A proxy-method direct solver for Helmholtz transmission problems with many inclusions compresses the linear system to O(ωD) size and runs in O(N^{1.5}) time using the PMCHWT formulation, outperforming Burton-Miller.
and Matsushima, K., Efficient LU factoriza- tion exploiting direct-indirect Burton–Miller equation for Helmholtz 28 transmission problems, arXiv preprint arXiv:2512.14193 (2025)
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An accelerated direct solver for scalar wave scattering by multiple transmissive inclusions in two dimensions
A proxy-method direct solver for Helmholtz transmission problems with many inclusions compresses the linear system to O(ωD) size and runs in O(N^{1.5}) time using the PMCHWT formulation, outperforming Burton-Miller.