High-order numerical methods for differential equations

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Computational methods underpin many aspects of science and modern technology. For example, climate and weather models rely on solving differential equations as does the design process of airplanes. In medicine, invaluable diagnostic information is provided by image reconstruction and processing, ultimately based upon mathematical algorithms. The overarching goal of my research program is to design, implement and analyze algorithms fundamental to applications. The research addresses the three tenets of numerical analysis: efficiency, accuracy and robustness of algorithms. The proposed research program involves two different areas. The first one is broadly concerned with design and analysis of computational methods for solving differential equations of the type arising in many scientific, engineering and other modelling applications, such as the modelling of electric circuits, electrochemical reactions or many problems in optimal control theory. To widen the application areas we will also consider mixed functional differential equations, for example, differential equations with a delay. The research is on high-order methods such as spectral methods and radial basis function methods. A high-order method is one which can approximate the solution of a problem to high accuracy with a relatively coarse discretization. One research focus is on adaptivity - improving efficiency and accuracy by choosing the computational mesh or various parameters adaptively, guided or dictated by features of the approximate solution - and sensitivity or robustness of the methods under investigation. The goal of this part of my research is to contribute to more accurate, stable and user-friendly methods for numerically solving differential equations. The second research area is computational medical imaging, in particular, methods for reconstructing static and dynamic images in single-photon emission computed tomography (SPECT) and magnetic resonance imaging (MRI). In static imaging, one single image is reconstructed from the data obtained during a patient scan; in the dynamic case, the same data are used to reconstruct a sequence of 3-D images (i.e., a 3D movie) to show the dynamic behaviour of uptake and wash-out of substances, indicating functional information about the imaged organs. Generally, there are not enough data to determine the unknowns. Hence, additional information must be incorporated into the solution algorithms to exclude mathematically feasible solutions that are not physically meaningful. Our approaches are either of a stochastic nature or are based on iterative methods with implicit enforcement of constraints. The goal of this research is to design algorithms that prove useful in clinical practice.