IAMOOC is a free MOOC on Interval Analysis with application to parameter estimation and robot localization.
Interval analysis concerns the methods which computes with intervals in place of real numbers.
There is a large number of applications in Control, Optimization, Robotics, Chemistry.
The registration is free.
What is Interval Analysis?
Interval Computation is a numerical tool which allows us to solve nonlinear problems in a guaranteed way.
One of the pioneers of interval computations is Ramon E. Moore. He contributed largely to its development and dissemination.
Nowadays, Interval Computations can be found in many fields: global optimization, set inversion, parameter estimation, localization of robots, etc.
Interval Computation offers a general numerical framework to easily get reliable results.
Interval Computation can deal with a huge class of problems involving equations or inequalities which can be non-smooth,
non-convex, or with different kind of variables.
Interval tools are easy to understand. They only require basic knowledge on mathematics and computer sciences.
The results provided by interval methods are guaranteed, which means that from correct assumptions.
Interval Computations will provide correct conclusions.
The results are guaranteed in any case. As a consequence, Interval Computations can be used to prove mathematical theorems.
For instance, W. Tucker has proven the conjecture of Lorenz and Thomas Hales the conjecture of Kepler, both using Interval Computations.
Registration
To register fill the form :
The registration is not mandatory to follow the MOOC and to see the videos.
But it is needed to have to get the diploma.
Required
To follow IAMOOC, you should have some basic notions in Python
and some knowledges in mathematics.
If you do not know Python, but any other programming language,
you may learn easily the required notions in this MOOC.
You will have to install Python 3 in your machine (see below).
Videos
A video with explanations related to each exercise is given as soon as the lesson opens.
You are not obliged to follow the method that is given in the video.
Diploma
To get the diploma, you need at least 12 points.
Therefore, doing all exercises is not mandatory.
The participants who got enough points will receive a diploma corresponding this MOOC.
This diploma can be used by students to obtain the corresponding
ECTS
from their PhD courses, or to comply with any other requests by their home university.
If needed by your doctoral school, I can also provide a 20 minutes Skype exam with a mark, once the MOOC is finished.
An example of the diploma you can obtain :
Dates
, 12:00. Chapter 0: Install all you need
, 12:00. Chapter 1: Interval Computation
, 12:00. Chapter 2: Set inversion
, 12:00. Chapter 3: Contractors
, 12:00. Chapter 4: Application to robot localization
, 12:00. All exercises should be posted.
, 12:00. Diplomas are sent by email
Post your work
For each lesson, you should send your solution by email to jaulin.iamooc@gmail.com
For the exercises that require the execution of a program,
You should provide the Python (or other) code.
You should also send a pdf file with some explanations and screen captures of the running program.
For some exercises, the solution corresponds to text and equations and no program is required.
In such a case, you should post a scan of your paper sheet,
(taken with you phone for instance) or any pdf file.
Files
Exercises to be done:
A small manual on Codac and Vibes:
A zip file with the slides of the lessons :
Lessons et vidéos
Errata
The interval package PyIbex has changed into Codac.
As a consequence from pyIbex import * should be replaced by from codac import *
Moreover, to start Vibes from your terminal, write
Vibes-viewer instead of vibes.
Some other modifications are described in the codac documentation, (in magenta)
Chapter 0. Install all you need
Open:
Abstract: The goal of Chapter 0 is to help you to install all what you need to follow IAMOOC.
Exercise 0. Send an image
You install all what you need in your computer. You have to send by email an image to jaulin.iamooc@gmail.com
with a subpaving (see below). The role of the exercise is to show that all is well installed and that you can start the mooc next week.
Install.
You need Python 3, codac and Vibes for graphics.
For the install, see:
Test the example given below :
from codac import *
from vibes import vibes
f = Function('x', 'y', '1+cos(x)+sin(y)^2')
S = SepFwdBwd(f, [1,2])
X0 = IntervalVector(2,[-2,2])
vibes.beginDrawing()
SIVIA(X0, S, 0.01)
vibes.axisAuto()
vibes.setFigureSize(500,500)
Chapter 1. Interval Computation
Open :
Abstract: The goal of Chapter 1 is to
introduce all the mathematics related to interval analysis. After
this chapter, you should be able to compute with uncertain numbers
represented by intervals and also to approximate a set by subpavings,
i.e., union of boxes. During the practice we will implement the
interval calculus under a Python environment.
What is an interval ?
Operators
Examples
Elementary functions
Examples
Boxes
Width
Subpavings
Inclusion functions
Monotonicity
Convergence
Natural inclusion function
Minimal inclusion function
The exercises have to be posted by email to jaulin.iamooc@gmail.com)
Exercise 1. Interval arithmetic (1 point)
For this exercise, you do not need to program anything, just computing with intervals.
Exercise 2. Intervals with Python (1 point)
Using overloading operators, we show how Python allows an implementation of your own interval arithmetic library.
Chapter 2. Set inversion
Open :
Abstract: Many problems of engineering such as
parameter estimation, tuning of a controller, etc., can be cast into
a set inversion problem. In this chapter, we will define the notions
of set inversion and illustrate these notions on some examples. An
algorithm for solving any set inversion problems will be given. This
algorithm, named SIVIA (Set Inverter Via Interval Analysis), will be
proposed. This algorithm will be implemented in
Python and some test-cases will be solved.
What is set inversion?
Example
Inclusion tests
SIVIA
Parameter estimation
Exercise 3. Minimization of a nonconvex function (1 point)
Interval analysis allows to solve non-convex global minimization problems without being trapped by a local minimum.
In this exercise, a simple interval-based algorithm is implemented to illustrate the principle.
Exercise 4. Parameter estimation (1 point)
This exercise illustrates how interval analysis can be used to solve set estimation problems
and how it can be robustified with respect to some outliers.
Chapter 3. Contractors
Open :
Abstract : A contractor is an operator which takes as an input a box X and contracts it into a subbox Y without
removing a single solution of the problem. Contractors are necessary to solve efficiently problems containing
a large number of unknowns. This chapter introduces the notion of contractor and
explains how they can be implemented in Python.
Motivations
What is a contractor?
Properties
Example
Contractor for z=x+y
For primitive equations
Decomposition
Forward-backward
Exercise 5. Electric circuit (1 point)
This exercise shows a very simple example related to bounded-error estimation which illustrates the power
of interval propagation.
Exercise 6. Forward-backward contractor (1 point)
This exercise shows how an interval forward-backward propagation makes it possible to build efficient contractors
for a huge class of constraints.
Exercise 7. Sine contractor (1 point)
This exercise gives an optimal contractor for the constraint y=sin(x) using symmetries.
Exercise 8. Separators (1 point)
We show here how a separator can be built easily from two contractors: an inner and an outer. The dual nature of
the separator allows us to compute an inner and an outer approximation of a set defined by several constraints.
Chapter 4. Application to robot localization
Open :
Abstract : This chapter considers a problem of localization inside an
environment with landmarks. No lessons is given in this chapter, only exercises.
The main problem to be treated here is SLAM (Simulataneous Localization And Mapping).
It will be decomposed into a list of 4 exercises with an increasing complexity.
Exercise 9. Contractors and separators with codac(1 point)
A separator is a pair of two contractors: an inner contractor and an outer contractor.
This exercise illustrates how Pybex (which is a Python extension of Ibex) can be used to
build easily separators for sets defined as union, intersection and complements of primitive sets.
Exercise 10. Estimation with codac(1 point)
Bounded error estimation problems can now be solved very easily and efficiently with the help of codac.
Exercise 11. Localization of a robot from landmarks using range and bearing measurements (2 points)
This exercise shows an important application which can be solved efficiently with interval analysis.
Without any linearization, we show that it is possible to localize a robot in a guaranteed manner.
Exercise 12. Simultaneous Localization And Mapping (SLAM) (2 points)
In a SLAM problem, many variables are involved with nonlinear constraints. This
problem is considered as difficult. We show here that it it possible to solve it easily and in a reliable way using intervals. Illustration of Interval SLAM for real applications..
List of the students who got the diploma
51 in 2015
ABADIE Moran
AUBERTOT Quentin
BARONI Kévin
BARONNIER Romain
BASSET Pierre
BEAUDOIN Maxime
BERNARDES Evandro
BOENNING Hannah
BOURGOIS Auguste
CHANU Simon
COTTEN Guillaume
DALIN Eloïse
EL ABDALAOUI Zacharie
ENNOUHI M'hamed Fadil
FONTANA Werner
GALLAND Alexandre
GY Morgan
KARKOUB El Wali
LE ROCH Gwenn
LEGAY Kevin
LI Ang
LIU Wanxin
MARTIN Pierre
MEHDI Nima
MILHEM Rémi
NEAU Guillaume
PERTIERRE DO MONTE
PLANCHOT Antoine
RAYNEAU Vincent
SOLA Yoann
SOULIE Camille
SUN Tithnara
TANGUY Florian
TERTRAIS Donatien
THIBAULT Adrien
TOMEZACH Julien
VADAINE Hugo
WELTE Anthony
ZHU Lei
ZIANE Mohamed Mahrez
BEN SAID Hela
BHIRI Bessem
BOUKALYassine
El JAWAD Alaa
MANSOUR Fatma
MESLEM Nacim
ORJUELA Rodolfo
RENAUDEAU Brice
ROUSSEAU Gauthier
TANGUY Noel
VANDERMOTTE Sylvain
34 in 2017
Raphael Abellan Romita
Alain Acevedo
Yacine Benhnini
Justine Bonnot
François C.
Antoine C.
Jean-Marie CODOL
Julien Damers
Hani Dbouk
Lionel Génevé
Gabriel GODEAU
Jean-Philippe Gras
Yoann Guguen
Fabrice Lallement
Philippe Lambert
Julien Langlois
Francois Leborne
Nisha Mahato
Fatma Mansour
Mohamad Mezher
Yasmine Najar
Mohamed Outahar
Clément Rolinat
Joris Tillet
Sophie Tuton
Nicolas Veylon
Raphael Voges
Jean Walter
Mohamed Ouadrhiri
Maha Abouzai
Yves Le Palud
Emilien Fournier
Philipe Miranda de Moura
Julien Brisset
29 in 2018
Arslanyan Antony
Noureddine Bensafi
Elias Aoun Durand
Denis Bacchus
Vanessa Baray
Philippe Chapuis
Alexandre Chenu
Evann Clavier
Maria Luiza Costa Vianna
Charles Coquet
georges daher
Sarah Delmas
Teva Temangeot
Maxime Do Rosario
Romain Dussot
Pierre Filiol
Mohamed Fnadi
Quentin François
Charlie Goutorbe
Alexandre Houdeville
Juliette Brugier
Jacques Kadima
Olivier Laurendin
Lucie Lefevre
Elodie Noëlé
Fabrice Poirier
Abbas Ramadan
Louis Valéry
Mihály Zsolt
25 in 2019
Maxens Achiepi-Autret
Hyacinthe Andrieux
Alexandre Argento
Aurélien Grenier
Kevin Bedin
Matthieu Bouveron
Philippe Capdepuy
Quentin Cardinal
Cyril Cotsaftis
Alexandre Courjaud
Nathan Fourniol
Jonathan Gomis
Corentin Jégat
Jean-Luc Kersulec
Arnaud Klipfel
Erwann Landais
Louis Lesieur
Luc Longin
Pierre-Yves Mauduit
John Nikas
Junaid Rasheed
Driss Tayebi
Suman Maiti
Florian Gaurier
Ayman Al-Khazraji
27 in 2020
Kévin Affraix
Victoria Alonso
Agathe Archet
Colin Baumgard
Jules Berhault
Quentin Brateau
Elliot Brendel
Jose Eduardo Carvajal Rubio
Mamadou Dembélé
Subham Dey
Aaronkumar Ehambram
Imad El Bouazzaoui
Alexandre Evain
Sylvain Guénon
Hamid Hacène
Maha Halimi
Paul-Antoine Le Tolguenec
Corentin Lemoine
Antonin Lizé
Badr Moutalib
Gwendal Priser
Alexander Radaev
Mohamed Sadiq Ikbal
Robin Sanchez
Bertrand Turk
Nicolas Torres Alberto
Yann Musellec
30 in 2021
Leobardo Valera
Timothée Guy
Jingyao Su
Sylvain Hernandez
Youssef El Gaouti
Mathilde Roques
Gabriel Adjouet
Angelica Ginnante
Julien Albrand
Yuehan Jiang
Hoang Nguyen
Estelle Arricau
Antonin Bétaille
Enzo-Loïd Essono Aubame
Martin Gounabou
Bernardo Hummes Flores
Jean-Vincent Klein
Katell Lagattu
Stéphane Ngnepiepaye Wembe
Yves Jordan Njamen Jamen Ngongang
Nicolas Odorico
Baptiste Orlhac
Hugo Piquard
Samuel Prouten
Hugo Sabatier
Julien Taran
Isaac-Andreï Witt
Thibault Jadoul
Yohann Gourret
Bilal Ghader
Alexis Stoven-Dubois
47 in 2022
Raphaël Bousigues
Alessandro Colotti
Marion Brateau
Danil Klyanchin
Luc Desbos
Artem Androsov
Chaolun Lu
Maxim Zvyagin
Dmitry Malyshev
Vilma Muco
Daan Wynen
Côme Butin
Alexandre L'Her
Chidiebere Millicent IGWEBUIKE
Anton Pisarenko
Patrick Auger
Muhammad Farhan AHMED
Alexandr Tereshchenko
Maksim Bezel
Mosaab Derbal
Jonatan Contreras
Ömer Gürsoy
Gabriel Betton
Théo Boutémy
Ermance Decaudaveine
Nicolas Defour
Laurent Droudin
Damien Esnault
Hugo Gace
Simon Gervaise
Mael Godard
Taddéo Guérin
Florian Jerram
Thibault Langlard
Augustin Morge
Virgile Pelle
Rémi Porée
Danut Pop
Chloe Potherat
Laurent Potin
Mirado Rajaomarosata
Hugo Reubrecht
Jonas Soueidan
Thomas Tacheron
Antoine Wanctin
Hugo Yverneau
Morgan Louédec
2023
Joan Burgalat
Hugo Rémin
Raúl Zaragoza
Hugo Duarte
Aditya Anand
Michele Grimaldi
Mattia Piras
Jiahao Zhang
Mihály Gencsi
Charly Peraud
Luis Eduardo Ramírez Montoya
Ibrahim Abdelrahman Sayed Sayed Emam
Abdelhaq Ouelli
Alexandre Batard
Victor Bellot
Léo Bernard
Johan Berrier-Gonzalez
Gwendal Créquer
Apolline De Vaulchier Du Deschaux
Alix Doineau
Marie Dubromel
Martin Galliot
Bastian Garagnon
Guillaume Garde
Louis Gillard
Clara Gondot
Adam Goux Gateau
Louis-Nam Gros
Hugo Hofmann
Tristan Le Floch
Jules Le Gouallec
Emilie Ledoussal
Hippolyte Leroy
Théo Massa
Marguerite Miallier
Ludovic Mustière
Océan Noel
Camilo Ortiz
Martin Pilon
Mathieu Pitaud
Aimé Randriamoramanana
Harendra Rangaradjou
Louis Ravain
Kevin Ren
Catherine Rizk
Louis Roullier
Etienne Roussel
Mathys Séry
Joachim Zafrana
Rania Ziane
