Conditioning and preconditioning of the optimal state estimation problem

Haben, S. A., Lawless, A. S. ORCID: https://orcid.org/0000-0002-3016-6568 and Nichols, N. K. ORCID: https://orcid.org/0000-0003-1133-5220, (2014) Conditioning and preconditioning of the optimal state estimation problem. Mathematics Preprint Series. 2014-08. Technical Report. Dept of Mathematics & Statistics, University of Reading pp21.

Abstract/Summary

Optimal state estimation is a method that requires minimising a weighted, nonlinear, least-squares objective function in order to obtain the best estimate of the current state of a dynamical system. Often the minimisation is non-trivial due to the large scale of the problem, the relative sparsity of the observations and the nonlinearity of the objective function. To simplify the problem the solution is often found via a sequence of linearised objective functions. The condition number of the Hessian of the linearised problem is an important indicator of the convergence rate of the minimisation and the expected accuracy of the solution. In the standard formulation the convergence is slow, indicating an ill-conditioned objective function. A transformation to different variables is often used to ameliorate the conditioning of the Hessian by changing, or preconditioning, the Hessian. There is only sparse information in the literature for describing the causes of ill-conditioning of the optimal state estimation problem and explaining the effect of preconditioning on the condition number. This paper derives descriptive theoretical bounds on the condition number of both the unpreconditioned and preconditioned system in order to better understand the conditioning of the problem. We use these bounds to explain why the standard objective function is often ill-conditioned and why a standard preconditioning reduces the condition number. We also use the bounds on the preconditioned Hessian to understand the main factors that affect the conditioning of the system. We illustrate the results with simple numerical experiments.

Item Type	Report (Technical Report)
URI	https://reading-clone.eprints-hosting.org/id/eprint/50580
Item Type	Report
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics Science > School of Mathematical, Physical and Computational Sciences > Department of Meteorology
Uncontrolled Keywords	Optimal state estimation, variational data assimilation, nonlinear least squares, condition number, preconditioning, correlation matrices, circulant matrices
Publisher	Dept of Mathematics & Statistics, University of Reading
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