EMSW21 - RTG: Numerical Mathematics for Scientific Computing

至少二十年来，关于美国数学科学研究健康状况的高层报告一直建议在这项研究和它的使用者之间建立一条更直接的途径。改善这种联系的最明显手段是软件，它在科学、工程、商业、医学以及日常生活中无处不在。从检测肿瘤到寻找最佳旅行路线，再到决定如何投资，软件中的数学可以迅速带来新的知识、见解和方法。软件作为数学科学的旗手的角色对工业来说尤其重要，因为工业中的数学发现往往只有在转化为软件后才会得到应用。成功生产高质量、及时的数学软件带来了一系列复杂的智力挑战，不仅需要在数学方面进行培训，还需要在软件开发和应用领域的需求方面进行培训。因此，这个研究训练组的目标是培养对数学算法及其在软件中的实例化有深刻、实际理解的数学科学家；此外，我们打算广泛提供培训计划的关键要素。参与这项活动的教师来自纽约大学科朗数学科学研究所和哥伦比亚大学应用物理和应用数学系，他们在应用和计算数学、数值算法和软件以及现实世界的应用方面有着丰富的经验。我们相信，这些研究人员的综合专业知识将为目前相对孤立的共同见解提供丰富的来源。研究培训组将为本科生、博士生和博士后研究人员提供：(1)深入探索数学建模、算法设计、软件选择和硬件环境之间关系的培训；(2)理解算法和软件设计如何受到问题的性质和需要解决问题的环境的影响，甚至驱动；(3)设计软件的实际经验，能成功就成功，不能成功就失败。在所有级别，参与者将体验个人和团队参与数学软件开发。该计划的本科部分将包括强化课程和夏季研究经验，将数学科学与现实世界的应用直接联系起来。博士生的培训将包括由在应用数学、计算数学和科学软件方面具有专长的教师联合指导，以两所大学的优势为基础的强化课程，以及在政府和工业研究实验室实习。博士后将参与以申请为重点的长期项目，并将参与本科生和研究生的培训。我们希望这个研究训练组的经验将有助于开发其他具有类似目标的项目。

For at least two decades, high-level reports on the health of mathematical sciences research in the United States have recommended creation of a more direct path between this research and its users. The most obvious means for improving this connection is software, which is ubiquitous in science, engineering, business, medicine, as well as everyday life. Mathematics embodied in software can quickly bring new knowledge, insights, and methods to bear in contexts ranging from detecting tumors to finding the best travel route to deciding how to invest. The role of software as standard-bearer for the mathematical sciences is especially important for industry, where mathematical discoveries tend to be applied only if they have been transformed into software. Successful production of high-quality, timely mathematical software brings an array of complex intellectual challenges requiring training not only in mathematics, but also in software development and the needs of application areas. Accordingly, the goal of this Research Training Group is to produce mathematical scientists with a deep, hands-on understanding of mathematical algorithms and their instantiation in software; in addition, we intend to make the crucial elements of the training program broadly available. The faculty involved with this activity, from the Courant Institute of Mathematical Sciences at New York University, and the Applied Physics and Applied Mathematics Department at Columbia University, represent a variety of experiences in applied and computational mathematics, numerical algorithms and software, and real-world applications. Our belief is that the combined expertise of these researchers will provide a rich source of common insights that currently exist in relative isolation. The Research Training Group will provide, for undergraduates, Ph.D. students, and postdoctoral researchers: (1) training that explores in depth the relationships among mathematical modeling, algorithm design, software choices, and the hardware environment; (2) understanding of how algorithm and software design may be affected, even driven, by the nature of problems and the contexts in which they need to be solved; (3) hands-on experience of designing software that succeeds when it can and fails gracefully when it cannot. At all levels, participants will experience individual and team involvement with mathematical software development. The undergraduate part of the program will include an enhanced curriculum and summer research experiences directly connecting the mathematical sciences with real-world applications. Ph.D. student training will include joint mentoring by faculty with expertise in applied and computational mathematics and scientific software, an enhanced curriculum that builds on the strengths of the two universities, and internships in government and industrial research laboratories. Postdocs will join longer-term projects with emphasis on applications, and will also be involved in undergraduate and graduate student training. Our hope is that the experiences of this Research Training Group will be helpful in developing other programs with similar goals.