Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

“…which is consistent with the Kronecker product A = A (1) ⊗ A (2) in the case d = 2 and R 1 = 1, and allows a natural multiplication with (24), returning the result in the same form.…”

“…Hackbusch and Grasedyck suggest the use of a multigrid scheme [31] that maintains the low-rank nature of the solution throughout the iteration. More reliable is the Krylov-plus-inverted-Krylov (KPIK) method, first developed in [24,80] for Lyapunov equations and later adapted for the case of Sylvester equations [81]. This method approximates the solution of the Sylvester equation in an extended Krylov subspace that involves two sequences with each of the system matrices C α and 1 √ γ I n − L in A. Additionally, the sequence requires the inverse or approximate inverse of both C α and 1 √ γ I n − L from (8) and (9).…”

“…Given a TT format (24) for the approximate solution y, we may collect its first, resp. last, k TT factors into the interface matrices,…”

A new investigation of the extended Krylov subspace method for matrix function evaluations

“…For particularly challenging problems, however, an unacceptably large approximation space may be required to obtain a sastisfactory approximation. This difficulty has lead to the study of enhancement techniques that aim either at enriching the approximation space or at making the overall procedure less expensive [4], [16], [18], [22], [33], [39], [40], [41], [46], [54].…”

“…In this paper we investigate the Extended Krylov subspace method, a technique that was proposed in [16]. By using recent algorithmic improvements developed in [50], we show that the method is competitive with respect to other enhancements techniques.…”