A Novel Iterative Algorithm for Regularized Separable Nonlinear Inverse Problems

The variable projection (VarPro) method is an effective approach for solving separable nonlinear inverse problems. It reformulates the original least squares problem into a reduced nonlinear least squares problem, which is typically solved using the Gauss–Newton (GN) method. However, for certain classes of problems, incorporating general form Tikhonov regularization introduces significant computational challenges, particularly in computing the Jacobian matrix. Although the reformulated problem depends solely on the nonlinear parameters, the linear parameters continue to influence the Jacobian computation, necessitating repeated application of the GN method. To address this, we employ the LSMR algorithm as an efficient inner solver to approximate the Jacobian, applying an appropriate stopping criterion. To balance accuracy and performance, numerical examples demonstrate the effectiveness and practicality of the proposed method.