Minimization of functionals on a large-scale ill-posed problem

Minimization of functionals on a large-scale ill-posed problem

We study the minimization of a linear functional defined on a set of approximate solutions of a linear discrete ill-posed problem. The primary application of interest is the computation of confidence intervals for components of the solution of such a problem. We exploit a technique introduced by Eld'en in 1990, and utilize a parametric programming reformulation involving the solution of a sequence of quadratically constrained least squares problems. The solution method is based on the connection between Lanczos bidiagonalization and Gauss-type quadrature rules which allows us to inexpensively bound certain matrix functionals. The method is well-suited for large-scale problems.