Solution through Optimization Sample Clauses

Solution through Optimization. Both of the applications described in the previous section require the solution of (3.2). In this section we consider a general nonlinear optimization scheme for simultaneous update of x and y. A Xxxxxx or Xxxxx-Xxxxxx approach can be quite difficult since it requires computation or approximation of the Hessian matrix, which is typically very large. Furthermore, another concern is the selection or estimation of a good regularization parameter. To reduce the computational effort, we discuss a variable projection ap- proach to take advantage of certain properties of the problem [51, 52, 85, 109, 117]. The basic idea is to exploit the following properties: the variables x and y separate, the problem is linear in x, and there are significantly fewer parameters defining y than there are defining x. More specifically, we mathe- matically eliminate one set of parameters to obtain a reduced cost functional, and a Xxxxx-Xxxxxx method is used for the reduced problem. This approach is a slight modification of the methods proposed by Xxxxx and Xxxxxxx [52] for separable nonlinear least squares problems and is similar to the approach used by Xxxxx, Xxxx and Xxxxxxxx [130] for phase diversity blind decon- volution. We are particularly interested in making the methods feasible for large-scale problems and incorporating a slick implementation for selecting good regularization parameters.