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Acceleration methods for the iterative solution of electromagnetic scattering problems


  • W. D. Murphy,

  • V. Rokhlin,

  • M. S. Vassiliou


We present a simple but effective technique for accelerating the convergence of iterative methods in the solution of electromagnetic scattering problems described by a second-kind integral equation (SKIE). We call the technique “complexification and extrapolation,” or simply “complexification.” It is based on the mathematical principle of limiting absorption, and it alleviates the difficulties arising from the interior resonances of this SKIE, thus allowing the efficient solution of scattering from electrically large objects. The technique involves introducing an imaginary part to the real wavenumber and solving the problem, then repeating with a different imaginary part and extrapolating the solutions linearly back to the real axis. For higher-order extrapolations we use additional complex wavenumbers. We have tested the method on a number of closed two-dimensional conducting scatterers, using this SKIE discretized by Nyström's method and solved by the fast multipole method. We use a variant of the conjugate gradient (CG) method that we call the pseudoconjugate gradient (PCG) method. The PCG method as we employ it performs only 1.2 matrix vector products on average per iteration, as opposed to two for the standard CG. Complexification gives excellent results. Solutions are fast and accurate. The condition number of the discrete matrix is asymptotically bounded for a given problem as the number of points per wavelength increases. The empirical evidence we have gathered thus far also suggests that the condition number is essentially asymptotically bounded as the electrical size of the scatterer increases, holding the number of points per wavelength fixed. Thus the technique has great potential in the solution of scattering from electrically large objects. Note that the technique of complexification is not limited to the fast multipole method and should be of broad applicability in the numerical solution of scattering problems.