Mathematical methods and models for complex engineering problems

我们的研究计划专注于开发现实模型和计算机算法，以解决复杂的工程问题。我们研究的应用问题具有以下复杂性：高维性、非线性和建模困难。我们将描述该程序的三个部分。图像的“无伪影”降噪方法：大多数降噪方法引入伪影，如果人们观察残差图像，这些伪影就会变得可见，这是原始噪声图像和去噪图像之间的差异。我们建议开发更好的方法，利用图像的精细局部分析和全局优化策略。这项工作应该在信号处理中产生在数量和质量上都更好的降噪性能，这反过来又应该对图像估计精度至关重要的应用产生重大影响，例如医学成像。OPTIMAL飞机轨迹：成本指数是控制飞行时间的参数，它确定燃料成本和延误惩罚(例如，由于乘客错过转机造成的)。出于经济和环境的原因，对最优成本指数的准确估计是非常重要的。我们建议开发一种快速而准确的成本指数估计方法。我们的方法将是使用我们的飞行轨迹优化系统，将出发和到达点连接起来的轨迹的子集。更准确地估计成本指数和更好地管理飞机的燃料储备应该会对燃料消耗和二氧化碳排放产生直接和重大的影响，这两个问题具有重大的经济和环境后果。微分方程式的类方法和几何方法：对工程师来说，解析解，即使是近似的，也比数值解有用得多，因为它们提供了对作为参数函数的现象的理解。我将研究一个与含水层测试有关的水文地质学问题，这是一套用于表征地下水的方法。一种这样的测试是抽水测试，它包括从井中抽水，并测量抽水井和观测井中的水位随时间的变化。我将研究无承压含水层抽水测试的解决方案。无承压含水层经常用于饮用水供应，最容易受到污染，目前尚无完整的解决方案。其瞬时响应的解析解应有助于实现对资源的更可靠的评估以及对污染情况的更有效的泵送和处理系统。

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.