Implicit particle filters for data assimilation generate high-probability samples by representing each particle location as a separate function of a common reference variable. This representation requires that a certain underdetermined equation be solved for each particle and at each time an observation becomes available. We present a new implementation of implicit filters in which we find the solution of the equation via a random map. As examples, we assimilate data for a stochastically driven Lorenz system with sparse observations and for a stochastic Kuramoto-Sivashinski equation with observations that are sparse in both space and time.

The implicit particle filter is a sequential Monte Carlo method for data assimilation that guides the particles to the high-probability regions via a sequence of steps that includes minimizations. A new and more general derivation of this approach is presented and the method is extended to particle smoothing as well as to data assimilation for perfect models. Minimizations required by implicit particle methods are shown to be similar to those that one encounters in variational data assimilation, and the connection of implicit particle methods with variational data assimilation is explored. In particular, it is argued that existing variational codes can be converted into implicit particle methods at a low additional cost, often yielding better estimates that are also equipped with quantitative measures of the uncertainty. A detailed example is presented.

Using the idealized integrable Maxwell-Bloch model, we describe random optical-pulse polarization switching along an active optical medium in the Λ-configuration with disordered occupation numbers of its lower energy sub-level pair. The description combines complete integrability and stochastic dynamics. For the single-soliton pulse, we derive the statistics of the electric-field polarization ellipse at a given point along the medium in closed form. If the average initial population difference of the two lower sub-levels vanishes, we show that the pulse polarization will switch intermittently between the two circular polarizations as it travels along the medium. If this difference does not vanish, the pulse will eventually forever remain in the circular polarization determined by which sub-level is more occupied on average. We also derive the exact expressions for the statistics of the polarization-switching dynamics, such as the probability distribution of the distance between two consecutive switches and the percentage of the distance along the medium the pulse spends in the elliptical polarization of a given orientation in the case of vanishing average initial population difference. We find that the latter distribution is given in terms of the well-known arcsine law.

Random optical-pulse polarization switching along an active optical medium in the Λ configuration with spatially disordered occupation numbers of its lower energy sublevel pair is described using the idealized integrable Maxwell-Bloch model. Analytical results describing the light polarization-switching statistics for the single self-induced transparency pulse are compared with statistics obtained from direct Monte Carlo numerical simulations.