  We analyze a solution method for minimization problems over a space of
$\mathbb{R}^d$-valued functions of bounded variation on an interval $I$. The
presented method relies on piecewise constant iterates. In each iteration the
algorithm alternates between proposing a new point at which the iterate is
allowed to be discontinuous and optimizing the magnitude of its jumps as well
as the offset. A sublinear $\mathcal{O}(1/k)$ convergence rate for the
objective function values is obtained in general settings. Under additional
structural assumptions on the dual variable this can be improved to a locally
linear rate of convergence $\mathcal{O}(\zeta^k)$ for some $\zeta <1$.
Moreover, in this case, the same rate can be expected for the iterates in
$L^1(I;\mathbb{R}^d)$.
