An Introduction to Solving the Least-Squares Problem in Variational Data Assimilation
For researchers in numerical linear algebra and geophysical data assimilation, this is an incremental tutorial that organizes existing knowledge without advancing the state of the art.
This paper provides a numerical linear algebra perspective on variational data assimilation for large-scale geophysical applications, focusing on solving the least-squares subproblems and discussing preconditioners. It offers a concise introduction and extensive bibliography but does not present new results or concrete numbers.
Variational data assimilation is a technique for combining measured data with dynamical models. It is a key component of Earth system state estimation and is commonly used in weather and ocean forecasting. The approach involves a large-scale generalized nonlinear least-squares problem. Solving the resulting sequence of sparse linear subproblems requires the use of sophisticated numerical linear algebra methods. In practical applications, the computational demands severely limit the number of iterations of a Krylov subspace solver that can be performed and so high-quality preconditioners are vital. In this paper, we present a numerical linear algebra perspective on variational data assimilation and discuss contemporary solution methods for the challenges posed by large-scale geophysical applications. The principal contribution is a focused treatment of the underlying linear algebraic subproblems, accompanied by a concise and clear introduction to the essential concepts of variational data assimilation and an extensive bibliography.