Numerical methods and software for applications in science, engineering and finance

我参与了几个研究项目，以开发、分析、测试和评估数值方法，并构建强大的数学软件。我希望这些项目最终将导致数学软件的改进，使科学家、工程师和金融建模人员能够解决以前难以解决的问题，或者比目前可能的情况更有效和/或更可靠地解决问题。 以前，我的研究主要集中在常微分方程组的初值问题(IVP)，但后来扩大到包括常微分方程组的边值问题(BVPS)，以及常微分方程组(PDE)和解决科学、工程和金融等实际问题的常微分方程技术的应用。我对数值线性代数，特别是由微分方程式引起的问题有次要的兴趣。在接下来的几年里，我将集中精力开展以下四个主要项目。 (1)计算金融。我希望，这些项目将带来改进的数值方法和软件，使金融工程师能够更好地定价、对冲和管理复杂金融衍生品的风险。 (2)带长期目标的常微分方程组数值解的保证误差界研究。 开发一个包含这些计划的强大、可靠、易于使用的一揽子计划。 (3)发展更可靠、更有效的随机生化动力学数值方法。需要这些方法来帮助科学家更好地了解涉及细胞内少量发生的物种的反应，例如涉及特定基因的反应。 (4)改进的计算机断层扫描(CT)数值方法--特别是考虑到CT扫描中使用的X射线的多色性的数值方法。

I am engaged in several research projects to develop, analyze, test and evaluate numerical methods and to construct robust mathematical software. I hope these projects will ultimately lead to improvements in mathematical software, enabling scientists, engineers and financial modelers either to solve previously intractable problems or to solve problems more efficiently and/or reliably than is currently possible. Previously, my research focused primarily on initial-value problems (IVPs) for ordinary differential equations (ODEs), but it has since broadened to include boundary value problems (BVPs) for ODEs as well as the application of ODE techniques to partial differential equations (PDEs) and to the solution of practical problems in science, engineering and finance. I have a secondary interest in numerical linear algebra, particularly problems arising from differential equations. Over the next few years, I will concentrate on the following four major projects. (1) Computational Finance. My hope is that these projects will result in improved numerical methods and software that will allow financial engineers to better price, hedge and manage risk for complex financial derivatives. (2) The study of guaranteed error bounds for the numerical solution of IVPs for ODEs, with the long term goal of developing a robust, reliable, easy-to-use package incorporating these schemes. (3) The development of more reliable and more efficient numerical methods for stochastic biochemical kinetics. These methods are needed to help scientists better understand reactions involving species that occur in low numbers within a cell, such as the reactions involving a specific gene. (4) Improved numerical methods for Computerized Tomography (CT) -- in particular numerical methods that will take into account the polychromatic nature of the x-rays used in CT scans.