What is your research field?
My research field lies at the intersection of modelling, analysis and scientific computing.
The problems I am more particularly interested in originate from the engineering sciences and very often from materials science. I study the modelling of materials at all scales. This covers their electronic structure, their mesoscopic properties and their mechanical properties. I study the mathematical well-posedness of the models (existence, uniqueness, regularity of the solution of the differential equations - most of the latter being partial differential equations, and also, increasingly frequently, differential equations with random terms). I also seek to understand how the models are related to one another, namely how a model at a given scale may be obtained as the limit of another model at a finer scale (for example using the so-called homogenization techniques). One prototypical problem is to understand mathematically and simulate numerically how the micro-structures of a composite material and the local properties of its constituent phases affect its global mechanical behavior.
I also try to understand the various approaches to simulate the solutions numerically, to make these approaches more efficient, and possibly develop alternative, more accurate and/or faster approaches.
The mathematical ingredients in my works come from analysis in all its forms (functional analysis, PDE analysis, calculus of variations, applied probability, numerical analysis...), and techniques of scientific computing (discretization approaches such as finite elements for example, etc).
To a certain extent, my affiliation as a researcher at Ecole des Ponts and as the scientific leader of a project at Inria is a symbol of this activity at the interface between disciplines.
Mathematical sciences, even when they are characterized as "applied", are fundamental. What does this statement mean to you?
I think that what characterizes mathematics as a field is the adamant concern for rigor. For example, in most applied engineering disciplines, impressive numerical simulations are nowadays performed. The point is however not to simulate. It is to guarantee that the result of the simulation is, within a quantified margin, close to the actual solution. It is up to the physicist to guarantee that the model closely represents the physical phenomenon. But the certification that the result of the simulation is close to the solution of the model is the responsibility of the applied mathematician. It may only be obtained using a mathematical approach, combining a theoretical analysis of the mathematical nature of the model and a numerical analysis of the algorithms implemented to approximate its solution.
The mathematician motivated by applications must therefore adopt a pragmatic and creative approach in terms of methods, embrace each particular case, and obtain a solution to the specific problem considered. The approach is thus close to that of an engineer or of an expert in an applied science. On the other hand, the mathematician must also look for the generality beyond the particular case, formulate the problem in an abstract form in order to solve it, prove general theorems on the objects he is dealing with. This makes the applied mathematician an actor in fundamental science. The breakthroughs in the applied sciences are indeed often obtained by a new, transverse, if not iconoclastic perspective, and much less on incremental improvements of the state of the art on a particular case.
In your opinion, what are the benefits of collaborative work?
The type of applied mathematics I like consists of the whole journey that leads from the problem to its numerical simulation. The stage where I personally feel the most useful is that of the mathematical formalization of a problem arising in the physical sciences. Once the formalization has been completed in the suitable terms, the mathematical work can either be relatively standard and involve well-established techniques (both theoretical and numerical), or alternately raise new questions requiring either to adapt classical approaches or to invent new ones.
Both stages, the mathematization stage and the resolution stage, are equally interesting. For the former, one must first understand the essence of the physical problem. To this end, it is crucial to have access to the right experts, with the right professional skills as physicists, mechanicians or chemists. These experts will educate you, teach you the existing models, their successes and their limitations. For example, they will tell you whether a given model poorly represents the physics or, in contrast, represents it satisfactorily but is too costly to simulate. The applied mathematician can certainly contribute to the common effort to improve the model in the case it is not accurate enough, but he has an instrumental role when the question is the simulation. An appealing aspect of the work is that one rarely knows beforehand what type of mathematical questions will be relevant. It is up to the mathematician to discover which techniques are required, and next to learn those. There again, the interaction between colleagues, mathematicians, is a key point. Identifying the appropriate tools, possibly coming from different mathematical areas, becomes key.
At a later stage, when the problem is formalized mathematically, well understood and now ready to be simulated, one should then discuss with experts in high performance computing (domain decomposition methods, parallelization techniques, data sciences, etc).
Given the variety and the technicality of contemporary scientific problems, there are fewer and fewer mathematicians capable of covering the whole spectrum of expertise necessary to efficiently solve a given problem. Combining various skills from different mathematically-related fields, all the way from the modeling of the physical problem to its numerical simulation, is the only possible successful approach.
Claude Le Bris is Civil Engineer-General, Research scientist at the Ecole Nationale des Ponts et Chaussées, Scientific leader of the MATHERIALS research team (formerly MICMAC project-team) on multiscale computational mechanics at INRIA.
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