- Introduction to IAMOOC
- Acces to Moodle (post your homework and access to the forum)
- Dates

- Exercises and lessons (en pdf)
- Lessons et vidéos
- References
- List of students who got the diploma

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.

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.

You will have to install Python 3 in your machine (see below).

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.

April 2018, Thursday 12. Chapter 0: Install all you need

April 2018, Thursday 19. Chapter 1: Interval Computation

April 2018, Thursday 26. Chapter 2: Set inversion

May 2018, Thursday 03. Chapter 3: Contractors

May 2018, Thursday 10. Chapter 4: Application to robot localization

May 2018, Thursday 24. All exercises should be posted.

May 2018, Thursday 31. Diplomas are sent by email

Exercises to be done:

A small manual on PyIbex and Vibes:

from pyIbex import * should be replaced by from pyibex import *

Some other modifications are described in the PyIbex documentation. (in magenta)

For the install of PyIbex and Vibes, see:

If you have Ubuntu, you should do as follows:

1) Install spyder 3: sudo apt-get install spyder3.

2) Install pip : sudo apt-get install python3-pip.

3) Install pyibex: sudo python3 -m pip install pyibex.

4) Install vibes: sudo python3 -m pip install vibes.

5) Download Vibes (i.e., the binary file VIBes-0.2.3-linux.AppImage) at https://github.com/ENSTABretagneRobotics/VIBES/releases/tag/0.2.3

6) Make VIBes-0.2.3-linux.AppImage executable (Right click on the icone + permissions).

Click on the icone of Vibes. You will see a small window with nothing inside, waiting for drawing orders.

7) Start Spyder and test the example given at http://benensta.github.io/pyIbex/

What is an interval ? |
Operators |
Examples |
Elementary functions |

Examples |
Boxes |
Width |

Subpavings |
Inclusion functions |
Monotonicity |

Convergence |
Natural inclusion function |
Minimal inclusion function |

Exercise 1.

For this exercise, you do not need to program anything, just computing with intervals.

Exercise 2.

Using overloading operators, we show how Python allows an implementation of your own interval arithmetic library.

What is set inversion? |
Example |
Inclusion tests |

SIVIA |
Parameter estimation |

Exercise 3.

Interval analysis allows to solve non-convex global minimization problems [Kea96] without being trapped by a local minimum [Nin11]. In this exercise, a simple interval-based algorithm is implemented to illustrate the principle.

Exercise 4.

This exercise illustrates how interval analysis can be used to solve set estimation problems [Kre97] and how it can be robustified with respect to some outliers [Jau96].

Motivations |
What is a contractor? |
Properties |
Example |

Contractor for z=x+y |
For primitive equations |
Decomposition |
Forward-backward |

Exercise 5.

This exercise shows a very simple example related to bounded-error estimation [Poi03] which illustrates the power of interval propagation.

Exercise 6.

This exercise how an interval forward-backward propagation [Ben99] makes it possible to build efficient contractors for a huge class of constraints.

Exercise 7.

We show here how a separator [Jau14] 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.

Exercise 8.

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 9.

Bounded error estimation problems can now be solved very easily and efficiently with the help of PyIbex.

Exercise 10.

This exercise shows an important application which can be solved efficiently with interval analysis [Mei02]. Without any linearization, we show that it is possible to localize a robot in a guaranteed manner.

Exercise 11.

In a SLAM problem, many variables are involved with nonlinear constraints [Jau15a,Jau15b]. This problem is considered as difficult. We show here that it it possible to solve it easily and in a reliable way using intervals [Jau11].

Illustration of Interval SLAM for real applications..

[Moo66] RE Moore (1966), *Interval analysis, * Prentice-Hall.

[Jau01] L. Jaulin, M. Kieffer, O. Didrit and E. Walter (2001), *Applied Interval Analysis
with Examples in Parameter and State Estimation, Robust Control and Robotics,* Springer-Verlag.

[Jau02] L. Jaulin and E. Walter (2002). Guaranteed robust nonlinear minimax estimation.
*IEEE Transaction on Automatic Control*.
Volume 47, number 11, pages 1857, 1864. pdf.

[Cha09] G. Chabert and L. Jaulin (2009), Contractor programming. *Artificial Intelligence*.
Vol. 173, pp 1079-1100. pdf.

[Kea96] B. Kearfott, Rigorous Global Search: Continuous Problems (Nonconvex Optimization and Its Applications), Kluwer, 1996.

[Nin11] J. Ninin et F. Messine. A metaheuristic methodology based on the limitation of the memory of interval branch and bound algorithms. Journal of Global Optimization (2011) 50:629-644.

[Jau15a] L. Jaulin, *Automation for robotics*, ISTE WILEY, 2015.

[Jau15b] L. Jaulin, *Mobile robotics*, ISTE WILEY, 2015.

[Dre14] V. Drevelle and J. Nicola. *VIBes: A Visualizer for Intervals and Boxes.
*Mathematics in Computer Science, 2014.

[Jau14] L. Jaulin and B. Desrochers (2014). Introduction to the Algebra of Separators with
Application to Path Planning. *Engineering Applications of Artificial Intelligence*
pdf.

[Mei02] D. Meizel, O. Lévêque, L. Jaulin and E. Walter (2002). Initial Localization by Set Inversion. *IEEE
Transactions on Robotics and Automation *. Volume 18, Number 6, pages 966-971.
pdf.

[Kie99] M. Kieffer, L. Jaulin, E. Walter and D. Meizel. Guaranteed mobile robot tracking using interval analysis, MISC'99 Workshop on Application of Interval Analysis to System and Control, Girona, 24-26 février 1999. pdf.

[Jau11] L. Jaulin (2011). Range-only SLAM with occupancy maps; A set-membership approach. *IEEE-TRO*.
Vol 27, Issue 5. pdf.

[Jau96] L. Jaulin, E. Walter and O. Didrit (1996). Guaranteed robust nonlinear parameter bounding, CESA'96 IMACS
Multiconference (*Symposium on Modelling, Analysis and Simulation*), Lille.
pdf.

[Kre97] V. Kreinovich, A.V. Lakeyev, J. Rohn, P.T. Kahl (1997). Computational complexity and feasibility of data processing and interval computations, Springer Science Business Media.

[Poi03] P Poignet, N Ramdani, O Vivas, Robust estimation of parallel robot dynamic parameters with interval analysis, CDC, 2003.

[Ben99] F. Benhamou, F. Goualard, L. Granvilliers, Revising hull and box consistency, Proceedings of the 1999 International Conference on Logic Programming.

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

Alain Acevedo

Yacine Benhnini

Justine Bonnot

François Cébron

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

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