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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...