This paper considers the problem of iterative Bayesian smoothing in nonlinear state-space models with additive noise using Gaussian approximations. Iterative methods are known to improve smoothed estimates but are not guaranteed to converge, motivating the development of methods with better convergence properties. The aim of this article is to extend Levenberg-Marquardt (LM) and line-search versions of the classical iterated extended Kalman smoother (IEKS) to the iterated posterior linearisation smoother (IPLS). The IEKS has previously been shown to be equivalent to the Gauss-Newton (GN) method. We derive a similar GN interpretation for the IPLS and use this to develop extensions to the IPLS, with improved convergence properties. We show that an LM extension for the IPLS can be achieved with a simple modification of the smoothing iterations, enabling algorithms with efficient implementations. We also derive the Armijo--Wolfe step length conditions for the IPLS enabling an efficient inexact line-search method. Our numerical experiments show the benefits of these extensions in highly nonlinear scenarios.

翻译：本文研究了在具有加性噪声的非线性状态空间模型中使用高斯近似的迭代贝叶斯平滑问题。已知迭代方法能够改进平滑估计，但不能保证收敛，这促使我们开发具有更好收敛特性的方法。本文旨在将经典迭代扩展卡尔曼平滑器（IEKS）的Levenberg-Marquardt（LM）和线搜索版本扩展到迭代后验线性化平滑器（IPLS）。先前研究已证明IEKS等价于高斯-牛顿（GN）方法。我们为IPLS推导了类似的GN解释，并利用此解释开发了具有改进收敛特性的IPLS扩展方法。我们证明通过对平滑迭代进行简单修改即可实现IPLS的LM扩展，从而获得具有高效实现的算法。我们还推导了IPLS的Armijo-Wolfe步长条件，实现了高效的非精确线搜索方法。数值实验表明，这些扩展方法在高度非线性场景中具有显著优势。