Innovative Weak Galerkin Finite Element Methods with Application in Fluorescence Tomography

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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.