  In order to develop fast inversion algorithms we have used overlap solvers in
two dimensions. Lattice QED theory with U(1) group symmetry in two dimensional
space-times dimensions has always been a testing ground for algorithms. By the
other side, motivated by our previews work that the two-grid algorithm converge
faster than the standard iterative methods for overlap inversion but not for
all quark masses, we thought to test this idea in less dimensions such as U(1)
gauge theory. Our main objective of this paper it is to implement and develop
the idea of a two level algorithm in a new algorithm coded in QCDLAB. This
implementation is presented in the preconditioned GMRESR algorithm, as our new
contribution in QCDLAB package. The preconditioned part of our algorithm,
different from the one of [18], is the approximation of the overlap operator
with the truncated overlap operator with finite N3 dimension. We have tested it
for 100 statistically independent configurations on 32 x 32 lattice background
U(1) field at coupling constant \b{eta}=1 and for different bare quark masses
mq = [0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1]. We have compared the
convergence history of the preconditioned GMRESR residual norm with another
overlap inverter of QCDLAB as an optimal one, such as SHUMR. We have shown that
our algorithm converges faster than SHUMR for different quark masses. Also, we
have demonstrated that it saves more time for light quarks compared to SHUMR
algorithm. Our algorithm is approximately independent from the quark mass. This
is a key result in simulations with chiral fermions in lattice theories. By the
other side, if we compare the results of [18] for quark mass 0.1 in SU(3),
results that our chosen preconditioned saves a factor of 2 but in U(1). Our
next step is to test this algorithm in SU(3) and to adopt it in parallel.
