Computational 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，或者比现在更有效和/或更可靠地解决问题。在我学术生涯的第一阶段，我的研究主要集中在数值方法的初值问题（IVP）和边值问题（BVP）的常微分方程（ODE）和应用ODE技术的数值解偏微分方程（PDE）。 我还致力于解决科学中的实际问题，如天气和气候预测。 然而，我的兴趣已经扩大到包括应用数值方法的问题，在金融，工程和医学。 我对数值线性代数有次要的兴趣，特别是微分方程产生的问题。今后几年，我将主要集中精力抓好以下三项工作。 * （1）计算金融。我在这个领域有几个项目。 我希望这些项目将改进数值方法和软件，使金融工程师能够更好地定价，对冲和管理与复杂金融衍生品相关的风险。(2)计算医学 我在这一领域的主要项目是开发高效，强大，可靠的数值方法来估计CT扫描的血流速度。 这将有助于医生诊断心血管疾病，并在必要时准备适当的医疗干预措施（如手术计划）。(3)扩展我早期关于常微分方程和偏微分方程的工作，以开发更好的数值方法 ** - 随机微分方程（SDEs）** - 随机生化动力学（SBKs）** - 延迟微分方程（DDE）** 例如，需要更好的SBK数值方法来帮助科学家更好地理解涉及细胞内少量物种的反应，例如涉及特定基因的反应。

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 modellers either to solve previously intractable problems or to solve problems more efficiently and/or reliably than is currently possible.****During the first phase of my academic career, my research focused primarily on numerical methods for initial-value problems (IVPs) and boundary-value problems (BVPs) for ordinary differential equations (ODEs) and the application of ODE techniques to the numerical solution of partial differential equations (PDEs). I also worked on the solution of practical problems in science, such as weather and climate prediction. However, my interests have since broadened to include applications of numerical methods to problems in finance, engineering and medicine. I have a secondary interest in numerical linear algebra, particularly problems arising from differential equations.****Over the next several years, I will concentrate mainly on the following three major projects. ****(1) Computational Finance. I have several projects in this area. 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 the risk associated with complex financial derivatives.****(2) Computational Medicine. My main on-going project in this area is the development of efficient, robust, reliable numerical methods to estimate blood-flow rates from CT scans. This will help doctors diagnose cardiovascular disease and prepare appropriate medical interventions (such as surgical plans), if necessary.****(3) Extension of my earlier work on ODEs and PDEs to develop better numerical methods for ** - stochastic differential equations (SDEs)** - stochastic biochemical kinetics (SBKs)** - delay differential equations (DDEs)**Better numerical methods for SBKs, for example, 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.****