Dynamical Systems: Modelling, Analysis, and Numerics

本文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，它们的参数必须根据物理测量进行校准，以便模型能够复制观察到的现象。在这里，数据通知模型。另一方面，对模型的分析可能会揭示由于参数的微小变化而导致的不同模型状态之间的突然转变。在这里，模型预测了现实。初步结果表明，北极气候可能存在这样的分叉，如果不大幅减少二氧化碳，可能会不可逆转地从目前的寒冷气候转变为数百万年前那里存在的温带气候。加强能源平衡气候模型，理解其预测，并确定其有效性是本提案的一部分。其他的例子是生物过程的模型。在与一个研究病毒治疗癌症的病理生物学实验室的合作下，这项提议将开发病毒治疗免疫反应的数学模型，包括细胞因子的产生和感染动力学。它将使用实验室的数据，并帮助深入了解人体对病毒治疗的反应所涉及的基本过程。其他生物现象也将被模拟和研究。物理和生物过程的建模引发了关于将模型与实验数据相匹配以及为此而采用的数值算法的问题。许多生物模型所涉及的参数的先验信息很少或根本没有。如果参数数量多或可用数据量少，则对模型进行拟合可能是一项非常困难的任务。该建议还旨在通过继续开发利用模型的单调性和模型方程的数值离散化的参数范围缩减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，该方法特别适用于更传统的点估计方案，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，并且它还提供了对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。

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.