"SCREMS" Computational Mathematics Research at UTM

来自得克萨斯大学奥斯汀分校数学系的研究人员参与的项目包括玻尔兹曼型动力学输运模型的确定性高阶数值方案的开发和非平衡统计力学的研究;与动力系统不稳定性的发生有关的分析中的计算机辅助定理证明;开发计算工具，用于研究不同尺度下DNA的材料和构象特性;计算二维或三维空间随机反应中的随机前沿，扩散方程和生成一个大的解决方案的样本空间，以分析前端统计;计算调查的大规模分布的预测阶的Tate-Shafarevich群的二次扭曲的一个固定的椭圆曲线;以及各种纠错码族中最小权重的增长。 这些项目中的每一个都需要大量不间断的计算周期。 在动力系统的分析中，一个计算机辅助的证明可以在1GHz奔腾级处理器上消耗超过3年的CPU时间。 动力学输运模型的数值方法测试的计算要求基本上是开放式的，研究人员目前使用所有可用的资源不断。 类似地，研究Tate-Shafarevich群的阶的大规模分布的计算要求实际上是无限的。 上述项目的范围和规模保证了为实现这些目标而部署大量专用计算设施的必要性。美国国家科学基金会为购买计算服务器集群提供的资金为积极追求上段所述的研究议程提供了相当大的动力。 所有这些项目都涉及计算数学中当前感兴趣的问题。 它们是理解在固态物理学、生物学、密码学和其他科学中起重要作用的现象背后的机制的努力的一部分。 此外，他们推进了计算数学的最新发展，这给数学分析带来了新的问题。 一些项目还涉及与软件开发人员的互动，这有助于为科学应用设计新软件。 该研究的应用范围广泛而深刻。 它们包括纳米电子和生物结构的建模;分析动力系统的稳定性，如聚变能量加速器，天体力学和大气动力学模型中的等离子体约束;了解森林火灾和湍流火灾前沿如何传播;以及调查计算机安全和量子计算。 研究结果通过讨论、讲座、参考期刊、预印本档案、网站等方式传达给尽可能广泛的受众。研究涉及研究生和本科生的参与和培训。

Investigators from the Department of Mathematics at The University of Texas at Austin are involved in projects that include the development of deterministic high order numerical schemes for Boltzmann type kinetic transport models and the study of non-equilibrium statistical mechanics; computer-assisted theorem proving in analysis related to the onset of instabilities in dynamical systems; development of computational tools for studying the material and conformational properties of DNA at various scales; computations of random fronts in two or three space dimensional stochastic reaction-diffusion equations and generation of a large solution sample space to analyze front statistics; computational investigations for large-scale distribution of predicted orders of the Tate-Shafarevich groups of the quadratic twists of a fixed elliptic curve; and the growth of the minimal weight in various families of error correcting codes. Each of these projects shares the need for large amounts of uninterrupted computation cycles. A single computer-assisted proof in the analysis of dynamical systems can consume over 3 years of CPU time on a 1GHz Pentium class processor. The computational requirements of testing numerical methods for kinetic transport models are essentially open ended; investigators currently use all available resources continuously. Similarly, the computational requirements of investigating the large-scale distribution of orders of Tate-Shafarevich groups are practically unlimited. The scope and the magnitude of the projects outlined above warrant the deployment of a substantial dedicated computational facility for the pursuit of these objectives.The NSF funds awarded for the purchase of a computational server cluster provide considerable momentum to the vigorous pursuit of the research agenda outlined in preceding paragraph. All of these projects deal with problems that are of current interest in computational mathematics. They are part of an effort to understand the mechanisms behind phenomena that play an important role in solid state physics, biology, cryptography, and other sciences. In addition, they advance the state of the art in computational mathematics, which brings new problems within the reach of mathematical analysis. Some of the projects also involve interactions with software developers, which contributes to the design of new software for scientific applications. The applications of this research are wide ranging and profound. They include the modeling of nano-electronic and biological structures; analyzing the stability of dynamical systems such as plasma confinement in fusion energy accelerators, celestial mechanics, and models for atmospheric dynamics; understanding how forest fires and turbulent fire fronts propagate; and investigating computer security and quantum computing. The results are communicated to as wide an audience as possible, through discussions, lectures, refereed journals, preprint archives, web sites, etc. The research involves the participation and training of graduate and undergraduate students.