An Introduction to Solving the Least-Squares Problem in Variational Data Assimilation

TL;DR

This paper reviews numerical linear algebra techniques for efficiently solving large-scale least-squares problems in variational data assimilation, crucial for Earth system modeling and forecasting.

Contribution

It provides a focused linear algebra perspective on variational data assimilation, highlighting solution methods and preconditioning strategies for large-scale applications.

Findings

Emphasizes the importance of high-quality preconditioners in Krylov solvers.

Provides a clear introduction to the linear algebraic structure of data assimilation problems.

Discusses contemporary numerical methods for large-scale geophysical applications.

Abstract

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.