Truncation Strategies for Optimal

yearly review of activities and projects CrosSCutS (triannually): newsletter featuring announcements relevant to our users as well as research highlights in the eld of high-performance computing Speedup Journal (biannually): proceedings of the SPEEDUP Workshops on Vector and Parallel Computing, published on behalf of the SPEEDUP Society User's Guide: manual to hardware and software at CSCS/SCSC To receive one or more of these publications, please send your full name and complete address to: Library CSCS/SCSC via Cantonale CH-6928 Manno Switzerland Fax: +41 (91) 610 8282 Abstract. Optimal Krylov subspace methods like GMRES and GCR have to compute an orthogonal basis for the entire Krylov subspace to compute the minimum 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 sub-spaces of the Krylov space were important for convergence so far, and exactly how important. This information is then used to select which subspace to keep for or-thogonalizing future search directions. Numerical results indicate this to be a very eeective strategy.