Variable projection methods for separable nonlinear inverse problems with general-form Tikhonov regularization

The variable projection (VarPro) method is an efficient method to solve separable nonlinear least squares problems. In this paper, we propose a modified VarPro method for solving separable nonlinear least squares problems with general-form Tikhonov regularization. In particular, we apply the Gauss-Newton method to the corresponding reduced problem and investigate its convergence when different approximations of the Jacobian matrix are used. For special cases when computing the generalized singular value decomposition is feasible or a joint spectral decomposition of both forward and regularization operators exists, we provide efficient ways to compute the Jacobians and the solution of the linear subproblems. For large-scale problems, where matrix decompositions are not an option, we compute a reduced Jacobian and apply projection-based iterative methods and generalized Krylov subspace methods to solve the linear subproblems. In all cases, the regularization parameter can be computed automatically at each iteration using generalized cross validation. Several numerical examples highlight the proposed approach's performance in the quality of the reconstructed image and the reconstructed forward operator, including large-scale two-dimensional imaging problems arising from semi-blind deblurring.

Automated summary

The paper proposes a modified VarPro method for solving separable nonlinear least squares problems with general-form Tikhonov regularization. The Gauss-Newton method is applied to the reduced problem with different approximations of the Jacobian matrix, and convergence is investigated. Efficient ways to compute the Jacobians and the solution of the linear subproblems are provided. For large-scale problems, projection-based iterative methods and generalized Krylov subspace methods are used, and the regularization parameter is computed automatically using generalized cross validation. Numerical examples demonstrate the performance of the proposed approach in image reconstruction and forward operator reconstruction, including large-scale two-dimensional imaging problems arising from semi-blind deblurring.