Computational Methods and Software for Applications in Science, Engineering and Finance

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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.****