  An open problem that arises when using modern iterative linear solvers, such
as the preconditioned conjugate gradient (PCG) method or Generalized Minimum
RESidual method (GMRES) is how to choose the residual tolerance in the linear
solver to be consistent with the tolerance on the solution error. This problem
is especially acute for integrated groundwater models which are implicitly
coupled to another model, such as surface water models, and resolve both
multiple scales of flow and temporal interaction terms, giving rise to linear
systems with variable scaling.
  This article uses the theory of 'forward error bound estimation' to show how
rescaling the linear system affects the correspondence between the residual
error in the preconditioned linear system and the solution error. Using
examples of linear systems from models developed using the USGS GSFLOW package
and the California State Department of Water Resources' Integrated Water Flow
Model (IWFM), we observe that this error bound guides the choice of a practical
measure for controlling the error in rescaled linear systems. It is found that
forward error can be controlled in preconditioned GMRES by rescaling the linear
system and normalizing the stopping tolerance. We implemented a preconditioned
GMRES algorithm and benchmarked it against the Successive-Over-Relaxation (SOR)
method. Improved error control reduces redundant iterations in the GMRES
algorithm and results in overall simulation speedups as large as 7.7x. This
research is expected to broadly impact groundwater modelers through the
demonstration of a practical approach for setting the residual tolerance in
line with the solution error tolerance.
