Spectral estimates for saddle point matrices arising in weak constraint four‐dimensional variational data assimilation

Abstract

Summary We consider the large sparse symmetric linear systems of equations that arise in the solution of weak constraint four‐dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can be written as saddle point systems with a 3 × 3 block structure but block eliminations can be performed to reduce them to saddle point systems with a 2 × 2 block structure, or further to symmetric positive definite systems. In this article, we analyse how sensitive the spectra of these matrices are to the number of observations of the underlying dynamical system. We also obtain bounds on the eigenvalues of the matrices. Numerical experiments are used to confirm the theoretical analysis and bounds.

Saddle point preconditioners for weak-constraint 4D-Var

“…Figure 3 also considers two observation networks. We see that using a larger number of observations requires a larger number of iterations to reach convergence and longer wallclock time, which coincides with the findings of [8] for the unpreconditioned case. However, the qualitative behaviour across the different choices of L and R is the same for both observation networks.…”

“…The spatial structure for B and Q introduced above is similar to numerical frameworks that have been considered previously for weak-constraint data assimilation experiments [8,10]. However, we note that for realistic data assimilation systems the true structure of Q is not well known.…”

“…The second problem of interest concerns the Lorenz 96 model [24], a non-linear problem that is often considered as a test problem for data assimilation applications (see for example [8,18] for use within the saddle point formulation for data assimilation). The Lorenz 96 model consists of s coupled ordinary differential equations which are discrete in space and continuous in time.…”

“…Standard Krylov subspace solvers can then be applied for the inner loops. An alternative approach involves solving the linearised inner loop using a saddle point formulation [15,13,18,8]. This has increased potential for parallelisation, which can allow full exploitation of modern computer architectures.…”

Numerical linear algebra in data assimilation

“…In the work [80,97] the Kronecker structure of the saddle point problem was used in order to compute low-rank solutions to GMRES. The convergence of the saddle point formulation of weak constrained 4D-Var was reviewed in [90], and spectral estimates for the saddle point system were obtained in [58].…”