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

TL;DR

This paper introduces a fast primal-dual-active-jump method for minimizing functionals over BV spaces, achieving sublinear convergence generally and linear convergence under certain conditions, with practical implications for functions with bounded variation.

Contribution

The paper proposes a novel primal-dual-active-jump algorithm for BV minimization problems, providing convergence analysis and rates under various assumptions.

Findings

Sublinear convergence rate $ ext{O}(1/k)$ for general cases.

Linear convergence rate $ ext{O}( ho^k)$ under structural assumptions.

Effective for functions of bounded variation with practical convergence guarantees.

Abstract

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 $L^1(I;\mathbb{R}^d)$.