  We study efficient importance sampling techniques for particle filtering (PF)
when either (a) the observation likelihood (OL) is frequently multimodal or
heavy-tailed, or (b) the state space dimension is large or both. When the OL is
multimodal, but the state transition pdf (STP) is narrow enough, the optimal
importance density is usually unimodal. Under this assumption, many techniques
have been proposed. But when the STP is broad, this assumption does not hold.
We study how existing techniques can be generalized to situations where the
optimal importance density is multimodal, but is unimodal conditioned on a part
of the state vector. Sufficient conditions to test for the unimodality of this
conditional posterior are derived. The number of particles, N, to accurately
track using a PF increases with state space dimension, thus making any regular
PF impractical for large dimensional tracking problems. We propose a solution
that partially addresses this problem. An important class of large dimensional
problems with multimodal OL is tracking spatially varying physical quantities
such as temperature or pressure in a large area using a network of sensors
which may be nonlinear and/or may have non-negligible failure probabilities.
