Professor Anders Forsgren
Optimization and Systems Theory
Department of Mathematics
Royal Institute of Technology (KTH)
SE-100 44 Stockholm, Sweden
The conjugate-gradient method is a well-known iterative method for minimizing a quadratic function where the Hessian is positive definite.
In this talk, we discuss some aspect of the method that are less well known. In particular, we study the behavior of the conjugate-gradient method on ill-conditioned problems, for which the Hessian has one set of eigenvalues that are large and the remaining are small. Our motivation is twofold: first, interior methods, where infinitely ill-conditioned matrices arise, and second, radiation therapy optimization, where ill-conditioned systems arising from discretized Fredholm equations of the first kind arise. We characterize the behavior of the residuals associated with the large eigenvalues throughout the iterations, and also characterize the behavior of the residuals associated with the small eigenvalues for the early iterations. Our results show that the residuals associated with the large eigenvalues are made small first, without changing very much the residuals associated with the small eigenvalues. Subsequently, the residuals associated with the small eigenvalues are reduced.
The motivation for this research comes from radiation therapy optimization.