Functional Analysis and Computational Methods in Imaging, Materials, and Atmospheric Sciences

研究人员、他们的学生和合作者研究图像分析、材料科学、大气和气候建模中出现的数学公式和有效的计算技术。这项多学科研究结合了计算数学、逆问题、图像分析、界面和自由边界以及大气科学等领域。他们研究了使用卡通和纹理分解的图像恢复，随机点扩散函数存在下的图像恢复，隐式开放曲线演化及其在材料科学自由边界问题中的应用，以及多仪器获得的大气记录的变分数据融合。研究小组开发了新的变分方法，迭代和数值分析技术来解决这些逆问题。拟议的活动提供了有效的数学公式、成像方法和气候和材料科学应用之间的联系，在这些领域还没有尝试过类似的方法。特别是，合并多个传感器获得的数据的能力是我们了解地球气候系统的关键部分，因此，在对气候变化和气候影响进行预测时非常重要。目前的大气数据融合方法在很大程度上是临时的，建立一个坚实的数学基础和计算方法来组合重要的记录，可以提高它们的科学可信度，并进一步扩大科学分析的范围。研究者促进多学科的教学、培训和学习。数学专业的学生接触到广泛的主题和技术：(i)应用和计算数学，图像处理和分析；（ii）数学以外的主题，包括材料和大气科学。

The investigators, their students and collaborators study mathematical formulations and efficient computational techniques for applications arising in image analysis, materials science, and atmospheric and climate modeling. This multidisciplinary research combines areas of computational mathematics, inverse problems, image analysis, interfaces and free boundaries, and atmospheric sciences. They study image restoration using cartoon and texture decompositions, restoration of images in the presence of a stochastic point spread function, implicit open curve evolution and applications to free boundary problems in materials sciences, and variational data fusion of atmospheric records acquired by multiple instruments. The research team develops novel variational approaches, iterative and numerical analysis techniques for solving these inverse problems.The proposed activity provides the link between efficient mathematical formulations, imaging approaches, and applications in climate and materials sciences, where similar approaches have not yet been attempted. In particular, capability for merging data acquired by multiple sensors is a key part to our understanding of the Earth's climate system, and therefore, is of importance when making projections about climate change and climate impacts. Current atmospheric data fusion methods are largely ad hoc and establishing a firm mathematical foundation and computational methods for combining important records enhances their scientific credibility and further a wide range of scientific analyses. The investigators promote multidisciplinary teaching, training and learning. Mathematics students are exposed to a broad range of topics and techniques: (i) in applied and computational mathematics, image processing and analysis; and (ii) topics outside mathematics, including materials and atmospheric sciences.