Truncation Strategies for Optimal Krylov Subspace Methods∗

Optimal Krylov subspace methods like GMRES and GCR have to compute an orthogonal basis for the entire Krylov subspace to compute the minimal residual approximation to the solution. Therefore, when the number of iterations becomes large, the amount of work and the storage requirements become excessive. In practice one has to limit the resources. The most obvious ways to do this are to restart GMRES after some number of iterations and to keep only some number of the most recent vectors in GCR. This may lead to very poor convergence and even stagnation. Therefore, we will describe a method that reveals which subspaces of the Krylov space were important for convergence thus far and exactly how important they are. This information is then used to select which subspace to keep for orthogonalizing future search directions. Numerical results indicate this to be a very effective strategy.