**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**32870

##### Optimal Relaxation Parameters for Obtaining Efficient Iterative Methods for the Solution of Electromagnetic Scattering Problems

**Authors:**
Nadaniela Egidi,
Pierluigi Maponi

**Abstract:**

The approximate solution of a time-harmonic electromagnetic scattering problem for inhomogeneous media is required in several application contexts and its two-dimensional formulation is a Fredholm integral equation of second kind. This integral equation provides a formulation for the direct scattering problem but has to be solved several times in the numerical solution of the corresponding inverse scattering problem. The discretization of this Fredholm equation produces large and dense linear systems that are usually solved by iterative methods. To improve the efficiency of these iterative methods, we use the Symmetric SOR preconditioning and propose an algorithm to evaluate the associated relaxation parameter. We show the efficiency of the proposed algorithm by several numerical experiments, where we use two Krylov subspace methods, i.e. Bi-CGSTAB and GMRES.

**Keywords:**
Fredholm integral equation,
iterative method,
preconditioning,
scattering problem.

**References:**

[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1968.

[2] J.C. Aguilar, Y. Chen, . A high-order, fast algorithm for scattering calculation in two dimensions, Comput. Math. Appl. 47(1), 1-11 (2004).

[3] S. Ambikasaran, C. Borges, L.M. Imbert-Gerard, and L. Greengard, Fast, adaptive, high-order accurate discretization of the Lippmann-Schwinger equation in two dimensions, SIAM J. Sci. Comput., 38 A1770-A1787 (2016).

[4] F. Andersson, A. Holst, A fast, bandlimited solver for scattering problems in inhomogeneous media, J. Fourier Anal. Appl., 11 471-487 (2005).

[5] X. Antoine, M. Darbas, An Introduction to Operator Preconditioning for the Fast Iterative Integral Equation Solution of Time-Harmonic Scattering Problems. Multiscale Sci. Eng. 3, 1-35 (2021).

[6] M.O. Bristeau, J. Erhel, Augmented conjugate gradient. Application in an iterative process for the solution of scattering problems, Numer. Algorithms 18 (1998) 71-90.

[7] O. P. Bruno and E. M. Hyde, An efficient, preconditioned, high-order solver for scattering by two-dimensional inhomogeneous media,J. Comput. Phys. 200 (2004), 670-694.

[8] O.P. Bruno, T. Yin, Regularized integral equation methods for elastic scattering problems in three dimensions, Journal of Computational Physics, 410, 109350, (2020).

[9] S.H. Christiansen, J.C. N´ed`elec, A preconditioner for the electric field integral equation based on Calderon formulas, SIAM J. Numer. Anal. 40 (2002) 1100-1135.

[10] D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1992.

[11] N. Egidi, P.Maponi, Preconditioning techniques for the iterative solution of scattering problems, Journal of Computational and Applied Mathematics, 218 (2008), 229-237.

[12] F. Liu, L. Ying, Sparsify and sweep: An efficient preconditioner for the Lippmann-Schwinger equation, SIAM Journal on Scientific Computing, 40(2),B379-B404, 2018.

[13] P. Maponi, L. Misici, F. Zirilli, A Numerical Method to Solve the Inverse Medium Problem: an Application to the Ipswich Data, IEEE Antennas & Propagation Magazine 39 (1997), 14-19.

[14] J. Rahola, Solution of dense systems of linear equations in the discrete-dipole approximation, Special issue on iterative methods in numerical linear algebra (Breckenridge, CO, 1994), SIAM J. Sci. Comput. 17 (1996) 78-89.

[15] Y. Saad, Iterative Methods for Sparse Linear Systems, PWS publishing, New York, 1996.

[16] J. L. Volakis, A. Chatterjee, L. C. Kempel, Finite Element Method Electromagnetics: Antennas, Microwave Circuits, and Scattering Applications, Wiley-IEEE Press, 1998.

[17] F. Xiao, H. Yabe, Solution of scattering from conducting cylinders using an iterative method, IEEE Transactions on Magnetics, 36(4) (2000) 884-887.