Modeling, analysis and computing for problems from idustry

这个建议是沿着工业数学的指导原则：它是问题驱动的和跨学科的。在应用方面，我们将研究神经生理学和生物流体力学中的一些问题，其中将开发新的模型来理解人脑中的皮层扩散性抑制，膜载蛋白与细胞膜和膜融合的相互作用，水，离子，在正常和病理条件下，brian-cell微环境和人体循环系统中的氧和其他物质。 风险管理和金融工程的一个重要领域是金融和保险部门的接口，在那里开发了新产品，但没有正确定价/理解。在拟议的研究中，我们将根据我们最近在生命周期模型和人力资本方面的工作，开发有效的分析和数值工具，供金融专业人士使用，以提供合理的财务建议。 从数学和计算的角度来看，许多这些问题（尽管应用领域的各种来源）可以通过偏微分方程建模。由于问题的性质和研究的重点，其中一些方程具有复杂的结构和广泛的新特征。求解这些方程往往带来巨大的挑战，迫切需要新的方法。在这个建议中，我们将分析和发展三类问题的新方法：多尺度反应扩散方程，浸没结构的传质问题和不完全相关的风险因素的随机最优控制问题。我们将通过探索降维和尺度分离的尺度特性，以及通过开发浸没结构问题的新计算方法，开发新的混合分析-数值方法。

This proposal is developed along the guiding principle of industrial mathematics: it is problem driven and interdisciplinary. On the application side, we will study a number of problems in neural physiology and biofuid mechanics where new models will be developed for understanding cortical spreading depression in the human brain, the interaction of membrane-borne proteins with cellular membrane and membrane fusion, the transport of water, ions, oxygen and other materials in the brian-cell micro-environment and in the human circulatory system under normal and pathological conditions. One of the important areas in risk management and financial engineering lies at the interface of financial and insurance sectors where new products are developed but not properly priced/understood. In the proposed research, we will develop efficient analytical and numerical tools which can be used by financial professionals for providing sound financial advices, based on our recent work on life-cycle models and human capital. From mathematical and computational points of view, many of these problems (despite the various sources of application areas) can be modeled by partial differential equations. Some of these equations have complex structures and a wide range of new features, due to the nature of the problems and the focus of the investigation. Solving these equations often imposes a great challenges and new methodologies are desperately needed. In this proposal, we will analyze and develop new methods for three classes of problems: reaction-diffusion equations with multiple scales, mass transfer problems with immersed structures and stochastic optimal control problems with imperfectly correlated risk factors. We will develop new hybrid analytical-numerical methods by exploring scaling properties for dimension reduction and scale separation, and by developing new computational methods for problems with immersed structures.