Rapidly-convergent, high-performance PDE solvers for materials-science and engineering applications: theory, implementation and applications.

研究者寻求开发和分析准确，快速收敛，高性能的求解器，用于当今材料科学应用中出现的具有挑战性的偏微分方程类型，以及使用这些求解器来设计具体设备和预测现实系统的特性。他创建算法来处理与势理论和波传播、复合材料和光学器件、弹性、声学和电磁学相关的某些类型的问题。在考虑材料科学问题的背景下，这些数值分析和计算科学的经典领域出现了重大困难，这些问题需要处理微观结构，非矫顽性系统，几何奇点，共振结构，高频，多重散射等。这些数学问题，尽管在上个世纪取得了重大的努力和进步，但在许多重要的实例中仍然难以捉摸。许多这样的挑战经常出现在材料科学配置的处理中，这使得材料问题难以解决。特别是，该项目中考虑的材料科学应用类型的大多数算法要么缺乏准确性，要么缺乏计算效率，要么两者兼而有之——这种情况阻碍了该领域的进展。研究者开发了有效的，高性能的计算方法来解决偏微分方程，这些偏微分方程模拟了各种物理系统的性质和行为，强调了材料的重要特征。该项目对社会感兴趣的几个领域产生了影响，包括医学领域（断层扫描，成像），电气工程（光学，电子），军用和民用遥感（雷达，声纳，隐身），通信（天线），大气科学等。该项目还为研究生和本科生以及博士后提供培训。

Bruno0707564 The investigator seeks development and analysis of accurate,rapidly convergent, high performance solvers for the types ofchallenging partial differential equations that arise inpresent-day materials science applications, as well as use ofsuch solvers for design of concrete devices and prediction ofproperties of realistic systems. He creates algorithms to treatcertain types of problems related to potential theory and wavepropagation, composite materials and optical devices, elasticity,acoustics, and electromagnetism. Significant difficulties arisein these classical fields of numerical analysis and computationalscience in the context of the materials science problemsconsidered, which require treatment of microstructures,non-coercive systems, geometric singularities, resonantstructures, high frequencies, multiple scattering, etc. Theseare mathematical problems that, in spite of significant effortsand progress over the last century, remain elusive in a widerange of important instances. The combination of a number ofsuch challenges often arises in the treatment of materialsscience configurations, which has made materials problemsdifficult to tackle. In particular, most algorithms availablefor the types of materials science applications considered in theproject lack in either accuracy, computational efficiency, orboth -- a circumstance that has hindered progress in the field. The investigator develops effective, high-performancecomputational methods for solving partial differential equationsthat model the properties and behaviors of various physicalsystems, emphasizing features important for materials. Theproject has consequences for several areas of societal interest,including the medical field (tomography, imaging), electricalengineering (optics, electronics), military and civilian remotesensing (radar, sonar, stealth), communications (antennas),atmospheric science, etc. The project also provides training forstudents at both graduate and undergraduate levels, as well asfor postdoctoral associates.