IDR Explained

The Induced Dimension Reduction (IDR) method is a Krylov spa ce method for solving linear systems that was developed by Peter Sonneveld around 1979. It was not iced by only a few people, and mainly as the forerunner of Bi-CGSTAB, which was introduced a decade late r. In 2007, Sonneveld and van Gijzen reconsidered IDR and generalized it to IDR (s), claiming that IDR(1) ≈ IDR is equally fast but preferable to the closely related Bi-CGSTAB, and that IDR(s) with s > 1 may be much faster than Bi-CGSTAB. It also turned out that whe n s > 1, IDR(s) is related to ML(s)BiCGSTAB of Yeung and Chan, and that there is quite some flexib ility in the IDR approach. This approach differs completely from tradition al approaches to Krylov space methods, and therefore it requires an extra effort to get familiar with it and to unde rstand the connections as well as the differences to better-known Krylov space methods. This expository paper a ims to provide some help in this and to make the method understandable even to non-experts. After presenti ng the history of IDR and related methods, we summarize some of the basic facts on Krylov space methods. Then we prese nt th original IDR(s) in detail and put it into perspective with other methods. Specifically, we analyze th e differences between the IDR method published in 1980, IDR(1), and Bi-CGSTAB. At the end of the paper, we discuss a recently proposed ingenious variant of IDR(s) whose residuals fulfill extra orthogonality conditions. Th ere we dwell on details that have been left out in the publications of van Gijzen and Sonneveld.