Numerical methods for Data Assimilation

• the CN is a real positive number or

• if is lipschitz continuous around , with lipschitz constant then

• Examples

  • for , for .

    For the polar factor of ,

  • , for

• is the best possible constant such that

• let be Fréchet différentiable at

• The norm is the operator norm induced by and

• The condition number is

• if i is implicitely defined by , and the asumption of the implicit function theorem hold at the condition number is

• Practical questions : compute closed formula, sharp estimates or simply bound

• CN considered so far is the  normwise absolute CN

• If , the relative condition number is

• The term can be replaced in the limsup definition by the quantity yielding a mixed condition number. Under the differentiability assumption, the absolute CN then , the relative counterpart being

• In linear algebra the data space is often a cartesian product

• example , and , where the matrix norm is induced by the vector norm ||{.}||. It is possible to show that

  

• from ( ) follows that for ,

  the CN satisfies ( and have same order of magnitude).

• differentiability assumption, , condition number

• and are finite dimension spaces. If , it might be interesting to evaluate the CN using the adjoint of

• example : find the worse-case perturbation, or samples in a smaller space

• aim of this part : show how duality results translate into CN estimation

• Let anf be two euclidean spaces

• The scalar products norms and dual norms are denoted by , i.e. .

• the adjoint of the linear operator is defined by for any ,

• it is easily shown that

• For the full rank least-squares problem,

  

• Scalar products : on and . The trace inner product is taken on and on the data space

• We denote by an absolute norm on . Example

  when is considered. Let be its dual w.r.t. the canonical scalar product on . Then the dual of is .

For the least-squares,

• 

• possible norm ; the adjoint is

• maximization over a vector space of dimension , instead of a maximization over a vector space of dimension

• in next part, the operator norm is directly computed using statistical methods based on sampling