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Variational Data Assimilation & Surrogate Models

Presentation

Variational Data Assimilation

Introduction to inverse problems : 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

  • Models – errors – observations (datasets)
  • Cost function, minimisation, regularization techniques. 
  •  Links with BLUE & filtering methods. 
  • Covariance operators derived from physical-based solutions (Green's functions).

 

Practical (Python-Fenics or FreeFem++).

 

Surrogate models

Variance-based sensitivity measures

Sampling-based estimators 

Meta (surrogate) models based approaches

Analytical computations of Sobol indices. Sobol indices error estimation 

 

Organisation:

Assimilation:  Flipped classed based on on-line ressources (videos SPOC-MOOC + detailed manuscript).

Surrogate models : lectures.

Assimilation & surrogate models : Tutorials + programming practicals.

Objectives

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

 Identification:

  • How to set up a complete modeling problem: datasets – models - errors (and corresponding inverse problems).
  • Parameter identification and /or model calibration based on the optimal control of the model.
  • The adjoint method.
  • Links between Variational Data Assimilation & BLUE – Kalman filtering.
  • Variance-based sensitivity measures
  • How to derive Meta (surrogate) models
  • How to compute Sobol indices; sensitivity analysis.

 

The student will be able to:

  • Fuse at best a PDE model with datasets and prior probabilistic behavior (if the uncertain parameters)
  • Control dynamic systems (PDE or ODE)
  • Derive and implement the adjoint model, the global optimization process (Variational Data Assimilation)
  • Analyse uncertainties propagation
  • Compute Sobol indices, sensitivity analysis. 
  • Build up a surrogate model.

Needed prerequisite

PDE equations [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...