Innovative Weak Galerkin Finite Element Methods with Application in Fluorescence Tomography

创新的弱伽辽金有限元方法在荧光断层扫描中的应用

• 批准号: 1905195
• 负责人: Chunmei Wang
• 金额: $ 1.45万
• 依托单位: Texas Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-10-05 至 2019-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1905195&HistoricalAwards=false
• 关键词: Innovative; Weak; Galerkin; Finite; Element

Fluorescence tomography (FT) is an emerging three-dimensional optical imaging modality that uses in vivo noninvasive depth-resolved localization and quantification of fluorescent-tagged inclusions. FT techniques have been extensively employed in early cancer detection, guidance of tumor resection, and drug monitoring and discovery. This research project aims to develop new numerical methods for computational problems arising in FT imaging. Besides fluorescence tomography, the numerical methods can be applied to solve partial differential equations that arise in various other disciplines. The models, theory, and computational methods under development in this project will be of great value in the fields of digital image processing, medical imaging, and numerical analysis, with direct applications in broad areas such as digital image and even construction industries. Students will be trained in this project.The goal of this research project is to develop and analyze efficient numerical methods -- weak Galerkin (WG) finite element methods -- to address challenges posed by fourth-order problems arising in fluorescence tomography theory. The research will explore algorithmic advancements, new convergence theory, and imaging technology improvement, by: (1) developing robust finite element methods for a fourth order partial differential equation in the primal variable formulation for which no existing method works; (2) establishing a stability and convergence, including superconvergence, theory for the newly developed finite element methods; (3) validating and verifying the methods through collaboration with domain-specific researchers; (4) analyzing a new weak Galerkin mixed finite element method for a fourth order partial differential equation; and (5) developing application-oriented software packages, tested and validated with collaborators in the area of FT.

荧光断层扫描（FT）是一种新兴的三维光学成像方式，使用体内无创深度分辨定位和定量荧光标记的内含物。FT技术已广泛应用于早期癌症检测、指导肿瘤切除、药物监测和发现等领域。本研究计划旨在发展新的数值方法，以解决傅立叶变换成像中出现的计算问题。除荧光层析成像外，数值方法还可用于求解其他学科中出现的偏微分方程。该项目所开发的模型、理论和计算方法在数字图像处理、医学成像和数值分析等领域具有重要价值，在数字图像甚至建筑行业等广泛领域具有直接应用价值。学生将在这个项目中接受培训。本研究项目的目标是开发和分析有效的数值方法-弱伽辽金（WG）有限元方法-以解决荧光层析成像理论中出现的四阶问题所带来的挑战。该研究将探索算法的进步，新的收敛理论和成像技术的改进，通过：(1)开发鲁棒的有限元方法来解决四阶偏微分方程在原始变量公式中没有现有方法有效；(2)为新发展的有限元方法建立了稳定性和收敛性，包括超收敛性理论；(3)通过与特定领域研究人员的合作，对方法进行验证和验证；(4)分析了求解四阶偏微分方程的一种新的弱Galerkin混合有限元法；(5)开发面向应用的软件包，并与FT领域的合作者进行测试和验证。