Developing reduced basis methods for Galerkin and Collocation framework

缩减基方法（RBM）是一种用于输入参数化偏微分方程快速可靠模拟的模型缩减框架。 许多应用需要模拟重复数万次，以研究参数对解决方案的影响。这种重复在计算成本方面可能是禁止的。RBM可以在可忽略的计算时间内提供替代解。一种类似的方法，称为缩减基元法（RBEM），可以用于计算一个复杂的域上的代理解决方案。该方法是区域分解和RBM的结合。在这项资助提案中，PI建议继续在这两个领域开展工作，为配置框架设计一个全新的RBM，并开发RBM和RBEM的（Galerkin）变体，适用于大量，宽范围参数和粗糙几何形状的散射问题的模拟。拟议的研究包括两个阶段。第一阶段的目标是建立一个坚实的理论基础，包括新的搭配为基础的RBM，一个新的错误估计过程RBEM，RBM算法设计的基础上有效的错误估计更广泛的弱配方的研究。该项目的第二阶段是将新开发的方法应用于具有相当高维参数和散射体几何形状不确定性的声/电磁散射。所提出的研究的智力价值在于其全面覆盖新颖的算法设计，坚实的分析和高效的实施。PI的工作具有超越当前提案的深远目标，因为该方法对科学和工程中具有重大影响的问题具有广泛的适用性。现实世界的应用领域包括（但不限于）国家安全（隐形技术的形状和材料的微调），可再生能源（太阳能电池的设计）和非破坏性传感。这项建议的更广泛影响将来自其科学影响和教育内容。调查结果将广为传播，守则将向公众公布。拟议的研究将包括严格的本科生和研究生培训和指导。将特别关注代表性不足的群体，包括少数群体。

Reduced basis method (RBM) is a model reduction framework for rapid and reliable simulations of input-parametrized partial differential equations. Many applications require simulations to be repeated tens of thousands of times to study the effect of the parameters on the solution. This repetition can be prohibitive in terms of computational cost. The RBM can provide a surrogate solution in negligible computational time. A similar approach, known as the reduced basis element method (RBEM) can be employed for computing the surrogate solution on a complicated domain. This method is a combination of domain decomposition and RBM. In this grant proposal, the PI proposes to continue his work in these two areas to design a completely new RBM for the collocation framework and to develop (Galerkin) variants of RBM and RBEM suitable for applications to simulations of scattering problems with large number, wide range of parameters, and rough geometries. The proposed research includes two phases. The first phase aims to build a solid theoretical foundation that includes study of the new collocation-based RBM, a novel error estimation procedure for RBEM, a RBM algorithm design based on efficient error estimation for a wider range of weak formulations. The second phase of this project is the application of the newly-developed methodologies to acoustic/electromagnetic scattering with rather high-dimensional parameter and uncertainties in the geometry of the scatterer. The intellectual merit of the proposed research lies in their comprehensive coverage of novel algorithm design, solid analysis, and efficient implementation. The PI's work has far-reaching goals beyond the current proposal because of the methods' broad applicability to problems of significant impact in science and engineering. The real-world application areas include (but are not limited to) national security (fine-tuning of the shape and material for stealth technology), renewable energy (design of solar cells), and non-destructive sensing. The broader impact of this proposal will result from its scientific impact and educational component. The results will be widely disseminated and the codes made publicly available. The proposed research will incorporate rigorous undergraduate and graduate student training and mentoring. Special attention will be paid to under-represented groups including minorities.