Recycling Techniques for Induced Dimension Reduction Methods Used for Solving Large Sparse Non-Symmetric Systems of Linear Equations

Abstract

Induced Dimension Reduction (IDR) methods are among the most stable and effi- cient iterative methods for solving large, sparse, non-symmetric systems of linear equa- tions known today. However, the exact reasons for their effectiveness are only partially understood. In recent work on recycling techniques for IDR it is shown that greater efficiency than theoretically possible for a Krylov subspace-method can be achieved by altering the auxiliary input of an IDR algorithm, perhaps turning the methods into a class of their own. In this thesis a self-contained derivation of IDR(s) and IDR(s)Stab(l) is presented, prior to an experiment driven investigation of the possibilities of recycling techniques. Issues with the numerical stability of these techniques will be addressed and theory is presented providing more insight in the matter. Moreover, an attempt is made to apply theory on spectral aspects of IDR-methods to justify the effectiveness of recycling, which also might be a starting point in gaining a further understanding of the convergence behavior of IDR methods in general.