SUMMARYThis paper presents a general preconditioning method based on a multilevel partial elimination approach. The basic step in constructing the preconditioner is to separate the initial points into two parts. The ÿrst part consists of 'block' independent sets, or 'aggregates'. Unknowns of two di erent aggregates have no coupling between them, but those in the same aggregate may be coupled. The nodes not in the ÿrst part constitute what might be called the 'coarse' set. It is natural to call the nodes in the ÿrst part 'ÿne' nodes. The idea of the methods is to form the Schur complement related to the coarse set. This leads to a natural block LU factorization which can be used as a preconditioner for the system. This system is then solved recursively using as preconditioner the factorization that could be obtained from the next level. Iterations between levels are allowed. One interesting aspect of the method is that it provides a common framework for many other techniques. Numerical experiments are reported which indicate that the method can be fairly robust.

No abstract

Batch maximum likelihood (ML) and maximum a posteriori (MAP) estimation with process noise is now more than thirty-five years old, and its use in multiple target tracking has long been considered to be too computationally intensive for real-time applications. While this may still be true for general usage, it is ideally suited for special needs such as bias estimation, track initiation and spawning, long-term prediction of track states, and state estimation during periods of rapidly changing target dynamics. In this paper, we examine the batch estimator formulation for several cases: nonlinear and linear models, with and without a prior state estimate (MAP vs. ML), and with and without process noise. For the nonlinear case, we show that a single pass of an extended Kalman smoother-filter over the data corresponds to a Gauss-Newton step of the corresponding nonlinear least-squares problem. Even the iterated extended Kalman filter can be viewed within this framework. For the linear case, we develop a compact least squares solution that can incorporate process noise and the prior state when available. With these new views on the batch approach, one may reconsider its usage in tracking because it provides a robust framework for the solution of the aforementioned problems. Finally, we provide some examples comparing linear batch initiation with and without process noise to show the value of the new approach.

No abstract

No abstract