Mathematical methods and models for complex engineering problems

我们的研究计划侧重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属性：高维性，非线性和建模困难。我们描述了这个程序的三个部分。用于噪声的“无伪像”噪声降低方法：大多数噪声降低方法引入伪像，如果观察残差图像，则伪像变得可见，残差图像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，利用精细的局部图像分析和全局优化策略。这项工作应导致在信号处理中的定量和定性的上级降噪性能，这反过来应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。最佳飞机轨迹：成本指数是控制飞行时间的参数，其确定燃料成本和迟到惩罚（例如由错过其转接航班的乘客引起的）。从经济和环境的角度来看，精确估计最佳成本指数是非常重要的。我们建议开发一个快速和精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与链接出发点和到达点的轨迹的子集。更准确地估算成本指数和更好地管理飞机的燃料储备，对燃料消耗和二氧化碳排放都有直接和重大的影响，这对经济和环境都有重大的影响。微分方程的经典和几何方法：解析解，即使是近似解，比数值解对工程师更有用，因为它们提供了对作为其参数的函数的现象的理解。我将研究与含水层测试有关的水文地质问题，含水层测试是一套用于描述地下水特征的方法。一种这样的测试是抽水测试，其包括从井中抽水并测量作为时间函数的抽水井和观测威尔斯中的水位。我将研究潜水含水层抽水试验的解决办法。对于无压含水层，目前还没有一个完整的解决办法，因为无压含水层经常用于饮用水供应，而且最容易受到污染。其瞬态响应的解析解应有助于实现更可靠的资源评估和更有效的泵和处理系统的污染情况。

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.