Interpreting IDR as a Petrov–Galerkin Method
Valeria Simoncini, Daniel B. Szyld · SIAM Journal on Scientific Computing · 2010
The induced dimension reduction (IDR) method of Sonneveld and van Gijzen [SIAM J. Sci. Comput., 31 (2008), pp. 1035–1062] is shown to be a Petrov–Galerkin (projection) method with a particular choice of left Krylov subspaces; these left subspaces are rational Krylov spaces. Consequently, other methods, such as BiCGStab and ML(s)BiCGStab, which are mathematically equivalent to some versions of IDR, can also be interpreted as Petrov–Galerkin methods. The connection with rational Krylov spaces inspired a new version of IDR, called Ritz-IDR, where the poles of the rational function are chosen as certain Ritz values. Experiments are presented illustrating the effectiveness of this new version.