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

科学模拟和可视化在科学发现和工程设计以及商业和社会科学中发挥着越来越重要的作用。在医疗保健行业，糖尿病、中风和癌症等常见疾病每年在加拿大造成大量支出。了解调控基因网络如何影响细胞功能将有助于治疗常见疾病。这样的研究记录了成千上万的图像在一个时间序列中，在每个帧中有几十到几百个细胞。计算模型为分析细胞图像和跟踪细胞行为提供了一种高效、强大和可靠的工具。*另一个例子是经济学研究。不断上涨的汽油价格引发了石油储备如何影响能源供应和市场价格的问题。石油工业可以被看作是寡头垄断的一个例子，其中竞争性市场由少数公司组成。企业通过战略性地设定最优产量来进行竞争，以最大化其预期利润。在智能手机市场也存在类似的情况，公司在战略上控制奖品（而不是数量）。对不同市场结构的动态研究在经济学中具有长期的重要性。当资源是可耗尽的（自然资源）时，这个问题变得更加有趣和具有挑战性。有效的模型和数值方法是至关重要的，它提供了一种强大的手段来精确计算（纳什）均衡，从而确定市场价格和数量。无论是医学还是经济学应用，它们都可以通过非线性数学方程进行研究和建模。数值建模和算法是计算科学的基本组成部分。然而，随着应用程序变得越来越复杂，求解这些方程变得越来越具有挑战性和耗时。拟议的研究将有助于快速和强大的数值求解器以及准确和有效的模型的发展。该研究计划的一个主要目标是为经济学和医学成像应用开发合理和预测性的科学模拟。该数值算法的研究扩展了当前计算工具的适用性。这项研究将推动该领域的发展，并为金融机构的从业人员提供快速算法。对于医学影像学研究，本研究将有助于开发一种能够自动处理细胞图像的图像处理软件。计算框架将快速分析细胞信息，从而促进对生物过程的更好理解，这将导致药物发现和新疗法的改进。

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.**