SWIM 08
Small Workshop on
Interval Methods
June 2008, Thursday 19 and
Friday 20,
LIRMM, Amphi St Priest
Université de Montpellier 2
Campus SaintPriest  Bât. 2860,
rue de Saint Priest 34 090
Montpellier Cedex 5, France
Contact: Nacim Ramdani: nacim.ramdani(at)inria(dot)
Luc Jaulin: jaulinlu(at)ensieta(dot)fr
Swim08 is now finished. We had 21 speakers. Several countries were
represented (
Next year Swim 09 will be. We don’t known yet where it will take
place.
2) How to participate to SWIM?
3) Where will SWIM take place?
4) Program
6) FAQ
Since about eight years, we have an active
French working group named MEA (Méthodes ensemblistes pour l'automatique),
which belongs to the larger structure GDR MACS. The MEA group
encloses 235
people at the moment.
The aim of our MEA group is to promote interval
analysis, constraint propagation and other setbased methods in order to apply
these tools to estimation, robotics, control and other engineering areas.
Four times a year, we have a oneday scientific
meeting (often in
For instance, the slides of our last meeting
are available at http://www.lirmm.fr/ensemble/spip.php?article12
This year, for the first time, we would like to
open ourselves to the international interval community by organizing a twoday
informal and free workshop in
It
suffices to send us (i.e to Nacim
Ramdani: nacim.ramdani@inria.fr and Luc Jaulin: jaulinlu@ensieta.fr)
an email. That’s all. If you want to present a contribution, (of course,
related to interval analysis, setmembership methods or constraint
propagation), please, send us a title and a 10 lines abstract before June 10.
Note that all slides will be made available in a pdf format on this site, after
the workshop.
The
workshop is free, but, nothing will be provided (no CDRoms, no banquet, no
meal…). Hopefully,
SWIM’08
took place at the following address:
Amphi St
Priest
Université Montpellier 2 –
Campus
SaintPriest  Bât. 2860,
rue de
Saint Priest
34 090 Montpellier Cedex 5
France
From the
center of
In case of
problem, you can call the organizers Nacim (06 17 83 35 42) or Luc (06 82 99 00
41).
============================================================
Thursday
June 19
9h0010h00
: Welcome. Coffee
=============================================================
10h12h :
Session CSP , nonlinear system
and intervals arithmetics
 Jamila
SamHaroud, Artificial Intelligence Laboratory,
(
Title:
Enhancing numerical constraint propagation using multiple
inclusion representations
 Gilles Trombettoni, Gilles Chabert.
Title:
Constructive Interval Disjunction: past, present, and future
 Ignacio
Araya, Bertrand Neveu, Gilles Trombettoni.
Title:
Exploiting Common Subexpressions in Numerical CSPs
 JeanPierre MERLET, INRIA Sophia Antipolis. Title: Influence of
uncertainties
on ultrasonic localisation systems

Title:
Computing the range of real eigenvalues of an interval matrix
 Nathalie
Revol, INRIA Rhones Alpes
Title :
Standardization and interval arithmetics
=============================================================
=============================================================
14h15h : Plenary Talk
Ned Nedialkov, Dept. of Computing and
Software,
Title:
Interval Tools for ODEs and DAEs
=============================================================
=============================================================
============================================================
15h16h30
: Session Continuous Reachability and Verification
 Nacim
Ramdani, INRIA SophiaAntipolis Mediterranee and LIRMM UMR 5506,
CNRS UM2
Title:
Reachability of uncertain nonlinear systems using a nonlinear
hybridization
 Olivier Bouissou, CEA Saclay  DRT/DTSI/SOL/MeASI GifsurYvette
Title:
Using guaranteed integration of ODEs for the verification of
embedded
software.
 Nicolas
Delanoue, LISA, Angers.
Title: :
Attraction domain of a nonlinear system using interval analysis.
 M.
Lhommeau, LISA Angers
Title:
Inner and outer approximation of the capture basin of a state
space
model using interval analysis
===========================================================
16h3017h30 : Session CSP again
 Dominique LOHEZ, ISENLille
Title:
Relational and Algebraic Concepts for
Automated Interval Reasoning

Guillaume Verger, LIRMM,
Title:
CSPs and Quantified CSPs
 Maëlenn
AUFRAY (ENSIACET 
Title: S.I.V.I.A. Applied to Dielectric
Spectroscopy: a Guaranteed
Parameter
Estimation Using Interval Analysis
===========================================================
20h00 Workshop Dinner.
===========================================================
===========================================================
Friday
June 20th
===========================================================
===========================================================
9h009h30 : Coffee
===========================================================
===========================================================
9h30 12h: Session Estimation, uncertainty and CSP
 Olivier
STRAUSS, LIRMM UMR CNRS Univ.
Title:
Imprecise estimation in signal processing.
 Emmanuel Bénazéra, Louise TraveMassuyes, LAAS, Toulouse.
Title: SetTheoretic Estimation of Hybrid
System Configurations.
 Farah
Mourad, Fahed Abdallah, Hichem Snoussi, Cédric Richard, UTC,
LM2S, UTT
Title:
Guaranteed boxed localization in MANETs by interval analysis and
constraints propagation techniques.
 Philippe LANGLOIS (DALI, Univ.
Title:
Accurate results with compensated algorithms and validated error bounds in
floating point computation: principles and applications.
 Jan
Sliwka, ENSIETA,
Title:
SAUC’ISSE : our interval robot

Alexandre Goldsztejn
Title: A
New Framework for Sharp and Efficient Resolution of NCSP with
Manifolds
of Solutions
 Luc
Jaulin and Gilles Chabert, ENSIETA.
Title:
Resolution of nonlinear interval problems using symbolic interval
arithmetic.
 Gilles
Chabert and Luc Jaulin, DTN, ENSIETA.
Title: :
Interval and Boolean constraint propagation for simultaneous
localization
and map building

Sébastien LENGAGNE, Nacim RAMDANI, Philippe FRAISSE (LIRMM / INRIA)
Title:
Application of Interval Analysis : safe path planning.
==========================================
==========================================
14h15h : Visit of Robotics Lab. LIRMM.
==========================================
15h16h : Round Table
==========================================
16h : End of Workshop
==========================================
==========================================
1)
Jan Sliwka, ENSIETA,
Title: SAUC’ISSE : our interval
robot. Slides.
Movie of Sauc’isse taken by an underwater camera.
Movie of Sauc’isse
for the experiment.
Data available for the localization.
Movie illustrating the interval
localization method..
Results obtained by the interval
localization method (txt file).
Abstract: Sauc’isse is an autonomous
underwater robot. It uses interval computation and constraint propagation tools
to localize itself. The robot is equipped with a sonar, a gyrometer and
accelerometers. The robot is assumed to know the exact shape of the environment
(here a swimming pool). Many outliers occur in the data provided by the sonar.
An interval method is then used to obtain a dynamic fusion of all data. The
associated method is fast (since it uses constraint propagation) and extremely
robust with respect to outliers. To the best of our knowledge, it is the first
time a robot embeds interval method to control itself. In this talk, the
principle of the localization method will be explained, the construction of the
robot will be shown and some videos will be displayed in order to illustrate
the good behaviour of the robot.
This robot has been built at
2) JeanPierre MERLET, INRIA Sophia Antipolis.
Title: Influence of uncertainties on ultrasonic
localisation systems. Slides.
Abstract: Ultrasonic emitter/receivers are frequently
used in robotics to locate objects. The basic principle is to have an emitter
that sends an ultrasonic wave and several receivers that receive the echo of
the sound wave. The time difference of arrival (TDOA) of the signal between two
receivers is then used to estimate the difference of distances between the
object and the receivers. Having at least 3 receivers usually allows to
determine the location of the object. However errors on the sound velocty,
frequency of the sound wave, respective location of the
receivers
and measurements errors on the TDOA, influence the result of the localisation.
In a first part we will illustrate the influence of these parameters while in the second part we will
show that interval analysis allows one to determine the possible locations of
the receivers to ensure that localisation error is less than a given threshold
over a given region. These locations are provided as a region, thereby allowing
to determine the possible tolerance on the receiver placement..
3) Emmanuel Bénazéra, Louise TraveMassuyes,, LASS, Toulouse
Title: SetTheoretic Estimation of Hybrid System
Configurations. Slides.
Abstract: Hybrid systems serve as a powerful modelling paradigm
for representing complex continuous controlled systems that exhibit discrete
switches in their dynamics. The system and the models of the system are non
deterministic due to operation in uncertain environment. Bayesian belief update
approaches to stochastic hybrid system state estimation face a blow up in the
number of state estimates. Therefore most popular techniques try to maintain an
approximation of the true belief state either by sampling or by maintaining a
limited number of trajectories. It appears these limitations can be avoided by
using bounded intervals to represent the state uncertainty. As a consequence,
the true system state can be captured by a finite number of hybrid
configurations. A set of dedicated algorithms is detailed that can compute
these configurations efficiently. Results are presented on two systems of the
hybrid system literature.
4)
Nacim Ramdani, INRIA SophiaAntipolis
Mediterranee and LIRMM UMR 5506, CNRS UM2
Title: Reachability of uncertain nonlinear systems
using a nonlinear hybridization. Slides.
Abstract: we investigate nonlinear reachability computation in presence of model uncertainty, via guaranteed set integration. We show how this can be done by using the classical Muller's existence theorem. The core idea developed is to no longer deal with whole sets but to derive instead two nonlinear dynamical systems which involve no model uncertainty and which bracket in a guaranteed way the space reachable by the original uncertain system. We give a rule for building the bracketing systems. In the general case, the bracketing systems obtained are only piecewise Ckcontinuously differential nonlinear systems and hence can naturally be modeled with hybrid automata. We show how to derive the hybrid model and how to address mode switching. An example is given with a biological process
5)
Farah Mourad, Fahed Abdallah, Hichem
Snoussi, Cédric Richard, UTC, LM2S, UTT.
Title: Guaranteed boxed localization in MANETs by
interval analysis and constraints propagation techniques. Slides.
Abstract: In this contribution, we propose an original
algorithm for selflocalization in mobile adhoc networks. The proposed technique,
based on interval analysis, is suited to the limited computational and memory
resources of mobile nodes. The incertitude about the estimated position of each
node is propagated in an interval form. The propagation is based on a state
space model and formulated by a constraints satisfaction problem. Observations
errors as well as anchor nodes imperfections are taken into account in a simple
and computationalconsistent way. A
simple Waltz algorithm is then applied in order to contract the solution, yielding
a guaranteed and robust online estimation of the mobile node's position.
Simulation results on mobile node group
trajectories corroborate the efficiency of the proposed technique and
show that it compares favorably to
particle filtering methods.
6) Luc Jaulin and Gilles Chabert, ENSIETA.
Title: Resolution of nonlinear interval problems
using symbolic interval arithmetic. Slides.
Abstract: An intervalvalued problem is a
problem where the unknown variables take interval values. Such a problem can be
defined by interval constraints, such as "the interval X=[a,b] is a subset
of X*X". Interval valued problems often appear when we want to analyze a
priori the behaviour of an interval solver. To solve intervalvalued problems,
we propose to transform the constraints on intervals into constraints on their
bounds. For instance, the previous interval constraint (X is a subset of X*X)
can be transformed into the following bound constraint "a>min(a*a,a*b,b*b)
and b<max(a*a,a*b,b*b)". Classical interval solvers can then be used to
solve the resulting bound constraints. The procedure which transforms interval
constraints into equivalent bound constraints can be facilitated by using
symbolic interval arithmetic. While classical intervals can be defined as a
pair of two real numbers, symbolic intervals can be defined as a pair of two
symbolic expressions. An arithmetic similar to classical interval arithmetic
can be defined for symbolic intervals. The approach will be motivated by
several examples related to estimation and experimental design.
7) Gilles Chabert and Luc Jaulin, DTN, ENSIETA.
Title: : Interval and Boolean constraint
propagation for simultaneous localization and map building. Slides.
Abstract.
The SLAM (Simultaneous localization and map building) problem asks if it is
possible for an autonomous robot to move in an unknown environment and build a
map of this environment while simultaneously using this map to compute its
location.
During
the talk, it will be shown that, when the landmarks are identical and when
outliers occur, the general SLAM problem can be cast into a constraint
propagation problem (CSP) where Boolean and numerical variables occur. The
corresponding CSP is nonlinear and classical nonlinear methods have some
difficulties to deal with this type of problems in a reliable way.
A
basic interval constraint propagation algorithm to solve the CSP will be
proposed.
The
efficiency of the approach will be illustrated on a twohour experiment where
an actual underwater robot is involved. This fourmeter long robot build by the
GESMA (Groupe d’étude sousmarine de l’Atlantique) is equipped with
many sensors (such as sonars, LochDoppler, gyrometers, …) which provide
the data.
8) Nicolas Delanoue, LISA, Angers.
Title: : Attraction domain of a nonlinear
system using interval analysis. Slides.
Abstract : Consider a given dynamical
system, described by \dot{x} = f (x) (where f is a nonlinear function) and [x0]
a subset of R^n. We present an algorithm, based on interval analysis, able to
show that there exists a unique equilibrium state x^\infty \in [x0] which is
asymptotically stable. The effective method also provides a set [x] (subset of
[x0 ]) which is included
in
the attraction domain of x^\infty. In a second time, the flow of the ordinary
differential equation \dot{x} = f (x) is discretized, inclusion methods are combined
with graph theory to compute a set which is included in the attraction domain
of x^\infty.
9) Olivier STRAUSS,
LIRMM UMR CNRS Univ.
Title: Imprecise estimation in signal
processing. Slides.
Abstract: A wide range of digital analysis
and signal processing procedures inherently rely on methods for reconstructing
a continuous underlying signal from a set of sampled corrupted values. These
values are usually uniformly sampled, since the measures come from systematic
observations. Kernels are essential tools in this context, since they are used
in reconstruction, impulse response modeling, resampling, interpolation, linear
or nonlinear transformations, stochastic or bandpass filtering, etc. In
digital signal processing, kernels are mainly used to derive discrete
algorithms from a continuous representation. Within most applications, a kernel
can be seen as a weighted neighborhood ensuring a smooth interplay between
continuous and discrete domains. They can be visualized as bumps that can be
shifted to any location of the signal domain, so as to absorb or spread the
information contained in the signal. They are often bounded, monomodal and
symmetric. Digital signal derivation is a typical example of such an application.
The classical finite differences method usually fails to perform the estimation
of the derivative of the signal, especially with a noisy signal. The
kernelbased method consists of computing the sampled derivative of an
estimation of the continuous signal. This estimation is obtained by convolving
the original discrete signal with a continuous kernel, chosen to lower the
impact of both acquisition noise and quantization effect. The implementation of
such a method simply consists
of
convolving the original sampled signal with the derivative of the chosen
kernel, which is also a kernel. Most of the kernels used in signal processing
are summative kernels, or linear combinations of summative kernels. A summative
kernel is a positive function, the integral of which equals
set
will lead to an intervalvalued response. In this talk, we propose to extend
the conventional expectation operator to the different nonadditive confidence
measures induced by using
maxitive
kernels instead of summative kernels. This shift from summative to maxitive
kernel aims at accounting for an illknowledge on the proper kernel to be used.
10) Ignacio Araya, Bertrand Neveu, Gilles Trombettoni.
Title: Exploiting Common Subexpressions in Numerical CSPs. Slides.
Abstract: It is acknowledged that the symbolic form of the equations is crucial for intervalbased solving techniques to efficiently handle systems
of (in)equations over the reals. However, only a few automatic transformations of the system have been proposed so far. Common Subexpression
Elimination (CSE) is the main feature in code optimization. In interval analysis, Vu, Schichl, SamHaroud, Neumaier exploit common subexpressions
by transforming the (in)equation system into a unique directed acyclic graph. They claim that the impact of common subexpressions on the gain in
CPU time would be only due to a reduction in the number of operations. This paper brings two main contributions. First, due to properties related
to interval arithmetics, we prove theoretically and experimentally that exploiting certain common subexpressions of equations not only reduces the
number of operations, but also brings additional filtering/contraction during propagation. Second, contrarily to existing approaches and based on a
better exploitation of nary plus and times operators, we propose a new algorithm ICSE that identifies and exploits ALL the ``useful'' common
subexpressions. We show on a sample of benchmarks that ICSE detects more useful common subexpressions than traditional approaches and leads
generally to significant gains in performance (of sometimes several orders of magnitude).
11) Gilles Trombettoni, Gilles Chabert.
Title: Constructive Interval Disjunction:
past, present, and future. Slides.
Abstract: We present two new filtering/contraction operators for numerical CSPs (systems with constraints over the reals) based on constructive disjunction,
as well as a new splitting heuristic. The fist operator (CID} is a generic algorithm enforcing constructive disjunction with intervals. The second
one (3BCID) is a hybrid algorithm mixing constructive disjunction and shaving, another technique already used with numerical CSPs through
the algorithm 3B. Finally, the splitting strategy learns from the CID filtering step the next variable to be split, with no overhead. Experiments
have been conducted with 20 benchmarks. On several benchmarks, CID and 3BCID produce a gain in performance of orders of magnitude
over a standard strategy. CID compares advantageously to the 3B operator while being simpler to implement. The last part of the talk describes
the most recent improvements and ideas to provide an adaptive version of CID.
12) Ned Nedialkov,
Dept. of Computing and Software,
Title: Interval Tools for ODEs and DAEs. Slides.
Abstract: An interval method for initial
value problems (IVPs) in ordinary differential equations (ODEs) has two
important advantages over
approximate
ODE methods: when it computes a solution to an ODE problem, it (1) proves that
a unique solution exists and (2) produces rigorous
bounds
that are guaranteed to contain it. Such bounds can be used to ensure that this
solution satisfies a condition in a safetycritical calculation,
to
help prove a theoretical result, or simply to verify the accuracy and
reliability of the results produced by an approximate ODE solver.
We
overview interval methods and software for IVP ODEs, discuss applications of
these methods, and present the author's interval ODE
solver
VNODELP.
Computing
rigorous bounds in IVPs for differentialalgebraic equations (DAEs) is a
substantially more challenging task than in ODEs.
A
promising approach is Pryce's structural analysis combined with
13) Guillaume Verger,
LIRMM,
Title: CSPs and Quantified CSPs. Slides.
Abstract: Constraint satisfaction has been
studied from the seventies (Montanary
1974,
Waltz 1975). Constraint programming has been applied in numerous domains such
as
computer
graphics, natural language, optimization problems, molecular biology etc...
I
will present a state of the art of techniques used in Constraint programming,
in the case of
finite discrete domains. These techniques concern constraint propagation algorithms
(Forward
Checking,ArcConsistency, Path ArcConsistency, Singleton ArcConsistency...)
and
search
techniques (Backtrack, BackJumping, Variable and Value Ordering
heuristics...).
Since 1995, interest in the Quantified Constraint Satisfaction Problem
is
growning. The QCSP can be seen as an extension of CSP in which variables are
either
existentially
or universally quantified. I will present the different techniques used to
solve such
problems,
some of them are adapted from techniques used in quantified Boolean formulas
(QBF),
the
quantified extension of SAT.
14) Jamila SamHaroud, Artificial Intelligence Laboratory,
Title: Enhancing numerical constraint
propagation using multiple inclusion representations. Slides.
Abstract:
Building tight and conservative enclosures of the solution set is of
crucial
importance in the design of efficient
complete solvers for
numerical
constraint satisfaction problems (NCSP).
The
talk will present an generic algorithm
enabling the cooperative
use,
during constraint propagation, of multiple enclosure techniques.
This
allows bringing into the constraint propagation framework the
strength
of techniques coming from different areas such as interval arithmetic, affine
arithmetic
or mathematical programming. Some experiments illustrating
the
potential of the approach will be reported.
15) Sébastien LENGAGNE, Nacim RAMDANI, Philippe FRAISSE (LIRMM / INRIA)
Title: Application of Interval Analysis :
safe path planning. Slides.
Abstract: In humanoid robotics field, path
planning is used to compute offline,
a
database of complex motions. Path planning problem is to find the
motion
that satisfies a set of equality and inequality constraints and
minimizes
a cost function. Some inequality constraints must be validated
over
the whole motion duration. Nevertheless optimization algorithms
deal
with a finite number of constraints. That is why, the
discretization
of the inequality constraints is necessary.
Usual
path planning methods use a timepoint discretization, that
computes
the values of the constraints over a timegrid. This will produce
a
solution that satisfies the constraints for the considered timepoint.
However
no information is given about the constraint validity elsewhere.
So,
we present the safe path planning method that uses classical
optimization
algorithms, but replaces the timepoint discretization of
the
constraint with a timeinterval discretization.
Experimental
results concerning the 2D model with 6 degrees of freedom
of
the HOAP3 humanoid robot prove that the classical path planning method
allows
some constraint violations, whereas our safe path planning method
generates
motions which ensure the balance and the integrity of the robot.
16) Olivier Bouissou, CEA Saclay  DRT/DTSI/SOL/MeASI GifsurYvette
Title: Using guaranteed integration of ODEs for the
verification of embedded software. Slides.
Abstract: In this talk, I will give an insight of why
we must take into account
the
physical environment of embedded programs when we deal with
their
verification. This implies the analysis of a wider system that
contains
two parts : a discrete program and a continuous
environment.
If the analysis of the discrete part is well studied, the
verification
of the continuous part and of the interactions between
both is
more recent. In this talk, I will focus on the analysis of
the
continuous subsystem using guaranteed integration techniques,
i.e. tools that
compute verified bounds on the solution of ODEs using
interval
methods. In particular, I will present a new algorithm for
this that
is based on a classical RungeKutta order 5 numerical method
for which
exact error bounds are computed. I will also present
experimental
results that compare this approach with the tool VNODE.
17)
Title: Computing the range of real eigenvalues of an
interval matrix. Slides.
Abstract: We propose an algorithm for computing the set
of eigenvalues of an
interval
matrix. It is based on the branchandprune method and exhibits
diverse
interval analysis techniques including regularity tests, among
others.
Particularly, the JanssonRohn algorithm turned out to be very
efficient
recognizing outer boxes, and its modification was developed to
recognize
inner boxes as well. Our approach enables not only to compute
the range
of eigenvalues with a given accuracy, but in the most of cases
we are
able to find out in reasonable time the exact boundary points.
Finally,
we demonstrate efficiency of the proposed method on a series of
numerical
experiments.
18) Maëlenn AUFRAY
(ENSIACET 
Title:
S.I.V.I.A. Applied to Dielectric Spectroscopy: a Guaranteed Parameter
Estimation
Using Interval Analysis. Slides.
Abstract: Dielectric spectra of materials are often
difficult to analyze since
the common
software algorithms and line shape functions do not always
provide
unambiguous data for the fitted parameters. In this
presentation,
a software, based on a global optimization algorithm which
uses
interval analysis, is presented. Taking into account the
experimental
error of each data point in the measured dielectric
spectrum,
the sofware provides a confidence interval for every parameter
of the
dielectric function implemented in the software. This is
demonstrated
for an epoxy monomer with a sum of Debye relaxators as
dielectric
line shape function. This software is also able to deliver
and
guarantee the number of relaxation processes even if they are in
part masked
by other phenomena like conductivity or electrode
polarization.
19)
Philippe
Langlois (DALI, Univ. Perpignan) and Nicolas
Louvet (Arénaire, INRIA and ENS Lyon)
Title: Accurate results with compensated algorithms
and validated error bounds in floating point computation: principles and
applications. Slides.
Abstract: In this talk we focus the reliability of floating point computation
where rounding errors worsen the significance of computed results
(data error is not here considered). Compensated algorithms allow us
to compute very accurate results of some classic algorithms (summation,
inner product, polynomial evaluation, triangular linear solution)
within a very realistic runningtime overcost. It is classic to
also compute a dynamic bound of the remaining error of
these accurate solutions. This a posteriori bound is validated
(whereas computed in floating point arithmetic), its overcost is
reasonable and the bound can be thin enough to decide, for example,
whether a computed result is a faithful rounding of the exact value.
We present some of these points being motivate by the following
question we ask to the interval arithmetic expert audience: when
these accurate, validated and fast algorithms can be an alternative to
interval arithmetic?
20) Jorge Florez, Mateu
Sbert, Josep Vehí (Mice, Girona,
Title: Improving ray tracing of implicit surfaces
using interval arithmetic. Slides.
Abstract: Ray tracing of implicit surfaces suffers of
reliability problems when
surfaces having
thin parts are rendered: such features are missed and not
appear in
the final image. Those problems are caused for the truncation
performed
in the floatingpoint representation in the computers.
Many
authors have used Interval Arithmetic in the intersection test to
perform a
guaranteed ray tracing. However, there are still two open
problems
in the ray tracing of implicit surfaces based on Interval
Arithmetic:
a) Ray tracing is slow, but Interval Arithmetic is slow too
which
reduces the efficiency in all the process. b) Interval Arithmetic
has been
applied to solve point sampling problems in the intersection
tests.
But the application of Interval Arithmetic to solve problems
related
with point sampling in the distribution of the rays inside the
pixel,
have not been studied. In this work, some algorithms to deal with
those
problems are presented.
21) Nathalie Revol (INRIA, Univ. Lyon, LIP,
Title:
Survey of Proposals for the Standardization of Interval Arithmetic. Slides.
Abstract: The
context of this work is the recent approval by IEEE of a committee
on the
standardization of interval arithmetic (bearing the number P1788).
A first
task of this working group will be to review existing proposals
and
points of view on interval arithmetic.
In this
talk, the need for a total and closed system is motivated. The
approach
adopted here is not to give a commented list of proposals.
Instead,
different points of view will be presented. A first convenient
distinction
consists in considering algorithms for constraint programming:
two
operating modes are adopted, the "forward" and the
"backward" modes,
and they
give rise to two different families of definitions.
Another
distinction between the existing definitions of interval arithmetic
concerns
the considered underlying set: either the reals, or extended reals
(ie.
reals and the infinities).
Then,
links with floatingpoint arithmetic, for implementation issues,
will be
discussed. Finally, a tentative list of operators will be given.
1)
How could I register the MEA group?
If you understand French, you can enter
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