The numerical properties of algorithms for finding the intersection of sets
depend to some extent on the regularity of the sets, but even more importantly
on the regularity of the intersection. The alternating projection algorithm of
von Neumann has been shown to converge locally at a linear rate dependent on
the regularity modulus of the intersection. In many applications, however, the
sets in question come from inexact measurements that are matched to idealized
models. It is unlikely that any such problems in applications will enjoy
metrically regular intersection, let alone set intersection. We explore a
regularization strategy that generates an intersection with the desired
regularity properties. The regularization, however, can lead to a significant
increase in computational complexity. In a further refinement, we investigate
and prove linear convergence of an approximate alternating projection
algorithm. The analysis provides a regularization strategy that fits naturally
with many ill-posed inverse problems, and a mathematically sound stopping
criterion for extrapolated, approximate algorithms. The theory is demonstrated
on the phase retrieval problem with experimental data. The conventional early
termination applied in practice to unregularized, consistent problems in
diffraction imaging can be justified fully in the framework of this analysis
providing, for the first time, proof of convergence of alternating approximate
projections for finite dimensional, consistent phase retrieval problems.Comment: 23 pages, 5 figure