A fast Primal-Dual-Active-Jump method for minimization in $\mathrm{BV}((0,T);{\mathbb{R}}^d)$

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 $\mathcal{O}(1\mathrm{/}k)$ 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 $L^1(I;{\mathbb{R}}^d)$.