Abstract
Hypersingular integral equations are applied in various areas of applied mathematics and engineering. The paper presents a method for solving the problem of diffraction of an electromagnetic wave on a perfectly conducting object of complex form. In order to solve the problem of diffraction with large wave numbers using the method of integral equations, it is necessary to calculate a large dense matrix.
In order to solve the integral equation, the author used low-rank approximations of large dense matrices. The low-rank approximation method allows multiplying a matrix of size \(N\times N\) by a vector of size N in \(\mathcal {O}(N\log (N))\) operations instead of \(\mathcal {O}(N^2)\). An iterative method (GMRES) is used to solve a system with a large dense matrix represented in a low-rank format, using fast matrix-vector multiplication.
In the case of a large wave number, the matrix becomes ill-conditioned; therefore, it is necessary to use a preconditioner to solve the system with such a matrix. A preconditioner is constructed using the uncompressed matrix blocks of a low-rank matrix representation in order to reduce the number of iterations in the GMRES method. The preconditioner is a sparse matrix. The MUMPS package is used in order to solve system with this sparse matrix on high-performance computing systems.
In our work we used “Zhores” supercomputer installed at Skolkovo Institute of Science and Technology [1]. The work was supported by the Russian Science Foundation, grant 19-11-00338.
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Stavtsev, S. (2020). Low Rank Structures in Solving Electromagnetic Problems. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2019. Lecture Notes in Computer Science(), vol 11958. Springer, Cham. https://doi.org/10.1007/978-3-030-41032-2_18
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DOI: https://doi.org/10.1007/978-3-030-41032-2_18
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