Nonlinear Waves

数学科学：非线性波[摘要]goodman本工作由几个应用数学和计算数学项目组成。一个项目侧重于各向异性自适应有限元方法。已经证明，各向异性细化导致更有效的计算薄层，如边界层。这些方法将扩展到三维问题，并在通用软件中编码。另一个项目涉及理解股票期权和其他衍生证券价值的数学技术。我们将研究更理想、更复杂的定价模式与目前使用的定价模式有何不同。如果多因素模型中引入的额外不确定性来源不是太大，我们可以使用摄动方法而不是它们似乎需要的大规模计算来估计它们对定价和对冲策略的影响。第三个项目涉及求解随机微分方程的数值方法。目标是开发一种与Milstein的方法一样精确的方法，不需要采样重复的Ito积分或显式地评估噪声系数的导数。实际意义的计算量，如结构应力、流体或热流等，往往受到计算机内存和速度的限制，因此降低计算成本的技术具有直接的实际应用价值。该项目的一部分涉及开发更有效的有限元算法，这是工程计算的核心工具。目前，自适应计算自动确定适合一个区域的分辨率，这样计算就不会在不需要的地方浪费高分辨率（例如远离应力源）。各向异性精化，这项工作的主题，更进一步，寻找不需要高分辨率的方向（例如平行于长裂缝）。这个项目的第二部分侧重于银行业中使用的数学方法。金融行业已经被计算机、通信和即时获取信息所改变。通过使用复杂的定量方法，发生了一场流动性革命。投资者、企业和银行家现在可以更好地理解风险，并通过衍生工具和市场，将风险卖给那些愿意承担风险的人。纽约金融界通过使用更复杂的定量市场模型，在分析和计算方面对金融专业人士和应用数学家提出了挑战，因此与其他外国银行业中心相比，纽约金融界具有优势。这个项目探讨了使复杂的金融模型易于分析的方法。

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.