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Clearly, the community of the computational and applied mathematicians has always taken advantage of the contributions of their more theoretical colleagues:
1) As a source of scientific problems worth to investigate; this is particularly true in the area of partial differential equations to which this workshop will be specifically dedicated.
2) As a source of results and methods allowing the analysis and justification of the computational methods employed for the solution of a given problem (proving the convergence of an iterative method, or deriving a priori or a posteriori error estimates, require most often the use of sophisticated mathematics). But the contribution of (non- computational) mathematicians goes much beyond providing a solid foundation to the work of their more applied colleagues. Indeed, quite often techniques used to prove mathematical results have lead to efficient computational techniques; examples in that direction abound, striking ones being the method of Galerkin and the mountain pass theorem, which were both created to prove the existence of solutions to partial differential equations. Actually, these methods have lead to powerful computational techniques (such as finite elements) allowing the numerical solution of difficult problems, many of them related to important applications in the Natural and Engineering Sciences (Health Sciences included).
Fortunately, the flux of knowledge is also going the other way around: We can claim that computers have made mathematics an experimental science. This was even true two centuries ago, when Gauss (a kind of human computer himself) conjectured the asymptotic prime number distribution on the basis of a very large list of prime numbers he had obtained by hand calculations (the conjecture was proved almost a century later by J. Hadamard and Ch. de la Vallée-Poussin). More recently computers have played an important role by helping to prove or disprove theorems and conjectures (we think of the computer assisted proof of the four color theorem and of a conjecture of J.P. Serre concerning the non-existence of integer solutions to some Diophantine equation; using a quasi-Newton algorithm, a Paris 7 mathematician found an integer solution to Serre's problem). Concerning the area of Partial Differential Equations we know of several instances where the analysis was driven by computational results. Let us mention two examples: The first one (mentioned by Peter Lax in his invited lecture at ICIAM 1987, in Paris) concerns the proof, by H. Brezis and J.M. Coron, of some properties of the solutions of a nonlinear elliptic equation from high energy physics; Brezis and Coron proved these properties, after computations, performed on their request by R. Glowinski, were suggesting that the properties they wanted to prove were indeed verified by the computed solutions. The second example involves again Brezis and Glowinski: while solving numerically a nonlinear partial differential equation from continuum mechanics, Glowinski observed that the solutions were smoother than expected; he mentioned this result to Brezis, who was able to prove that the regularity property observed in a particular case was in fact general.
Back to the workshop, it will focus on nonlinear elliptic partial differential equations, a topic with many ramifications and motivations from Physics, Mechanics and Differential Geometry. Analysts will interact with Computational and Applied Mathematicians through formal presentations, one or two round tables and free discussions. Some successful interactions of that type already exist, the idea being to create more, revisiting old topics and trying to identify new ones. If successful, this kind of experience will be repeated.
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