99083 - DATA ASSIMILATION FOR DYNAMICAL SYSTEMS

Learning outcomes

The student will learn the foundation of dynamical systems theory for ordinary differential equations, with a focus on chaotic dynamics. The student will acquire knowldge of data assimilation, the term used in geoscience to refer to state estimation theory. Data assimilation is common practice in numerical weather prediction, but its application is becoming widespread in many other areas of climate, atmosphere, ocean and environment modelling. The student will learn the formulation of the problem from a Bayesian perspective and two popular families of Gaussian based approaches, the Kalman-filter/-smoother and the variational methods. The student will be exposed to specific challenges that data assimilation has encountered to deal with high-dimensional chaotic systems, such as the atmosphere and ocean, and the countermeasures that have been taken and which have driven the recent dramatic development of the field. The student will acquire knowldege of machine learning methods and their use in numerical weather predictions and data assimilation.

Course contents

Part I - Modelling the world: Overview on dynamical Systems and Probabilities

● Linear dynamical systems

○ Maps

○ Ordinary differential equations: Matrix exponential, Existence and uniqueness of solution, Fundamental matrix solutions and the model resolvant

● Nonlinear chaos

  ○ Linear stability analysis, invariant manifold

  ○ The Liouville equation

  ○ Attractors (fixed points, limit cycles ...) and bifurcations

  ○ Strange attractors, nonlinear stability, invariant manifolds

  ○ Multiplicative ergodic theorem: Lyapunov vectors and exponents

  ○ Entropy

 ● Stochastic dynamics

  ○ Probability theory and stochastic processes

  ○ Discrete stochastic dynamics and stochastic differential equations

  ○ The Fokker-Planck equation

Part II - Making sense of data using models: Data Assimilation

 ● Posing the problem under a Baysiean framework

  ○ Representation of the physical and of the observational systems

○ The three estimation problems: Prediction, Filter and Smoother

  ○ Statistical interpolation 

 ● Linear estimation theory 

  ○ Gauss-Markov Models

  ○ Observability and controllability

  ○ Minimum variance formulation - Kalman filter and smoother

○ Maximum a-posteriori formulation - Variational formalism 

  ○ Joint state-parameter estimation

  ○ Filtering versus smoothing

  ○ Expectation maximization

 ● Nonlinear estimation theory: the ensemble Kalman filter and 4DVar

○ Minimum Variance approaches: 

   ● The extended Kalman filter

  ● The ensemble Kalman filter and smoother

  ● Stochastic and Deterministic EnKF

  ● Filter stability and divergence

  ● Making the EnKF works: Inflation and localization

● Nonlinear least squares

○ Gauss-Newton

  ○ Adjoint-based minimization

  ○ 3D- and 4D-Var

  ○ Hybrid ensemble-variational techniques and other iterative methods

● Fully Bayesian estimation: Particle filters

● Data assimilation and Chaos

 Part III - Data driven data assimilation using machine learning: An Overview

 ● Overview of machine learning methods to retrieve time-evolving dynamics

 ● Data assimilation and machine learning similarities and key differences

  ○ Estimating a model using ML

  ○ Estimating a model using DA

 ● Combining DA and ML

 

Teaching methods

Lectures are given in person in the classroom in hybrid modes. Students can also attend remotely.

The course does also include up to three guest lectures from Prof Geir Evensen who is visiting fellow at University of Bologna.

Assessment methods

The final assessment will be under the form of an oral exam (~45 mins) where the student will be be posed a number of questions aimed at inspecting the student's degree of understanding of the concepts, methods, and problems explained in the course. 

Teaching tools

Blackboard, projected slides and computer simulations

Office hours

See the website of Natale Alberto Carrassi