Krylov Methods and Preconditioning in Computational Economics Problems

Krylov subspace methods have proven to be powerful methods for solving sparse linear systems arising in several engineering problems. More recently, these methods have been successfully applied in computational economics, for instance in the solution of forward-looking macroeconometric models (Gilli and Pauletto and Pauletto and Gilli), dynamic programming problems (Mrkaic) and pricing of financial options (Gilli, Kellezi and Pauletto). Since Krylov methods can suffer from slow convergence, one can modify the original linear system in order to improve convergence properties. This is known as preconditioning. In this paper, we investigate the effects of several preconditioning techniques in the framework of dynamic programming problems and financial option pricing. Very few theoretical results on preconditioning are known and experiments have to be conducted to recognize which classes of problems can be best solved using a given Krylov method and a given preconditioner.