Theory of Krylov subspace methods based on the Arnoldi process with inexact inner products

Several Krylov subspace methods are based on the Arnoldi process, such as the full orthogonalization method (FOM), GMRES, and in general all the Arnoldi-type methods. In fact, the Arnoldi process is an algorithm for building an orthogonal basis of the Krylov subspace. Once the inner products are performed inexactly, which cannot be avoided due to round-off errors, the orthogonality of Arnoldi vectors is lost. In this paper, we presented a new analysis framework to show how the inexact inner products influence the Krylov subspace methods that are based on the Arnoldi process. A new metric was developed to quantify the inexactness of the Arnoldi process with inexact inner products. In addition, the proposed metric can be used to approximately estimate the loss of orthogonality in the practical use of the Arnoldi process. The discrepancy in residual gaps between Krylov subspace methods employing inexact inner products and their corresponding exact counterparts was discussed. Numerical experiments on several examples were reported to illustrate our theoretical findings and final observations were presented.