Société de Calcul Mathématique SA
Robust mathematical modeling -2-
We continue here the description of our joint Research Program with several Companies, Institutions and Universities.
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. We usually do it several times a year, on various subjects. Previous seminars are listed on our web site and the lectures are available for downloading.
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 ? Quite often, we are asked to conduct a general survey, for some domains, and see what the strengths and weaknesses are, for the existing models in these domains. Do these models handle correctly the imprecisions upon the data, the laws, the objectives ?
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", which was later developed in a series of contracts with IRSN (Institut de Radioprotection et de Sûreté Nucléaire). 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. See the book "Probabilistic Information Transfer", by Olga Zeydina and Bernard Beauzamy.
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.
Another example in which we seek to build a robust model is the evolution of opinions ; it will be described in detail in the fourth chapter (robust4.htm).
To see the basic rule of real-life mathematics, click here