What are the objectives of our program "Robust Mathematical Modeling" ?

There are three objectives :

1. To understand better what are the needs of the users

As we said earlier, what the users want is often unclear, even for themselves. For instance, a short term economy may result in a loss in the long term. Forgetting to take into account elements such as human factors, recruitments, formation, and so on, may result in very poor resuts : a real life problem does not only involve figures, data, statistics ; it also involves men and women.

So, a better understanding of the users' needs can be achieved only by direct contact : reading is not enough.

We, at SCM, understood that long ago, and we keep organizing seminars. The most recent one, organized jointly with EdF, was held as a commemoration of our tenth anniversary. It gave us the opportunity to invite numerous actors in very various fields ; all of them talked about the weaknesses of present day models, in their respective domains.

Kent State University, starting Fall 2005, will organize a regular colloquium, where representatives of various companies or agencies will speak. This is a unique opportunity to have direct access to the true needs of these companies and agencies. This colloquium will meet once a month. The colloquium chairman is Prof. Olaf P. Stackelberg.

Both in Kent and in Paris, the contents of each conference will be made available, as a short summary on line, so that we can share the experiences.

2. Analyze and criticize the existing models

A very useful task is to review the existing models, in each domain, and ask the question : are they robust ? We plan to conduct a general survey, domain by domain, and see what the strengths and weaknesses are. Do these models handle correctly the imprecisions upon the data, the laws, the objectives ?

At present, SCM, in a joint contract with Bertin Technologies and Prof. Lucien Abenhaim, conducts a critical review of mathematical models used in epidemiology. The need was expressed by the French Ministère de la Défense, Centre d'Analyse de Défense.

Another example is species cohabitation. Click here to download this example (pdf format).

(other examples to be inserted)

3. To construct new robust models

The third task is of course to construct robust models. This can be done only in good coordination with the users' needs, so as to make sure they actually fit their needs. The key concept is of course the validation of the model : it should work in practice and bring results.

In application of a contract with Framatome-ANP, SCM has built a new concept, called « Experimental Probabilistic Hypersurface ». Indeed, any physical experience provides only a small number of results, whereas a large number of parameters may vary. What is the “value” of the information obtained this way, if, for instance, we have 300 measures, where the whole experiment may involve 50 parameters, thus leading to 10E50 possible states, if each parameter can take 10 values? Can we, from this very limited amount of information, predict the result of a new experience ? The Experimental Probabilistic Hypersurface allows us to represent the information obtained from any number of measures, in a physical experiment or in a computational code. This information is stored as a density of probability, « above » each point in the configuration space.

The applications are multiple. The EPH is a « storage » of information, which grows and becomes more precise when more and more experiments are performed. It allows you to get immediately « local » results : which regions or points are dangerous, which are safe, and so on. The EPH is intended to replace both the deterministic methods (for instance interpolation between existing values), which are artificial, and the statistical methods, which are only global. The EPH gives local results, but still keeps the global characteristics.

A cooperation with EdF is also developed upon these matters.

A theoretical description of the EPH can be obtained (pdf format) by clicking here.

Another example in which we seek to build a robust model is the evolution of opinions : click here. It will be described in detail in the fourth chapter (robust4.htm).

(more references and more papers to be added here)

To see the basic rule of real-life mathematics, click here

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