# 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.
• 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...