 # Variational Data Assimilation

## Presentation

Program (detailed contents):

Variational Data Assimilation

Examples of inverse problems : least-squares, optimal command, parameter identification, data assimilation.

Optimal control:

ODE Linear-Quadratic case, maximum principle, Hamiltonian.

PDE non-linear, adjoint equations, optimality system, Lagrangian.

Variational Data Assimilation.

Cost function, optimisation, regularisations..

Linear case : links between VDA, BLUE, sequential methods, Bayesian view.

Model learning

Learning a model (ODE or PDE) from large datasets; next derive a surrogate model.

Programming practical (Python-Fenics or FreeFem++): advection-diffusion models or Burgers equation.

Organisation

Variational Data Assimilation (VDA)

Flipped classes based on a detailed manuscript (140 pages) and/or 8 videos.

Tutorials + lab tutorials.

Modeling of a simple industrial (or geophysical) problem. Programming in Python- Fenics  (or FreeFem++ software).

## Objectives

At the end of this module, the student will have understood and be able to explain (main concepts):

Variational Data Assimilation

• Fuse at best a PDE model with datasets.
• The optimal control of dynamic systems (ODE) and PDE models.
• The adjoint method (Hamiltonian for ODE).
• Algorithms of parameters identification, model calibration, Variational Data Assimilation (VDA).
• Introduce prior information e.g. covariances between the unknown parameters.
• Links between VDA, BLUE, Kalman filtering and Bayesian view.

Model learning

• Learning a model (ODE or PDE) from large datasets

The student will be able to:

Set up the equations and the complete modeling chain to perform parameters identification / model calibration / Variational Data Assimilation for PDE models.

Learn a model (ODE or PDE) from large datasets; next derive a surrogate model.

## Needed prerequisite

Variational Data Assimilation

PDE & ODE [I4MMNP71] [I4MMNE81]

Optimization [I4MMMO71] [I4MMMO81]

Probabilities, Statistics [I4MMMS71]

Fundamentals : analysis-differential calculus, functional analysis, numerical schemes, programming.

## Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...