## 1. Introduction

[2] Numerical solutions of surface integral equation formulations of time-harmonic electromagnetic radiation and scattering from perfectly conducting targets involve solving linear systems of the form [*Peterson et al.*, 1998]

where **Z** is the impedance matrix, the vector **J** contains the coefficients of the basis functions used to represent the electric currents on the surface of an obstacle, and the vector **E**^{i} is determined by samples of an impressed electric field.

[3] The use of surface integral equations to determine equation (1) usually leads to a full impedance matrix. As a result, the computational costs associated with standard solutions of equation (1) are prohibitive for electrically large problems, and it is often necessary to solve equation (1) iteratively using compression algorithms for the impedance matrix [*Chew et al.*, 2001]. However, the computational costs of fast iterative solvers can be significant when the impedance matrix is poorly conditioned and when solutions are required for a large number of excitations.

[4] A general scheme for addressing these limitations should have two properties. First, it should provide a computationally efficient representation for the inverse of the impedance matrix, **Z**^{−1}. Second, to be generally applicable, it should also provide an efficient procedure to determine the compressed representation of the inverse from more directly available information, such as equation (1).

[5] In this paper we restrict our attention to only the first of these two requirements. Furthermore, instead of seeking a strategy to compress the inverse of the impedance matrix, we present a sparse representation for a related problem. For convex targets, the algorithm we report is observed to provide a compressed representation of the plane wave response matrix (P matrix), which specifies the currents excited on the surface of a target by a spectrum of incident plane waves. To the best of our knowledge, this is the first demonstration of the existence of a sparse, error-controllable representation of the P matrix having an asymptotic complexity that is less than *O*(*N*^{2}) at high frequencies. This demonstration, as well as the associated computational algorithms and concepts, comprise the primary contributions of this paper. These results were initially presented by *Adams et al.*, [2005a].

[6] While it may well be possible to efficiently determine such compressed representations of the P matrix from sparse representations of **Z**, this is beyond the scope of this paper. In the following we obtain the P matrix by first using standard *O*(*N*^{3}) matrix inversion techniques to determine **Z**^{−1}. Section 7 indicates directions for future research that are expected to help overcome this *O*(*N*^{3}) bottleneck. (*Adams et al.* [2006a] discuss an alternative sparse strategy to determine the inverse of the impedance matrix.)

[7] The remainder of this paper is organized as follows. A class of convex, two-dimensional scattering problems is defined in section 2. These problems are used to illustrate the properties of the proposed sparse representation. The plane wave response matrix is defined in section 3, and its relationship to the formulations discussed in section 2 is identified. The multilevel spatial groupings used are defined in section 4. The new compression strategy for the P matrix is summarized in section 5, and numerical results obtained by applying this method to convex scattering problems are reported in section 6. A summary and discussion of the P matrix and the compression algorithm are provided in section 7.