Automatic Preconditioning by Limited Memory Quasi-Newton Updating
José Luis Morales, Jorge Nocedal · SIAM Journal on Optimization · 2000
This paper proposes a preconditioner for the conjugate gradient method (CG) that is designed for solving systems of equations A x =b i with different right-hand-side vectors or for solving a sequence of slowly varying systems A k x = b k . The preconditioner has the form of a limited memory quasi-Newton matrix and is generated using information from the CG iteration. The automatic preconditioner does not require explicit knowledge of the coefficient 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 offer when these matrix-vector products are expensive to compute and when low accuracy in the solution is required. The effectiveness of the preconditioner is tested within a Hessian-free Newton method for optimization and by solving certain linear systems arising in finite element models.