  A possible solution of the notorious sign problem preventing direct Monte
Carlo calculations for systems with non-zero chemical potential is to deform
the integration region in the complex plane to a Lefschetz thimble. We
investigate this approach for a simple fermionic model. We introduce an easy to
implement Monte Carlo algorithm to sample the dominant thimble. Our algorithm
relies only on the integration of the gradient flow in the numerically stable
direction, which gives it a distinct advantage over the other proposed
algorithms. We demonstrate the stability and efficiency of the algorithm by
applying it to an exactly solvable fermionic model and compare our results with
the analytical ones. We report a very good agreement for a certain region in
the parameter space where the dominant contribution comes from a single
thimble, including a region where standard methods suffer from a severe sign
problem. However, we find that there are also regions in the parameter space
where the contribution from multiple thimbles is important, even in the
continuum limit.
