A low-rank approach to the solution of weak constraint variational data assimilation problems

Abstract

Weak constraint four-dimensional variational data assimilation is an important method for incorporating data (typically observations) into a model. The linearised system arising within the minimisation process can be formulated as a saddle point problem. A disadvantage of this formulation is the large storage requirements involved in the linear system. In this paper, we present a low-rank approach which exploits the structure of the saddle point system using techniques and theory from solving large scale matrix equations. Numerical experiments with the linear advection-diffusion equation, and the non-linear Lorenz-95 model demonstrate the effectiveness of a low-rank Krylov subspace solver when compared to a traditional solver.

Guaranteeing the convergence of the saddle formulation for weakly constrained 4D‐Var data assimilation

“…Armed with these concepts and notation, we may now consider the original saddle technique as discussed in Fisher and Gürol () (and also used in (Freitag and Green, )). It is outlined as Algorithm 3.1, where we define …”

“…Armed with these concepts and notation, we may now consider the original saddle technique as discussed in Fisher and Gürol (2017) (and also used in Freitag and Green, 2018). It is outlined as Algorithm 3.1, where we define…”

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

“…An active research topic in this area is the weak constraint four-dimensional variational (4D-Var) data assimilation method. [8][9][10][11][12][13][14] It is employed in the search for states of the system over a time period, called the assimilation window. This method uses a cost function that is formulated under the assumption that the numerical model is not perfect and penalizes the weighted discrepancy between the analysis and the observations, the analysis and the background state, and the difference between the analysis and the trajectory given by integrating the dynamical model.…”

“…The resulting large symmetric linear systems are solved using Krylov subspace solvers. 14,17,18 One criteria that affects their convergence is the spectra of the coefficient matrices. 18 We derive bounds for the eigenvalues of the 3 × 3 block matrix using the work of Rusten and Winther.…”

Optimality Properties of Galerkin and Petrov–Galerkin Methods for Linear Matrix Equations

“…that have arisen as a natural algebraic model for discretized partial differential equations, possibly including stochastic terms or parameter dependent coefficient matrices [4,8,28,30], for PDEconstrained optimization problems [39], data assimilation [13], and many other applied contexts, including building blocks of other numerical procedures [23]; see also [35] for further references. The general matrix equation (1.2) covers two well known cases, the (generalized) Sylvester equation (for = 2), and the Lyapunov equation…”