Modeling, Analysis, and Computation for Problems from Biology and Industry

我的长期研究目标是开发基于演化微分方程的数学模型和数值方法，用于工业应用和其他科学学科的问题。随着这些问题的根源不断演变，我的研究计划在继续现有的研究课题和开发新的研究方向之间保持平衡，这反映在当前的提案中。该提案旨在分析和开发三类具有复杂结构的问题的新数学模型，方法和算法：（i）应用于生理学和病理学的反应和电扩散方程的多尺度系统，（ii）应用于耦合质量传递和流体流动问题的传输方程，以及（iii）应用于金融最优控制的Hamilton-Jacobi-Bellman方程。1.脑细胞电活动的扩散抑制（SD）对我来说是一个相对较新的领域，但从建模的角度来看，也是一个欠发达的领域。SD期间的细胞膜动力学（1A）和局部场电位（1B）具有理论意义。Poisson-Nernst-Planck（PNP）系统的渐近分析（1C）和均匀化（1D）是具有多时空尺度的挑战性问题。*-离子通道模型和计算（1 E）的目的是推导出比当前Hodgkin-Huxley和Goldman-Hodgkin-Katz模型更准确的近似通量公式。*2。聚合物纤维的拉伸流动是一个新的方向。*-将从理论上分析一个组合的初值和边值问题，该问题尚未研究过（2A）。*-针对温度和化学反应的影响进行稳定性分析（2B）。*3。将浸没边界框架扩展到传质问题是我正在进行的研究课题之一。*-将开发一种方法用于溶剂和溶质都可渗透的界面（3A：Navier-Stokes方程与传质方程耦合）。这种方法将扩展到离子输运与浸没界面处的流体流动耦合（3B：Navier-Stokes与PNP方程）。*4。个人理财和退休计划中的风险建模与控制是我长期的研究课题。*-产品分配（4A）和税收和借款约束（4 B）与典型的最优控制生命周期框架相比在技术上具有挑战性，但仍然可以使用Hamilton-Jacobi-Bellman方程来制定。*-将收入/健康冲击纳入生命周期模型（4C）涉及建模，并且更加开放。尽管如此，我们还是希望从简单的跳跃过程（如冲击的泊松过程）和分离风险厌恶和跨期替代的递归效用函数入手，取得一些进展。

My long-term research objective is to develop evolution differential equation based mathematical models and numerical methods for problems motivated by industrial applications and other scientific disciplines. As the source of these problems evolves, my research program maintains a balanced approach between continuing existing research topics and developing new research directions, which is reflected in the current proposal.***This proposal aims at analyzing and developing new mathematical models, methods and algorithms for three classes of problems with complex structures: (i) multi-scale systems of reaction- and electro-diffusion equations applied to physiology and pathology, (ii) transport equations applied to coupled mass transfer and fluid flow problems, and (iii) Hamilton-Jacobi-Bellman equation applied to optimal control in finance.***1. Spreading depression (SD) of brain cell electric activities is a relatively new area for me, but also an under-developed area from modeling point of view.***- Cell membrane dynamics (1A) and local field potential (1B) during SD are of theoretical interest.***- Asymptotic analysis (1C) of Poisson-Nernst-Planck (PNP) system and homogenization (1D) of PNP system are challenging problems with multiple time and spatial scales. ***- The objective of ion channel model and computation (1E) is to derive approximate flux formula that are more accurate than the current Hodgkin-Huxley and Goldman-Hodgkin-Katz models. ***2. Extensional flow of polymer fibers is a new direction. ***- A combined initial- and boundary-value problem will be theoretically analyzed, which has not yet been investigated (2A). ***- Stability analysis will be carried out for the effects of temperature and chemical reaction (2B). ***3. Extension of the immersed boundary framework to mass transfer problems is one of my ongoing research topics.***- A method will be developed for interface permeable to both solvent and solute (3A: Navier-Stokes equations coupled with mass transfer equation).***- This method will be extended to ion transport coupled with fluid flows at immersed interfaces (3B: Navier-Stokes with PNP equations). ***4. Risk modeling and control in personal finance and retirement planning is a long-term research topic of mine. ***- Product allocation (4A) & taxation and borrowing constraints (4B) are technically challenging compared to the typical optimal control lifecycle framework but can still be formulated using the Hamilton-Jacobi-Bellman equation. ***- Incorporating income/health shocks into a lifecycle model (4C) involves modelling and is more open ended. Nevertheless, we hope to make some progress by starting with simple jump processes such as Poisson one for shocks and a recursive utility function that separates risk aversion and intertemporal substitution.**