Numerical methods for Data Assimilation

Quick solvers

In the Incremental 4D Var approach, we have to solve a series of linear systems
.
the linear systems are large ( ) and dense.
For large problems (atmosphere DA, ocean DA), solving these linear systems is the dominant computational part of the algorithm.
need for efficient methods (memory, cpu).

Linear system occuring in the method will be denoted , or if no-confusion possible.

Two main approaches :

direct solvers: compute a factorization and use the factors to solve the linear system,
iterative solvers : build a sequence .

Express the matrix as the product of matrices having simple structures (i.e. diagonal, triangular).

for unsymmetric matrices, (Cholesky) for symmetric positive

definite matrices, for symmetric indefinite matrices.

Advantages	Drawbacks
accurate - backward stable,	complex for sparse matrices,
perform the factorization once but use it for many solves.	can become prohibitive (CPU, memory) for large problems.