Low Rank Structures in Solving Electromagnetic Problems

  • Stanislav StavtsevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)


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.


Parallel algorithm Fast matrix method Preconditioner Electromagnetic scattering 


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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Marchuk Institute of Numerical Mathematics Russian Academy of SciencesMoscowRussia

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