High-order numerical methods for differential equations

计算方法是科学和现代技术的许多方面的基础。例如，气候和天气模型依赖于求解微分方程，飞机的设计过程也是如此。在医学中，无价的诊断信息是通过图像重建和处理提供的，最终是基于数学算法。我的研究计划的首要目标是设计、实现和分析应用程序的基本算法。本研究探讨了数值分析的三个原则：算法的效率、准确性和稳健性。拟议的研究计划涉及两个不同的领域。第一种方法广泛地涉及到求解在许多科学、工程和其他建模应用中出现的那种类型的微分方程的计算方法的设计和分析，例如电路的建模、电化学反应或最优控制理论中的许多问题。为了扩大应用领域，我们还将考虑混合泛函微分方程，例如具有时滞的微分方程。研究的是高阶方法，如谱方法和径向基函数方法。高阶方法是一种用相对粗略的离散化方法将问题的解逼近到高精度的方法。其中一个研究重点是适应性--通过自适应地选择计算网格或由近似解的特征指导或支配的各种参数来提高效率和精度--以及正在研究的方法的敏感性或稳健性。我这部分研究的目的是为更准确、更稳定、更方便的数值求解微分方程组的方法做出贡献。第二个研究领域是计算医学成像，特别是在单光子发射计算机断层扫描(SPECT)和磁共振成像(MRI)中重建静态和动态图像的方法。在静态成像中，从患者扫描期间获得的数据重建单个图像；在动态情况下，相同的数据被用于重建一系列3D图像(即3D电影)，以显示物质摄取和洗出的动态行为，指示关于成像器官的功能信息。一般来说，没有足够的数据来确定未知数。因此，必须将附加信息合并到求解算法中，以排除物理上没有意义的数学上可行的解。我们的方法要么是随机性质的，要么是基于隐式强制约束的迭代方法。这项研究的目标是设计在临床实践中被证明有用的算法。

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.