## 1. Introduction

[2] Electromagnetic (EM) issues are the key to a much larger than perceived host of systems, applications, and services in present and future information and communication technology, as well as in fields like medical diagnosis and telemedicine, Earth observation and monitoring, noninvasive electromagnetic sensing, and so on. In all these applications, numerical methods are ubiquitous for their ability to reduce the prototyping and/or testing time. Increase in application demand and in the availability of computer power have pushed computational electromagnetism to phenomenal achievements in the last decade. However, practical applicability of EM solvers still remains challenging or unattainable in a variety of situations of definite industrial or scientific interest. If we take into account also the trend toward the use of higher and higher frequencies, it is clear that the need for accurate numerical tools cannot be answered by relying only on the increase of computers' speed and memory capacity. This situation has prompted active research on ways to reduce the numerical load required to solve large-scale problems.

[3] Asymptotic techniques, which can be very efficient in some cases, are not of general applicability and lose effectiveness in the case of complicated structures, wherein multiple interactions must be taken into account. Moreover, it is difficult to assess and control their accuracy. Accordingly, the only general and flexible approach is to adopt some kind of “full wave” numerical technique, turning the solution of the problem into that of a linear system in a finite dimensional space.

[4] As is well known, the numerical approaches to EM problems can be divided into two broad classes: those based on a differential formulation (DF) and those based on an integral equation formulation (IF) of the boundary value problem for Maxwell equations. To the former class belong the finite element method (FEM) and the finite difference time domain (FDTD) methods, while the so-called method of moments (MOM) is the most widely employed denomination for the latter class. The two classes have essentially complementary features, and an analysis of the existing literature and commercial packages indicates that IF methods still lead the way for electrically large problems with linear media, as is typical of wave scattering and antenna problems, and possibly also for complex structures not completely enclosed in metal casings.

[5] DF methods entail unknowns distributed inside the entire volume of the considered computational geometrical domain (volume discretization), as opposed to unknowns typically distributed only on surfaces (e.g., of conductors or dielectric discontinuities or in apertures of enclosures and antennas) in IF ones; differential methods require artificial truncation of the computational domain (via “absorbing” boundary conditions) in open problems, whereas IF methods incorporate radiation at infinity into the equation kernel. As a counterweight, direct discretization of differential equations leads to very large but extremely sparse matrices, or to no matrix manipulation at all (in conventional FDTD), while the (much smaller) IF matrices are fully populated. Moreover, the standard FDTD is typically beset with (artificial) dispersion problems that blow up for electrically large structures. A general comparison between FDTD and IF is difficult because they operate in dual domains (time and frequency), while FEM and IF both operate in the frequency domain and allow an easier comparison. Matrix entries of the IF method can be assigned an individual physical meaning in terms of fields and waves (field radiated by a basis function source and “received” in a given region), which is suggestive of physical and mathematical “tricks” to reduce the numerical complexity. Another key feature which allows such a reduction is that the system matrix can have a much larger size than that needed to store the required quantity of information, even for an accurate solution.

[6] Because of the above, and since the much smaller matrix size of IF methods favors their scalability to large problems, in the following we will concentrate on these methods. Following the path first presented by *Bucci* [2004], the aim of the paper is to answer the following fundamental questions: (1) What is the minimal computational complexity required to solve, with a given precision, a general scattering problem? and (2) is it possible to devise numerical techniques approaching this ideal limit?

[7] To this end, in section 2, the problem at hand is mathematically stated, the factors affecting the effectiveness of any numerical technique are discussed, and the main approaches for increasing numerical efficiency are briefly reviewed. This puts in evidence the crucial relevance of assessing the minimal size of a scattering problem and the minimal computational cost required to solve a problem of given size.

[8] The first point is faced in section 3, wherein it is shown how the problem size can be rigorously defined and estimated in a quite general way. Section 4 is devoted to assessment of the minimal computational cost and to discussion of possible ways to approach this ideal limit. Finally, in section 5 it is demonstrated that in the case of large, smooth metallic scatterers in free space, the method of auxiliary sources could provide a basis for developing a suboptimal algorithm.