 # Variational Data Assimilation & Model Learning

## Presentation

Variational Data Assimilation

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

Similarities with Artificial Neural Networks.

Optimal control:

ODE Linear-Quadratic case, maximum principle, Hamiltonian.

PDE non-linear, gradient computation via the adjoint equations, optimality system, Lagrangian.

Variational Data Assimilation.

Cost function, optimisation, regularisations.

Links between VDA, BLUE, sequential methods, Bayesian approach.

Model learning

Learning a model (ODE system or scalar PDEs) from large datasets.

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Few programming practicals in Python: inverse problems  based on linear and non-linear advection-diffusion models.

Organisation

Variational Data Assimilation (VDA)

50% of classes are on flipped mode. Lectures – exercises – Python codes.

A complete Moodle page is available with:

-        A more than complete course manuscript (140 pages with “to go further sections”),

-       exercises with their corrections,

-       programming practical(s) (lab tutorials) with Python codes.

The marked programming practical consists to solve a challenging inverse problem arising in Earth Sciences (spatial hydrology). Programming in Python (with Fenics library).

## 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.

- To compute a gradient via the adjoint method.

- The basic principles of Automatic Differenciation.

- Algorithms of parameters identification, model calibration, Variational Data Assimilation (VDA).

- Introduce prior information via covariances matrix

- Links between VDA, BLUE, Kalman filtering and Bayesian approach.

Model learning

- Learning a model, ODE or (scalar)  PDE from datasets and an a-priori given dictionary.

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.

Identify physical based model terms (e.g. ODE terms) from datasets.

## Needed prerequisite

Variational Data Assimilation :

PDE (& ODE) models, differential calculus, optimisation, basis of 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...

## Bibliography

- "Variational data assimilation & Model Learning " by J. Monnier. INSA - Univ Toulouse. Course manuscript 144 pages

http://www.math.univ-toulouse.fr/jmonnie/Enseignement/VDA.html

- Asch, M., Bocquet, M. and Nodet, M., 2016. Data assimilation: methods, algorithms, and applications. Society for Industrial and Applied Mathematics.

- “Computational Methods for Data Evaluation and Assimilation”.Dan Gabriel Cacuci, Ionel Michael Navon, Mihaela Ionescu-Bujor. Chapman and Hall, 2013.

- “Data Assimilation for Scientists and Engineers” : a Toulouse University pedagogical project. O. Thual, S. Gratton, G. Monnier, O. Pannekoucke. http://pedagotech.inp-toulouse.fr/130107/