  This paper discusses the error and cost aspects of ill-posed integral
equations when given discrete noisy point evaluations on a fine grid. Standard
solution methods usually employ discretization schemes that are directly
induced by the measurement points. Thus, they may scale unfavorably with the
number of evaluation points, which can result in computational inefficiency. To
address this issue, we propose an algorithm that achieves the same level of
accuracy while significantly reducing computational costs. Our approach
involves an initial averaging procedure to sparsify the underlying grid. To
keep the exposition simple, we focus only on one-dimensional ill-posed integral
equations that have sufficient smoothness. However, the approach can be
generalized to more complicated two- and three-dimensional problems with
appropriate modifications.
