Innovative Weak Galerkin Finite Element Methods with Application in Fluorescence Tomography

荧光断层成像(FT)是一种新兴的三维光学成像方式，它使用体内荧光标记的包裹体的非侵入性深度分辨定位和定量。FT技术已广泛应用于癌症早期检测、肿瘤切除指导、药物监测和发现等领域。该研究项目旨在为傅立叶变换成像中的计算问题开发新的数值方法。除了荧光层析成像外，数值方法还可以用来求解其他学科中出现的偏微分方程组。该项目正在开发的模型、理论和计算方法将在数字图像处理、医学成像和数值分析领域具有重要价值，并将直接应用于数字图像甚至建筑行业等广泛领域。学生将在这个项目中接受培训。这个研究项目的目标是开发和分析有效的数值方法--弱Galerkin(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带来的挑战。这项研究将探索算法的进步、新的收敛理论和成像技术的改进，具体包括：(1)开发求解四阶偏微分方程的稳健有限元方法，现有方法对其无效；(2)为新开发的有限元方法建立稳定和收敛的、包括超收敛的理论；(3)通过与特定领域的研究人员合作来验证和验证方法；(4)分析四阶偏微分方程的一种新的弱Galerkin混合有限元方法；以及(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。

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.