Mathematical methods and models for complex engineering problems

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Our research program focuses on the development of realistic models and computer algorithms for the solution of complex engineering problems. The applied problems that we study share the following complexity attributes: high-dimensionality, nonlinearity and modeling difficulties. We describe three parts of this program.******"ARTEFACT-FREE" NOISE REDUCTION METHODS FOR IMAGES: most noise reduction methods introduce artefacts that become visible if one looks at the residual image, which is the difference between the original noisy image and the denoised image. We propose to develop better methods that exploit a fine local analysis of the image and a global optimization strategy. This work should lead to both quantitatively and qualitatively superior noise reduction performance in signal processing, which in turn should have a major impact on the applications for which the image estimate accuracy is crucial, e.g. medical imaging.******OPTIMAL AIRCRAFT TRAJECTORIES: The cost-index is a parameter that controls flight time, which determines both fuel cost and late-arrival penalties (caused e.g. by passengers that miss their connecting flights). A precise estimation of the optimal cost-index is very important for economical and environmental reasons. We propose to develop a fast and precise cost-index estimation method. Our approach will be to use our flight trajectory optimization system with a subset of the trajectories that links departure and arrival points. A more accurate estimate of the cost-index and a better management of the fuel reserve for aircrafts should have a direct and significant impact on both fuel consumption and carbon dioxyde emissions, which have major economic and environmental consequences.******CLASSICAL AND GEOMETRICAL METHODS FOR DIFFERENTIAL EQUATIONS: Analytical solutions, even approximate, are much more useful to engineers than numerical solutions because they provide an understanding of the phenomenon as a function of its parameters. I will study a hydrogeology problem related to aquifer testing, which is a set of methods that are used to characterize groundwater. One such test is the pumping test, which consists in pumping water from a well and measuring the water level in the pumping well and observation wells as a function of time. I will study the solution of a pumping test for unconfined aquifers. No complete solution is known for unconfined aquifers, which are frequently used for drinking water supply and are the most vulnerable to contamination. Analytic solutions of their transient responses should help to achieve more reliable assessment of the resources and more effective pump-and-treat systems for contamination cases.