Numerical methods for Data Assimilation

• Saddle point (SP) associated with a function is a point such that

  (SP1)

• An equivalent definition of a SP is

  (SP2)

• For the minimization problem

  

• We consider the SP of the augmented Lagrangian associated with the optimization problem

  

• For a convex differentiable function we consider the convex optimization problem

  

• We have

• Theorem: The saddle points of are exactly the points such that

  1. is a solution of , and

  2. and

• We consider the saddle point problem for the Lagrangian of the data assimilation problem.

• From definition SP2, we introduce the direct problem and the adjoint problem .

From

we get

This yields the infsup result

where the inf is a min by convexity.

Direct explicitation of the constraint leads to the problem

whose solution is given by the normal equations .

Consider the infimum problem

The problem is convex differentiable. Zeroing the partial derivative wrt and , we get

which yields

Keeping in mind SP2, we consider

that leads to the adjoint maximization problem

The solution is that yields .

We see that the alternative formula obtained from the Sherman-Morrison formula can be obtained from duality theory.