Société de Calcul Mathématique, SA




Robust mathematical modeling

 

A joint Research Program with several Companies, Institutions and Universities, since 2005

 

The Robust Mathematical Modeling program is rather ambitious. Indeed, it has two aims: The first one is scientific, the second one deals with organization.

  1. Scientific

    To develop tools (both algorithms and software) which can handle, at the conceptual level, the three difficulties that are usually met in any real world project:

    • The data are missing or corrupted;
    • The laws describing the phenomena are not completely known;
    • The objectives are multiple and contradictory.

    These remarks apply to industrial programs, as well as to insurance, banking, finance, epidemiology, and so on. They apply to every real-life situation, that is any situation which is not of academic nature. They apply in fact to the laws of nature as well, as we will see below.

    Let's take some examples:

    • You need to deliver some goods, but some of the trucks are late, or under repair;
    • You want to foresee the spreading of a disease, but the law of propagation is uncertain;
    • You want to hire some people, but you cannot define correctly your objectives: of course, young people are cheaper, but the older ones have useful knowledge...

    The solutions brought by academic tools consist in ignoring these difficulties: one choses a specific situation, constructs a precise algorithm, then a precise software, which gives a precise answer to a precise question, often after several hours of computation. Then the question comes: what if the precise initial data are not the correct ones ? One has to start the computation all over again, with other data.

    The main danger of a precise algorithm is the impression of security it gives. Since we have a precise answer, we have the impression it is correct. We hide this way the fact that the three main components (data, laws, objectives) have been artificially chosen. This is dangerous, if our tool is to be used by someone who has to take a decision.

    Scientific approach of the RMM program

    Our approach is culturally quite different from the usual patterns. In real life problems, people are aware that data are insufficient and imprecise and they are generally aware of the fact that their problem can be put in various forms. So what they want is in fact a Quick Acceptable Solution (QAS), not an optimum.

    • Acceptable means that is satisfies their constraints or, more precisely, rough versions of their constraints;
    • Quick means that the computer should give a solution within minutes, not hours.

    When a Quick Acceptable Solution has been found, you can either:

    • be satisfied with it ;
    • or use it as a starting point for a QAS of a refined problem.

    Let's take an example, which comes from EDF (French Electricity). Everyday, you have to decide which plants will be used to produce electricity. There are about 200 plants, with various capacities, and extremely complex constraints: not all productions are possible in a given range by each plant, maintenance should be taken into account, and so on. The demand in electricity, depending on the weather conditions, should be satisfied, with minimal cost. So you have an extremely complicated optimization problem, which the existing software poorly solves within hours. A combinatorial approach is infeasible, since there are too many possibilities to examine.

    What EDF wants is in fact a coarse solution, a Quick Acceptable Solution, of the following type: is there a configuration of the plants which gives a cost smaller than a given cost (for instance last year's cost) ? If yes, find it quickly. We might later refine it, by diminishing the cost or adding more constraints. If not, we increase the cost, and we ask for a new QAS.

    In the search of a QAS, there is no optimization any more: everything is treated as a constraint. We ask the computer to find the first solution satisfying our constraints, the first and not the best. So this is usually very quick.

    Guidelines for the search of a QAS

    • The constraints must be simplified, in order to take into account the imprecision. They are taken to be piecewise linear, with coarse (simple) values;
    • The number of constraints must be reduced to the essential ones, at least at the beginning;
    • The optimization request disappears, as we said earlier, and is itself treated as a coarse constraint;
    • All unknown data or imprecise data are treated using probability laws.

    Let's give an example of such an approach. In 2004-2005, we worked for the CNES (French "Centre National d'Etudes Spatiales"). The question was: where will fall the debris coming from the disintegration of a satellite which reenters into the atmosphere ? Our answer was a "probabilistic map" : we divided the land into squares (for example of 1km x 1 km) and for each of them we indicated the probability to receive a debris. All data in the problem were treated by means of probability laws. For instance, air density at various heights is not precisely known, so it was considered as a random variable, following a uniform distribution between some bounds. The same for the weight of the debris, their air resistance, and so on.

    This use of probability laws is quite appropriate in many situations where risks analysis is requested, because the Safety Authorities do not accept a precise answer: they want a confidence interval. You say that the temperature will not be over 1000C, but can you compute the probability to go over 950C ?

    In order to answer this question, asked in France to Framatome-ANP by the Safety Authorities, SCM developed a specific tool, the "Experimental Probabilist Hypersurface", which later became the topic of the book "Probabilistic Information Transfer", by Olga Zeydina and Bernard Beauzamy. This is an example of a tool developed in the frame of the Robust Mathematical Modeling Program.

    What Nature does

    Quite obviously, in the development or evolution of species, Nature never looks for an "optimum". For instance, the shape of the pavilion of the ear, in the human race, differs from one person to the next, though this pavilion has a technical objective (to collect the sounds). This is true the same way for all organs. Nature builds each of them within some limits, considered as "appropriate", but keeps some variety within these limits. So we see that the result is not a consequence of an optimization process, which would bring a single result, identical in all cases.

    The reason why Nature does not bring a unique solution, but a large variety, is probably that a single solution might become inappropriate, if the conditions were to change. Instead, from the diversity of existing solutions, more modifications, variations, adaptations, may be sought. A large diversity of solutions is more robust than a single solution, even if this single solution seems to be optimal at a given time, under specific circumstances.

  2. Organization

    We also want to provide information about the real needs of the users. This is done by a series of colloquia, held in all participating institutions. The idea is to invite specific classes of users (for instance logistics, epidemiology, garbage collecting, and so on) and ask them to talk about the problems they meet in their professions. When we know these problems, we then build corresponding answers (academia usually does the converse : they build tools first, hoping that they will find a problem for the tool later). See SCM's archives in order to see the seminars which were previously organized and download some of the lectures.

Of course, in order to satisfy these two aims, the RMM program establishes contacts between the members, allows exchanges of students and researchers, visits from one group to another, and so on.

The RMM program also provides room for publication: papers are posted and may be criticized. The papers are written in a way that makes them understandable outside the mathematical community. In particular, engineers may find an interest in reading the papers and the books published within the RMM program.

What is a model ? Click here.

Mathematical description of the objectives : Click here.

The basic rule of real-life mathematics : Click here.

Can we model everyday life preoccupations ? Click here.

Probabilistic methods : Click here.

The four steps or a RMM program : Click here.

List of participating people and institutions : Click here.

Ongoing events : Click here.

Documents to download : Click here.

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