Abstracts: Nancy Nichols
Conditioning and Preconditioning of Problems in Data Assimilation
Data assimilation is a technique for estimating the current and future states of a dynamical system from observations of the system together with a prior estimate, or model prediction. Applications arise in very large environmental problems where the number of state variables is O(10^7 - 10^8) and the number of observations is O(10^4 - 10^6). The errors in the prior estimate and in the observations are assumed to be random with known distributions and the solution to the data assimilation problem is taken to be the maximum a posteriori likelihood estimate. With the aid of Bayes Theorem, the problem reduces to a very large nonlinear least squares problem, which is treated in practice using a Gauss-Newton iterative method. At each iteration a linearized least squares problem is solved, usually using a conjugate gradient method.
The conditioning of the linearized least squares problem depends on the error correlations in the states and in the observations. The correlation structures depend on length scales, which essentially determine the range of influence of a prior state estimate or an observation on the a posteriori estimates of the states. We derive estimates for the conditioning of the problem as a function of the correlation length scales and demonstrate how the conditioning is altered as a function of the available observations. Different correlation structures are compared. A commonly used preconditioning technique, in which the prior states are transformed to uncorrelated variables, is also analysed.
(This is joint work with S.A. Haben and A.S. Lawless.)