High-order numerical methods for differential equations

微分方程的高阶数值方法

• 批准号: RGPIN-2020-04663
• 负责人: Trummer, Manfred
• 金额: $ 1.31万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740104
• 关键词: order; numerical; methods; differential; equations

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.

计算方法是科学和现代技术许多方面的基础。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。在医学上，无价的诊断信息是通过图像重建和处理提供的，最终以数学算法为基础。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。该研究解决了数值分析的三个原则：效率，准确性和算法的鲁棒性。拟议的研究计划涉及两个不同的领域。第一类广泛涉及解决微分方程的计算方法的设计和分析，这些微分方程在许多科学，工程和其他建模应用中出现，例如电路，电化学反应的建模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，带延迟的微分方程。研究主要集中在谱法、径向基函数法等高阶方法上。高阶方法是一种可以用相对粗糙的离散化来近似问题的解并达到较高精度的方法。一个研究重点是自适应-通过自适应地选择计算网格或各种参数来提高效率和准确性，由近似解的特征引导或决定-以及所研究方法的灵敏度或鲁棒性。我这部分研究的目的是为数值求解微分方程提供更准确、稳定和用户友好的方法。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，相同的数据被用来重建一系列3D图像（即3D电影），以显示物质摄取和冲洗的动态行为，表明有关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，必须将附加信息纳入解决算法，以排除数学上可行的解决方案，而这些解决方案在物理上没有意义。我们的方法要么是随机的，要么是基于迭代方法的隐含强制约束。这项研究的目标是设计在临床实践中证明有用的算法。