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Fierro, I., & Jerez-Hanckes, C. (2020). Fast Calderon preconditioning for Helmholtz boundary integral equations. J. Comput. Phys., 409, 22 pp.
Abstract: Calderon multiplicative preconditioners are an effective way to improve the condition number of first kind boundary integral equations yielding provable mesh independent bounds. However, when discretizing by local low-order basis functions as in standard Galerkin boundary element methods, their computational performance worsens as meshes are refined. This stems from the barycentric mesh refinement used to construct dual basis functions that guarantee the discrete stability of L-2-pairings. Based on coarser quadrature rules over dual cells and H-matrix compression, we propose a family of fast preconditioners that significantly reduce assembly and computation times when compared to standard versions of Calderon preconditioning for the three-dimensional Helmholtz weakly and hyper-singular boundary integral operators. Several numerical experiments validate our claims and point towards further enhancements. (C) 2020 Elsevier Inc. All rights reserved.
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Kleanthous, A., Betcke, T., Hewett, D. P., Escapil-Inchauspe, P., Jerez-Hanckes, C., & Baran, A. J. (2022). Accelerated Calderon preconditioning for Maxwell transmission problems. J. Comput. Phys., 458, 111099.
Abstract: We investigate a range of techniques for the acceleration of Calderon (operator) preconditioning in the context of boundary integral equation methods for electromagnetic transmission problems. Our objective is to mitigate as far as possible the high computational cost of the barycentrically-refined meshes necessary for the stable discretisation of operator products. Our focus is on the well-known PMCHWT formulation, but the techniques we introduce can be applied generically. By using barycentric meshes only for the preconditioner and not for the original boundary integral operator, we achieve significant reductions in computational cost by (i) using “reduced” Calderon preconditioners obtained by discarding constituent boundary integral operators that are not essential for regularisation, and (ii) adopting a “bi-parametric” approach [1,2] in which we use a lower quality (cheaper) H-matrix assembly routine for the preconditioner than for the original operator, including a novel approach of discarding far-field interactions in the preconditioner. Using the boundary element software Bempp (www.bempp.com), we compare the performance of different combinations of these techniques in the context of scattering by multiple dielectric particles. Applying our accelerated implementation to 3D electromagnetic scattering by an aggregate consisting of 8 monomer ice crystals of overall diameter 1cm at 664GHz leads to a 99% reduction in memory cost and at least a 75% reduction in total computation time compared to a non-accelerated implementation. Crown Copyright (C) 2022 Published by Elsevier Inc. All rights reserved.
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