  We present a class of reduced basis (RB) methods for the iterative solution
of parametrized symmetric positive-definite (SPD) linear systems. The essential
ingredients are a Galerkin projection of the underlying parametrized system
onto a reduced basis space to obtain a reduced system; an adaptive greedy
algorithm to efficiently determine sampling parameters and associated basis
vectors; an offline-online computational procedure and a multi-fidelity
approach to decouple the construction and application phases of the reduced
basis method; and solution procedures to employ the reduced basis approximation
as a {\em stand-alone iterative solver} or as a {\em preconditioner} in the
conjugate gradient method. We present numerical examples to demonstrate the
performance of the proposed methods in comparison with multigrid methods.
Numerical results show that, when applied to solve linear systems resulting
from discretizing the Poisson's equations, the speed of convergence of our
methods matches or surpasses that of the multigrid-preconditioned conjugate
gradient method, while their computational cost per iteration is significantly
smaller providing a feasible alternative when the multigrid approach is out of
reach due to timing or memory constraints for large systems. Moreover,
numerical results verify that this new class of reduced basis methods, when
applied as a stand-alone solver or as a preconditioner, is capable of achieving
the accuracy at the level of the {\em truth approximation} which is far beyond
the RB level.
