Automatic Preconditioning by Limited Memory

The paper proposes a preconditioner for the conjugate gradient method CG that is designed for solving systems of equations Ax bi with di erent right hand side vec tors or for solving a sequence of slowly varying systems Akx bk The preconditioner has the form of a limited memory quasi Newton matrix and is generated using infor mation from the CG iteration The automatic preconditioner does not require explicit knowledge of the coe cient matrix A and is therefore suitable for problems where only products of A times a vector can be computed Numerical experiments indicate that the preconditioner has most to o er when these matrix vector products are expensive to compute and when low accuracy in the solution is required The e ectiveness of the preconditioner is tested within a Hessian free Newton method for optimization and by solving certain linear systems arising in nite element models