Acquisition of Computer Equipment for Development of Algorithms for Scientific Computing

这些设备将用于支持数学科学的研究以及研究生和本科生教育。卡内基梅隆大学数学系有一个强大的数值分析课程，它吸引了我们博士项目中一些最有才华的研究生。该设备将由首席研究人员、来访者、博士后研究员和他们的研究生使用，以加强正在进行的计算材料科学和多层次计算技术的研究计划。材料科学中的CMU程序在计算呈现复杂细尺度现象的材料的性质方面有应用。这些问题出现在冶金、磁学、优化设计等经常观察到细微尺度变化的研究中。最近的数学进步表明，这些振荡是由能量因素决定的，而且通常可以用一个宏观的(缓慢变化的)量--杨度量来表征。这些表征为这些材料及其性质的有效数值模拟和近似提供了基础。本提案考虑了解决此类问题的算法的开发、分析和测试。高效算法特别是多层算法的研究计划旨在使声学、电磁学、形状优化和无损检测中的优化设计计算在计算上可行。这类问题的有效解决需要底层问题的多个解，以便分离出最优解：如果没有正在研究的那种强大的计算算法，我们考虑的问题将是不切实际的。多层计算技术在科学和工程的各个领域都是非常有效的求解方法。作者一直致力于开发这些方法来解决各种问题，并取得了巨大的成功。该研究涉及非椭圆型方程组边界相关优化问题和双曲型演化问题的高效离散化和多层时间推进技术的新发展。这些技术的应用包括电磁、微波器件设计、声学降噪、无损检测和流体流动控制等方面的问题。

This equipment will be used to support research and graduate and undergraduate education in the mathematical sciences. The Department of Mathematics at Carnegie Mellon has a strong program in numerical analysis which attracts some of the most talented graduate students in our Ph.D. program. This equipment will be used by the principal investigators, visitors, postdoctoral fellows, and their graduate students to enhance ongoing research programs in computational material science and multilevel computational techniques. The CMU program in material science has application in computing the properties of materials that exhibit complex fine scale phenomena. These problems appear in the study of metallurgy, magnetism, optimal design, etc. where fine scale (microscopic) variations are frequently observed. Recent mathematical advances suggest that these oscillations are governed by energy considerations, and can often be characterized by a macroscopic (slowly varying) quantity, the Young measure. Such characterizations provide a basis for the efficient numerical modeling and approximation of these materials and their properties. The development, analysis, and testing of algorithms for the solution of such problems is considered in this proposal. The research program on efficient algorithms - especially multilevel algorithms- is aimed at making optimal design calculations in acoustics, electromagnetism, shape optimization, and nondestructive testing computationally feasible. Efficient solution of such problems requires multiple solutions for the underlying problem in order to isolate an optimal solution: the problems we consider would be impractical without powerful computation algorithms of the kind under investigation. Multilevel computational techniques are very effective solution methods in a variety of different fields of science and engineering. The authors have been involved in developing these methods for a variety of problems with great success. The proposed research deals with new developments of these techniques for boundary related optimization problems governed by non-elliptic systems, and for the efficient discretization and multilevel time marching techniques for hyperbolic evolution problems. Applications of these techniques include problems arising in electromag netism, design of microwave devices, acoustic noise reduction, non-destructive testing, and the control of fluid flows.