  We address the problem of autonomous tracking and state estimation for marine
vessels, autonomous vehicles, and other dynamic signals under a (structured)
sparsity assumption. The aim is to improve the tracking and estimation accuracy
with respect to classical Bayesian filters and smoothers. We formulate the
estimation problem as a dynamic generalised group Lasso problem and develop a
class of smoothing-and-splitting methods to solve it. The Levenberg--Marquardt
iterated extended Kalman smoother-based multi-block alternating direction
method of multipliers (LM-IEKS-mADMM) algorithms are based on the alternating
direction method of multipliers (ADMM) framework. This leads to minimisation
subproblems with an inherent structure to which three new augmented recursive
smoothers are applied. Our methods can deal with large-scale problems without
pre-processing for dimensionality reduction. Moreover, the methods allow one to
solve nonsmooth nonconvex optimisation problems. We then prove that under mild
conditions, the proposed methods converge to a stationary point of the
optimisation problem. By simulated and real-data experiments including
multi-sensor range measurement problems, marine vessel tracking, autonomous
vehicle tracking, and audio signal restoration, we show the practical
effectiveness of the proposed methods.
