Acceleration methods for the iterative solution of electromagnetic scattering problems

Abstract

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 Nystr6m'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. 2 MURPHY ET AL.: ACCELERATION METHODS FOR THE ITERATIVE SOLUTIONand extrapolating back to the real axis. We refer to this technique as "complexification and extrapolation," or simply as "complexification." Complexification sidesteps difficulties arising from resonance for problems described by certain second-kind integral equations (SKIEs) and ensures bounded condition numbers for the discrete linear system to be solved.The numerical method that serves here as a platform for exploring complexification is an extremely efficient one known as the Fast Multipole Method (FMM). The FMM, first developed by Rokhlin, [1985a, 1990] for acoustic problems and later applied by us [see Engheta et al., 1992] to electromagnetic scattering problems, performs the matrix vector product directly without need to store the matrix. Both the operation count and the storage requirement of the FMM is 0(n4/3), where n is the number of unknowns or the size of the discrete linear system. This can be improved to O(n log n) [see ]. Small, bounded condition numbers ensure that the number of iterations required for converg...

A combined field approach to scattering from infinite elliptical cylinders using the method of ordered multiple interactions

“…This is physically interpreted as the case of zero multiple scattering, an example of which occurs in the MFIE formulation of scattering from an infinite PEC planar surface. In this case (22) and the integral term in (20) provides no contribution to the total surface current. This results for the flat surface scattering problem because the magnetic field radiated to an observation point on the surface by sources that are also on the surface has no tangential component.…”

“…This leads to (20) where (21) Discretization of this equation is discussed in the Appendix. The resulting matrix equation can be put in the form of (9).…”

“…This results for the flat surface scattering problem because the magnetic field radiated to an observation point on the surface by sources that are also on the surface has no tangential component. Thus, the MFIE for which MOMI was originally developed can be viewed as the specialization of (20) to the rough surface case where is selected such that is identically zero in the unperturbed geometry of the rough surface scattering problem.…”

“…This corresponds to the value of which minimizes the maximum modulus eigenvalue of . A physically intuitive way of determining this optimal choice is to minimize the contributions to the total field in (20), which are due to the integral term. Observe that if is chosen such that then no iteration is required to obtain the exact solution to the scattering problem.…”

“…Thus (24) and the contribution of the integral term to the total field in (20) is minimized by choosing (25) where denotes an average of the source point over the surface of the cylinder. Using the small argument forms of the zero and first order Hankel functions [23] ( 26) in (25) gives (27) which is seen to be inversely related to the size of the cylinder.…”