Numerical Algorithms for Nonlinear Models with Applications in Economics and Medical Image Processing

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Scientific simulation and visualization have been playing an increasingly central role in scientific discoveries and engineering designs, as well as in business and social sciences. In the heath care industry, common diseases such as diabetes, stroke, and cancer results in substantial expenditure in Canada every year. Understanding how the regulatory gene networks affect cell function will facilitate treatment of common disorders. Such studies record thousands of images in a time series, with tens to hundreds of cells in each frame. Computational models provide an efficient, robust, and reliable tool to analyze cell images and the tracking of cellular behaviors. Another example is in economics research. The ever increasing price of gasoline has raised the question of how oil reserves impact the energy supply and market prices. The oil industry can be viewed as an example of oligopoly in which the competitive market consists of a small number of firms. The firms compete with each other by strategically setting the optimal quantity in order to maximize their expected profit. A similar circumstance exists in the case of the smart phone market where the firms strategically control the prize (rather than the quantity). The study of the dynamics of different market structures is of longstanding importance in economics. The problem becomes more interesting and challenging when the resources are exhaustible (natural resources). Effective models and numerical methods are crucial to provide a powerful means to compute accurately the (Nash) equilibriums that set the market prices and quantities. Whether it is a medical or economics application, they can be both studied and modelled by nonlinear mathematical equations. Numerical modelling and algorithms are fundamental building blocks in computational science. However, as applications become more complex, solving these equations become more challenging and time consuming. The proposed research will contribute to the development of fast and robust numerical solvers as well as accurate and effective models. A major goal of this research program will be to develop sound and predictive scientific simulations for applications in economics and medical imaging. This numerical algorithm research expands the applicability of the current computational tools. This research will advance the development in the field and also offer fast algorithms for practitioners in financial institutions. For the medical imaging research, the proposed research will help develop an image processing software that is capable of processing cell images automatically. The computational framework will analyze the cell information quickly, thus facilitating better understanding of biological processes which will lead to improvements in drug discovery and new therapies.