Developing reduced basis methods for Galerkin and Collocation framework

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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.