A remark on Domain Decomposition approaches solving Three Dimensional Variational Data Assimilation models

TL;DR

This paper reviews and proves the equivalence of two domain decomposition methods for solving the large-scale 3D variational data assimilation problem, enhancing computational approaches for complex system state estimation.

Contribution

It introduces and compares two domain decomposition approaches for 3D-Var data assimilation, establishing their theoretical equivalence.

Findings

Proves the equivalence between functional DD and Multiplicative Parallel Schwarz methods.

Provides insights into efficient large-scale 3D-Var DA problem solving.

Enhances understanding of domain decomposition techniques in data assimilation.

Abstract

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.