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

我从事几个研究项目，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能够改进数学软件，使科学家、工程师和金融建模者能够解决以前难以解决的问题，或者比目前更有效和/或更可靠地解决问题。 以前，我的研究主要集中在初值问题（IVP）的常微分方程（ODEs），但它已经扩大到包括边值问题（BVP）的常微分方程以及应用常微分方程技术偏微分方程（PDE）和解决实际问题的科学，工程和金融。我有一个次要的兴趣在数值线性代数，特别是问题所产生的微分方程。在未来几年，我将集中精力做好以下四项工作。 (1)计算金融。 我希望这些项目能够改进数值方法和软件，使金融工程师能够更好地为复杂的金融衍生品定价、对冲和管理风险。 (2)以长期为目标的常微分方程IVP数值解的保证误差界的研究 开发一个强大的，可靠的，易于使用的包，包括这些计划。 (3)随机生化动力学更可靠和更有效的数值方法的发展。 需要这些方法来帮助科学家更好地了解涉及细胞内少量物种的反应，例如涉及特定基因的反应。 (4)改进计算机断层扫描（CT）的数值方法-特别是考虑到CT扫描中使用的X射线的多色性质的数值方法。

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.