Speaker
Description
The two-level orthogonal Arnoldi algorithm, abbreviated as TOAR, proposed by Lu, Su and Bai, is a Krylov method for the solution of large sparse quadratic eigenvalue problems (QEPs). Traditionally, such eigenproblems are first linearised with an appropriate companion form, then fed into the standard Arnoldi algorithm. This approach has the advantage of being simple, but suffers from large memory requirements when storing the basis. In contrast, TOAR works compactly, requiring (to highest order) half the memory of Arnoldi. Similar algorithms existed prior to TOAR, for example the SOAR (second-order Arnoldi) algorithm, which had the same major benefit as TOAR but was prone to numerical instability.
Interest has been shown in block Arnoldi methods for some time because of their ability to exploit BLAS 3 subroutines. Standard block Arnoldi is widely used and some block SOAR algorithms have shown up in the literature, but not block TOAR. In my work I have extended TOAR to block form and have shown that it has similar numerical stability to non-block TOAR.
Stewart's Krylov-Schur algorithm for restarting Arnoldi is an improvement on the implicitly restarted Arnoldi method (IRAM) in terms of numerical stability. Campos and Roman provide an extension of Krylov-Schur to the TOAR algorithm, which I have further extended to block TOAR.
In this talk, I will briefly explain the TOAR algorithm, then cover my work extending TOAR to block form and talk about a numerical instability issue with the restart method.