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

我从事多个研究项目，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、工程师和金融建模者能够解决以前难以解决的问题，或者比目前可能的更有效和/或更可靠地解决问题。

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.