Variational Data Assimilation : from optimal control to large scale data assimilation

Language

This course is given in english a-priori, excepted if all attendants understand french perfectly.

Goal of this course

- Formulate real-like numerical modeling problems combining: the mathematical equations - conservation laws (PDE models) - numerical method - data / measurements - errors.

- Optimal control of a system governed by a PDE.

- Adjoint equations, 4D-var like algorithms and its variants.

- Sensitivity analysis (local), calibration, identification, data assimilation.

- Relationships with basic stochastic methods (linear case).

- Automatic differentiation (source-to-source code transformation).

At the end, the students are supposed to be able to set up a simple modeling problem (eg in fluid mechanics, geophysical fluid flows, structural mechanics and in presence of measurements / observations), write an adjoint model, elaborate an optimal control process on the optimality system, in view to perform local sensitivities analysis, calibrate the model or identify uncertain parameters.

Content

* Examples of inverse problems : optimal command, parameters identification, data assimilation, optimum design.

* Basics of optimal control:

- ODEs, linear-quadratic case, maximum principle.

- Non-linear PDE case (steady and unsteady), adjoint equations, optimality system, Lagrangian.

* Variational data assimilation: examples, errors. Cost function, regularization, sensitivities.

* Algorithms, minimization, 4D-var and variants.

* Linear case : link with basic sequential method and filtering.

* Exercices.

* Practical work / application to an advective-diffusive phenomena or Burgers equation using Scilab, Matlab or Comsol.

* Examples of recent research studies applied to complex geophysical flows (glaciers, river hydraulics - flood).

* Automatic differentiation (source-to-source, Tapenade software).

Keywords

PDE models, optimal control, adjoint method, variational data assimilation, un- certainties, sensitivities, automatic differentiation.

Prerequisites PDEs, ODEs, differential calculus, numerical schemes, optimization, programming.

Volume

At INSA Toulouse department, mathematical modeling department (GMM), the course is planned for 16H of plenary classes with exercises, plus 4H of practical work , plus the same volume of home work.

The students must study the detailed manuscript course BEFORE each class.

Examination

A practical work with a written report (including the details of the equations derived).