Kirk M. Soodhalter,
"Block Krylov subspace methods for shifted systems with different right-hand sides"
, in arXiv/Review by Journal, 2014
Block Krylov subspace methods for shifted systems with different right-hand sides
Sprache des Titels:
Many Krylov subspace methods for shifted linear systems take advantage
of the invariance of the Krylov subspace under a shift of the matrix.
However, exploiting this fact introduces restrictions; e.g., initial
residuals must be collinear and this collinearity must be maintained
at restart. Thus we cannot simultaneously solve (in general) shifted
systems with unrelated right-hand sides using this strategy, and
all shifted residuals cannot be simultaneously minimized over a Krylov
subspace such that collinearity is maintained.
We present two methods which circumvent this problem. Block Krylov
subspaces are shift invariant just as their single-vector counterparts.
Thus by collecting all initial residuals into one block vector, we
can generate a block Krylov subspace. Due to shift invariance, we
can define block FOM- and GMRES-type projection methods to simultaneously
solve all shifted systems. These are not block versions of the shifted
FOM method of Simoncini [BIT '03] or the shifted GMRES method of
Frommer and Gl\assner [SISC '98]. These methods are compatible with
unrelated right-hand sides, and residual collinearity is no longer
a requirement at restart. Furthermore, we realize the benefits of
block sparse matrix operations which arise in the context of high-performance
In this paper, we show that the block Krylov subspace built from an
appropriate block starting vector is compatible with solving individual
shifted systems and use this to derive our block FOM and GMRES methods
for shifted systems. Numerical experiments demonstrate the effectiveness
of the methods.