Abstract

We study the use of Krylov subspace recycling for the solution of
a sequence of slowly-changing families of linear systems, where each
family consists of shifted linear systems that differ in the coefficient
matrix only by multiples of the identity. Our aim is to explore the
simultaneous solution of each family of shifted systems within the
framework of subspace recycling, using one augmented subspace to
extract candidate solutions for all the shifted systems. The ideal
method would use the same augmented subspace for all systems and
have fixed storage requirements, independent of the number of shifted
systems per family. We show that a method satisfying both requirements
cannot exist in this framework.
As an alternative, we introduce two schemes. One constructs a separate
deflation space for each shifted system but solves each family of
shifted systems simultaneously. The other builds only one recycled
subspace and constructs approximate corrections to the solutions
of the shifted systems at each cycle of the iterative linear solver
while only minimizing the base system residual. At convergence of
the base system solution, we apply the method recursively to the
remaining unconverged systems. We present numerical examples involving
systems arising in lattice quantum chromodynamics.

Links

• http://arxiv.org/abs/1301.2650
• doi:10.1016/j.apnum.2014.02.006

BibTeX (Download)

@article{SSX.2014,
title = {Krylov subspace recycling for sequences of shifted linear systems},
author = {Kirk M. Soodhalter and Daniel B. Szyld and Fei Xue},
url = {http://arxiv.org/abs/1301.2650},
doi = {10.1016/j.apnum.2014.02.006},
year  = {2014},
date = {2014-01-01},
urldate = {2014-01-01},
journal = {Applied Numerical Mathematics},
volume = {81C},
pages = {105\textendash118},
abstract = {We study the use of Krylov subspace recycling for the solution of 
	a sequence of slowly-changing families of linear systems, where each 
	family consists of shifted linear systems that differ in the coefficient 
	matrix only by multiples of the identity. Our aim is to explore the 
	simultaneous solution of each family of shifted systems within the 
	framework of subspace recycling, using one augmented subspace to 
	extract candidate solutions for all the shifted systems. The ideal 
	method would use the same augmented subspace for all systems and 
	have fixed storage requirements, independent of the number of shifted 
	systems per family. We show that a method satisfying both requirements 
	cannot exist in this framework. 
	As an alternative, we introduce two schemes. One constructs a separate 
	deflation space for each shifted system but solves each family of 
	shifted systems simultaneously. The other builds only one recycled 
	subspace and constructs approximate corrections to the solutions 
	of the shifted systems at each cycle of the iterative linear solver 
	while only minimizing the base system residual. At convergence of 
	the base system solution, we apply the method recursively to the 
	remaining unconverged systems. We present numerical examples involving 
	systems arising in lattice quantum chromodynamics.},
keywords = {published},
pubstate = {published},
tppubtype = {article}
}