Numerical solution of differential equations with applications to computational finance and numerical weather forecasting

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。 在未来的几年里，我将集中精力于以下两个主要项目：（1）常微分方程IVP数值解的保证误差界的研究，长期目标是开发一个强大的，可靠的，易于使用的软件包，包括这些计划;（2）金融数值方法研究，特别强调计算可转换债券价格和信用衍生品价格的有效计算技术。

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 two major projects: (1) 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; (2) the study of numerical methods for finance, with particular emphasis on effective computational techniques for computing the price of convertible bonds and the price of credit derivatives.