A remark on Domain Decomposition approaches solving Three Dimensional Variational Data Assimilation models
For researchers in data assimilation and domain decomposition, this provides a theoretical unification of two existing approaches, but the contribution is incremental.
This paper reviews two domain decomposition approaches for solving the 3D-Var data assimilation problem and proves their equivalence. The result is a theoretical equivalence between functional DD and discrete multiplicative parallel Schwarz methods for least squares problems.
Data Assimilation (DA) is a methodology for combining mathematical models simulating complex systems (the background knowledge) and measurements (the reality or observational data) in order to improve the estimate of the system state. This is a large scale ill posed in- verse problem then in this note we consider the Tikhonov-regularized variational formulation of 3D- DA problem, namely the so-called 3D- Var DA problem. We review two Domain Decomposition (DD) approaches, namely the functional DD and the discrete Multiplicative Parallel Schwarz, and as the 3D-Var DA problem is a least square problem, we prove the equivalence between these approaches.