Developing reduced basis methods for Galerkin and Collocation framework

简化基法（RBM）是一种用于快速、可靠地模拟输入参数化偏微分方程的模型简化框架。许多应用程序需要重复模拟数万次，以研究参数对解决方案的影响。就计算成本而言，这种重复可能令人望而却步。RBM可以在可忽略不计的计算时间内提供替代解决方案。一个类似的方法，称为简化基元法（RBEM），可以用于计算复杂域上的代理解。该方法将区域分解和RBM相结合。在这份拨款申请中，PI建议继续他在这两个领域的工作，为搭配框架设计一个全新的RBM，并开发RBM和RBEM的（Galerkin）变体，适用于模拟大量、宽范围参数和粗糙几何的散射问题。本研究分为两个阶段。第一阶段旨在建立坚实的理论基础，包括研究新的基于搭配的RBM，一种新的RBEM误差估计方法，一种基于更大范围弱公式的有效误差估计的RBM算法设计。该项目的第二阶段是将新开发的方法应用于具有高维参数和散射体几何不确定性的声/电磁散射。所提出的研究的智力优点在于它们全面涵盖了新颖的算法设计，可靠的分析和有效的实现。PI的工作具有比当前提案更深远的目标，因为这些方法广泛适用于在科学和工程中具有重大影响的问题。实际应用领域包括（但不限于）国家安全（用于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成分。调查结果将广为散发，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。将特别注意包括少数民族在内的代表性不足的群体。

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.