Dynamical Systems: Modelling, Analysis, and Numerics

动力系统：建模、分析和数值

• 批准号: RGPIN-2020-03954
• 负责人: Willms, Allan
• 金额: $ 1.31万
• 依托单位: University of Guelph
• 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=759168
• 关键词: Dynamical; Systems; Modelling; Analysis; Numerics

This proposal is to study dynamical models of biological and physical processes and especially the relationship of experimental data with both qualitative and quantitative features of these models. Energy balance models of climate are an example. For these models to be useful, their parameters must be calibrated to physical measurements in order for the model to replicate observed phenomena. Here the data informs the model. In the other direction, an analysis of the model may reveal sudden transitions between different model states as a result of small changes in parameters. Here the model predicts the reality. Preliminary results have shown that the Arctic climate may have such a bifurcation and that, without significant CO2 reduction, may switch, irreversibly, from the present cold climate to a temperate climate as existed there millions of years ago. Enhancing the energy balance climate model, understanding its predictions, and ascertaining their validity is part of present proposal. Other examples are models of biological processes. In a collaboration with a pathobiology lab that studies viral treatments for cancer, this proposal will develop mathematical models of the immune response to viral treatment, including cytokine production and infection dynamics. It will use data from the lab and help provide insight into the fundamental processes involved in the body's response to viral treatments. Other biological phenomena will also be modelled and studied. Modelling of physical and biological processes prompts questions about fitting models to experimental data and the numerical algorithms employed to do so. Many biological models involve parameters for which little or no prior information is available. If the number of parameters is large or the amount of available data is small, fitting the model can be a very difficult task. This proposal also aims to expand on the tools available for model fitting by continuing to develop a parameter range reduction scheme that utilizes monotonicity properties of the model and numerical discretizations of the the model equations. The method is especially useful as a pre-conditioner for more traditional point estimation schemes when the initial parameter ranges are large, since it identifies a much smaller region in parameter space where the optimal estimate must be located, and it also provides a good estimate of this optimal value from which the traditional scheme will most likely converge rapidly to the optimal value.

本课题旨在研究生物和物理过程的动力学模型，特别是实验数据与这些模型的定性和定量特征之间的关系。气候的能量平衡模型就是一个例子。为了使这些模型有用，它们的参数必须校准到物理测量值，以便模型能够复制观察到的现象。在这里，数据为模型提供信息。在另一个方向上，对模型的分析可能会揭示由于参数的微小变化而导致不同模型状态之间的突然转变。这里，模型预测了现实。初步结果表明，北极气候可能有这样一个分叉，如果没有显著的二氧化碳减少，它可能不可逆转地从目前的寒冷气候转变为数百万年前存在的温带气候。加强能量平衡气候模式，理解其预测，并确定其有效性是本建议的一部分。其他的例子是生物过程的模型。在与一个研究癌症病毒治疗的病理生物学实验室的合作中，该提案将开发对病毒治疗的免疫反应的数学模型，包括细胞因子的产生和感染动力学。它将使用来自实验室的数据，并有助于深入了解人体对病毒治疗反应的基本过程。其他生物现象也将被建模和研究。物理和生物过程的建模提出了关于将模型拟合到实验数据和为此使用的数值算法的问题。许多生物学模型涉及的参数很少或根本没有先验信息。如果参数数量很大或可用数据量很少，则拟合模型可能是一项非常困难的任务。本建议还旨在通过继续开发一种利用模型单调性和模型方程数值离散化的参数范围缩小方案来扩展可用于模型拟合的工具。当初始参数范围较大时，该方法作为更传统的点估计方案的前置条件特别有用，因为它在参数空间中识别出最优估计必须位于的小得多的区域，并且它还提供了该最优值的良好估计，传统方案最有可能从该最优值快速收敛到最优值。