Adaptively Blocked Particle Filtering with Spatial Smoothing in   Large-Scale Dynamic Random Fields
  The typical particle filtering approximation error is exponentially dependent on the dimension of the model. Therefore, to control this error, an enormous number of particles are required, which means a heavy computational burden that is often so great it is simply prohibitive. Rebeschini and van Handel (2013) consider particle filtering in a large-scale dynamic random field. Through a suitable localisation operation, they prove that a modified particle filtering algorithm can achieve an approximation error that is mostly independent of the problem dimension. To achieve this feat, they inadvertently introduce a systematic bias that is spatially dependent (in that the bias at one site is dependent on the location of that site). This bias consequently varies throughout field. In this work, a simple extension to the algorithm of Rebeschini and van Handel is introduced which acts to average this bias term over each site in the field through a kind of spatial smoothing. It is shown that for a certain class of random field it is possible to achieve a completely spatially uniform bound on the bias and that in any general random field the spatial inhomogeneity is significantly reduced when compared to the case in which spatial smoothing is not considered. While the focus is on spatial averaging in this work, the proposed algorithm seemingly exhibits other advantageous properties such as improved robustness and accuracy in those cases in which the underlying dynamic field is time varying. 
Francesco Bertoli and Adrian N. Bishop
math.OC