Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such as the space of Radon measures, are much less developed than in Hilbert spaces. Most numerical algorithms for point source localisation are based on the Frank-Wolfe conditional gradient method, for which ad hoc convergence theory is developed. We develop extensions of proximal-type methods to spaces of measures. This includes forward-backward splitting, its inertial version, and primal-dual proximal splitting. Their convergence proofs follow standard patterns. We demonstrate their numerical efficacy.

翻译：点源定位通常被建模为测度空间上的Lasso型问题。然而，在非希尔伯特空间（如Radon测度空间）中的优化方法远不如在希尔伯特空间中成熟。大多数点源定位的数值算法基于Frank-Wolfe条件梯度方法，为此发展了专门的收敛理论。本文将近端型方法扩展到测度空间，包括前向-后向分裂法、其惯性版本以及原始-对偶近端分裂法。这些方法的收敛证明遵循标准范式。我们展示了其数值有效性。