Nonlinear Waves

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NSF Award Abstract - DMS-0073077Mathematical Sciences: Nonlinear WavesAbstract0073077 GoodmanThis work consists of several projects in applied and computational mathematics. One project focuses on anisotropic adaptive finite element methods. It has been demonstrated that anisotropic refinement leads to more efficient computations in thin layers, such as boundary layers. These methods will be extended to three dimensional problems and encoded in general-purpose software. Another project involves mathematical techniques for understanding the value of stock options and other derivative securities. We will examine how more desirable, sophisticated pricing models differ from those in use today. If the extra sources of uncertainty introduced in multi-factor models are not too large, we can estimate their effect on pricing and hedging strategies using perturbation methods rather than the large scale computation that they seem to require. A third project concerns numerical methods for solving stochastic differential equations. The goal is to develop a method as accurate as Milstein's method that does not require sampling repeated Ito integrals or explicitly evaluating derivatives of noise coefficients.Computation of quantities of practical interest, such as stresses in structures, fluid or heat flows, etc., are often limited by computer memory and speed, so techniques that reduce computational cost have immediate practical application. One part of this project concerns development of a more efficient finite-element algorithm, a central tool of engineering computation. Currently, adaptive computations automatically determine the resolution appropriate to a region, so that the computation will not waste high resolution where it is not needed (e.g. far from the source of a stress). Anisotropic refinement, the topic of this work, goes one step further, seeking directions in which high resolution is not needed (e.g. parallel to a long crack). A second part of this project focuses on mathematical methods used in the banking industry. The finance industry has been transformed by computers, communications, and instant access to information. A liquidity revolution has occurred through use of sophisticated quantitative methods. Investors, businesses and bankers can now understand risk much better, and, through derivative instruments and markets, sell the risk to those who wish to assume it. The New York financial community has an edge on other foreign banking centers through the use of more sophisticated quantitative market models, which pose analytical and computational challenges for finance professionals and applied mathematicians. This project examines ways to make complicated financial models amenable to analysis.