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39.3 Regularized Nonlinear Least-Squares

In cases where the Jacobian J is rank-deficient or singular, standard nonlinear least squares can sometimes produce undesirable and unstable solutions. In these cases, it can help to regularize the problem using ridge or Tikhonov regularization. In this method, we introduce a term in our minimization function which is designed to damp the solution vector x, or give preference to solutions with smaller norms. Here, the regularization matrix L is often set as L = \lambda I, for a positive scalar \lambda, but can also be a general m-by-p (where m is any number of rows) matrix depending on the structure of the problem to be solved. If we define a new (n+m)-by-1 vector or, in the weighted case, then which is in the same form as the standard nonlinear least squares problem. The corresponding (n+m)-by-p Jacobian matrix is or for weighted systems While the user could explicitly form the \tilde{f}(x) vector and \tilde{J} matrix, the fdfridge interface described below allows the user to specify the original data vector f(x), Jacobian J, regularization matrix L, and optional weighting matrix W, and automatically forms \tilde{f}(x) and \tilde{J} to solve the system. This allows switching between regularized and non-regularized solutions with minimal code changes.