Modeling, analysis and computing for problems from idustry

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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.