Fast and stable high-frequency electromagnetic simulations via generalized integral equations and advanced preconditioning techniques

• Applicant: Professor Dr.-Ing. Simon Adrian
• Subject Area: Communication Technology and Networks, High-Frequency Technology and Photonic Systems, Signal Processing and Machine Learning for Information Technology
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 544296727

Method of moment solution of surface integral equations is a powerful and established approach for electromagnetic scattering and radiation analysis. However, the moment system matrix is dense and thus expensive to construct, store, and use. When stemming from the modeling of realistic systems, which include multiple wavelength scales, it also suffers from both dense-discretization and high-frequency breakdowns. These acute manifestations of ill-conditioning lead to slow (or no) convergence of iterative solvers and to severe inaccuracy of direct solvers (which avoid iterations altogether). Fast integral equation solver research has been focusing on the development of efficient moment matrix representations and well-conditioned formulations. Hierarchical structures of algebraically low-rank compressed blocks are an appealing kernel-independent, geometry-indifferent, and convenient to implement choice of matrix representation. They are also the foundation for fast direct solvers. Yet, they are limited by the rapid asymptotic growth of the block ranks with frequency, which prevents asymptotic cost reduction. For curing some types of ill-conditioning, Calderón-type preconditioners have been proven preferable. The proposed project aims at enabling the fast algebraic compression-based simulation of electrically large and finely modeled electromagnetic systems by modifying the integral equations to have increased compressibility and curing their ill-conditioning using suitable efficient preconditioners. In one thrust of the work, integral equation formulations of reduced and slow-scaling ranks will be derived alongside fast algorithms for harnessing their favorable properties for developing fast solvers. The 3-D electromagnetic generalized integral equations will utilize modified dyadic integral kernels that radiate only weakly in the broadside direction. This reduces the effective dimensionality of interactions between problem regions, increases the rank deficiency and enables faster block compression. Effective and easy to compute such modified dyadic kernels and fast algorithms for the efficient and accurate compression the resulting matrix blocks will be developed. A second thrust of the work focuses on developing suitable preconditioners that enable usage of the new formulations for densely-discretized objects and at high frequencies. It will include the analytical and numerical study of the spectral properties of the modified integral operators for determining their susceptibility to familiar and new sources of ill-conditioning. For curing them, refinement-free Calderón preconditioning techniques will be designed by taking advantage of free parameters of both the preconditioners and the modified kernels for achieving well-conditioning at high frequency. The interplay of the developed fast solver and its components and the preconditioner will be examined and optimized.

• DFG Programme: Research Grants
• International Connection: Israel
• International Co-Applicant: Professor Dr. Yaniv Brick, Ph.D.