  Often in applications ranging from medical imaging and sensor networks to
error correction and data science (and beyond), one needs to solve large-scale
linear systems in which a fraction of the measurements have been corrupted. We
consider solving such large-scale systems of linear equations
$\mathbf{A}\mathbf{x}=\mathbf{b}$ that are inconsistent due to corruptions in
the measurement vector $\mathbf{b}$. We develop several variants of iterative
methods that converge to the solution of the uncorrupted system of equations,
even in the presence of large corruptions. These methods make use of a quantile
of the absolute values of the residual vector in determining the iterate
update. We present both theoretical and empirical results that demonstrate the
promise of these iterative approaches.
