  We analyze an exchange algorithm for the numerical solution total-variation
regularized inverse problems over the space M($\Omega$) of Radon measures on a
subset $\Omega$ of R d. Our main result states that under some regularity
conditions, the method eventually converges linearly. Additionally, we prove
that continuously optimizing the amplitudes of positions of the target measure
will succeed at a linear rate with a good initialization. Finally, we propose
to combine the two approaches into an alternating method and discuss the
comparative advantages of this approach.
