Parallel Fast Iterative H-Matrix Locally Corrected Nyström Discretization of Integral Equations with an Inaccurate H-matrix Preconditioner

In this research, a new method is presented for addressing the Combined Field Integral Equation (CFIE) using the Locally Corrected Nyström (LCN) technique, focusing on speed, error control, and parallel processing features of the implementation. The method involves utilizing a less precise preconditioner derived from the coefficient matrix, which accelerates the whole process despite its reduced accuracy. However, this fast approach has some challenges in trading-off between solution speed and accuracy since we are approximating H-LU decomposition as well. This is particularly affecting the residual and precision of the solution when dealing with CFIE problems. The method implemented uses MPI parallelization for coefficient filling, and it uses HLibPro, which is OpenMP parallelized, to construct H-Matrices and perform H-LU factorization. To reach the desired compression, ACA method is used to compress the matrices to the tolerance required. Giving larger tolerance to preconditioner matrix and H-LU factorization, makes the problem hard to converge to the desired accuracy. As a reason, it is necessary always to trade-off between accuracy, the speedup and compression of the matrices. To illustrate its efficacy, consider Nasa Almond with 84,000 unknowns (where $l=240 \mathrm{~m}$) exposed to an z-polarized dipole located in the x direction in the far-field. The operating frequency is set at 3 MHz, with geometric dimensions spanning two wavelengths. As it is shown in Fig. 1, the Compression Rate (CR) achieved for the preconditioner H-matrix is $91.56 \%$, while for the coefficient matrix, it is $85.44 \%$. The relative error of the computational problem, $|A x-b|$, is quantified at $4.1023 \mathrm{e}-09$, which converges in four iterations. Moreover, the CPU time required to build both H-matrices is 9555.36 seconds, while it takes 4130.66 seconds to do H-LU factorization with 40 cores working in parallel.

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