newton direction should be discarded or damped. The analysis considers for each component the reduction of the objective function and the contribution to the search direction, restricting the computed search direction to be within a trust region. The result is a truncated SVD-like approach to choosing the search direction. However, we do not necessarily discard components in order of decreasing singular value, and some components may be scaled to maintain fidelity to the trust region model. Our algorithm uses the Basic Trust Region Algorithm from [A.R. Conn, N.I.M. Gould, and Ph. L. Toint, Trust-Region Methods, SIAM, Philadelphia, 2000]. We prove that our algorithm is globally convergent to a critical point. Our numerical results show that the new algorithm generally outperforms competing methods applied to the DOT imaging problem with parametric level sets, and regularly does so by a significant factor.