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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758588
• 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.

计算方法支撑着科学和现代技术的许多方面。 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，最终基于数学算法的图像重建和处理提供了宝贵的诊断信息。我的研究计划的总体目标是设计、实现和分析应用程序的基础算法。该研究解决了数值分析的三个原则：算法的效率、准确性和鲁棒性。 拟议的研究计划涉及两个不同的领域。 第一个广泛涉及计算方法的设计和分析，用于求解许多科学、工程和其他建模应用中出现的微分方程，例如电路、电化学反应或最优控制理论中的许多问题的建模。为了扩大应用领域，我们还将考虑混合函数微分方程，例如带有延迟的微分方程。研究的是谱法、径向基函数法等高阶方法。高阶方法是一种可以通过相对粗略的离散化以高精度逼近问题的解决方案的方法。研究重点之一是自适应性 - 通过自适应地选择计算网格或各种参数，由近似解的特征引导或指示来提高效率和准确性 - 以及所研究方法的灵敏度或鲁棒性。我这部分研究的目标是为微分方程的数值求解提供更准确、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，根据患者扫描期间获得的数据重建一幅图像；在动态情况下，相同的数据用于重建一系列 3D 图像（即 3D 电影），以显示物质摄取和冲洗的动态行为，指示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方案算法中，以排除在数学上可行但没有物理意义的解决方案。我们的方法要么具有随机性，要么基于隐式执行约束的迭代方法。这项研究的目标是设计在临床实践中证明有用的算法。