New bounds on the condition number of the Hessian of the preconditioned variational data assimilation problem

Tabeart, J. M., Dance, S. L. ORCID: https://orcid.org/0000-0003-1690-3338, Lawless, A. S. ORCID: https://orcid.org/0000-0002-3016-6568, Nichols, N. K. ORCID: https://orcid.org/0000-0003-1133-5220 and Waller, J. A. (2022) New bounds on the condition number of the Hessian of the preconditioned variational data assimilation problem. Numerical Linear Algebra with Applications, 29 (1). e2405. ISSN 1099-1506 doi: 10.1002/nla.2405

Abstract/Summary

Data assimilation algorithms combine prior and observational information, weighted by their respective uncertainties, to obtain the most likely posterior of a dynamical system. In variational data assimilation the posterior is computed by solving a nonlinear least squares problem. Many numerical weather prediction (NWP) centers use full observation error covariance (OEC) weighting matrices, which can slow convergence of the data assimilation procedure. Previous work revealed the importance of the minimum eigenvalue of the OEC matrix for conditioning and convergence of the unpreconditioned data assimilation problem. In this article we examine the use of correlated OEC matrices in the preconditioned data assimilation problem for the first time. We consider the case where there are more state variables than observations, which is typical for applications with sparse measurements, for example, NWP and remote sensing. We find that similarly to the unpreconditioned problem, the minimum eigenvalue of the OEC matrix appears in new bounds on the condition number of the Hessian of the preconditioned objective function. Numerical experiments reveal that the condition number of the Hessian is minimized when the background and observation lengthscales are equal. This contrasts with the unpreconditioned case, where decreasing the observation error lengthscale always improves conditioning. Conjugate gradient experiments show that in this framework the condition number of the Hessian is a good proxy for convergence. Eigenvalue clustering explains cases where convergence is faster than expected.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/99302
Item Type	Article
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > National Centre for Earth Observation (NCEO) Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics Science > School of Mathematical, Physical and Computational Sciences > Department of Meteorology
Publisher	Wiley
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