Computational complexity in the solution of large antenna and scattering problems


  • Ovidio M. Bucci

    1. Department of Electronic and Telecommunications Engineering, Università degli Studi di Napoli Federico II, Naples, Italy
    2. Institute for Environmental Electromagnetic Sensing, Consiglio Nazionale delle Ricerche, Naples, Italy
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[1] The main factors affecting the overall efficiency of any numerical procedure for the solution of large antenna or scattering problems, that is, the problem size, the memory occupation, and the computational cost, are introduced and are briefly discussed. It is shown how the size can be rigorously defined and estimated and the corresponding minimum, ideal computational cost is determined. Then the problem of developing algorithms approaching the ideal limit is examined, and possible ways to achieve the goal are enumerated. In particular, it is shown that in the case of large metallic scatterers in free space, the method of auxiliary sources, coupled to some kind of multilevel fast multipole algorithm, can allow development of numerical procedures whose effectiveness approaches the ideal limit.

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.

2. Statement of the Problem

[9] A generic antenna analysis (or scattering) problem can be simply stated as follows: Given the geometrical and electromagnetic characteristic of the antenna (or scattering) system and the primary excitation (primary sources, incident field, etc.), determine the radiated (or scattered) field in a given region of interest.

[10] In the IF methods, this is usually performed in two steps: (1) determination of the (true or equivalent) currents induced on the antenna or scattering system by the primary excitation, and (2) evaluation of the field radiated by these currents over the region of interest. From the mathematical viewpoint, in the frequency domain, the first step amounts to solving the “source equation”

equation image

where J is the induced current density, F0 is the forcing field (impressed by the primary excitation), and G is a linear (integral or integrodifferential) operator, whose kernel depends on the antenna structure and the adopted formulation: electric field integral equation (EFIE), magnetic field integral equation (MFIE), mixed integral equation (EMFIE), etc. J and F0, whose domain of definition is the antenna support, belong to properly defined Hilbert spaces [Taylor, 1958].

[11] The second step amounts to evaluating the radiation integral

equation image

over the region of interest, the kernel of the linear integral operator G0 being the appropriate free space Green dyadic.

[12] In the MOM approach, the problem is turned into a finite dimensional one, to be solved numerically. In the case of large systems, this is a challenging, or even unattainable, task because of computer time and memory requirements, as well as possible ill conditioning. Accordingly, as stressed in section 1, the development of effective, that is, accurate and efficient, techniques is a crucial need.

[13] The overall effectiveness of any particular numerical technique is dictated by the following main factors: (1) the size of the problem, that is, the rank of the matrices which approximate the operators G and G0; (2) the total memory occupation (matrix size times its filling percentage) and the computational cost (number of flops) required for filling the matrix; and (3) the computational cost required for solving the linear system (1) and evaluating the required field values via (2).

[14] Any attempt to reduce computational complexity must face at least one of these points, also ensuring a global significant improvement, taking into account the stability with respect to round-off noise and other error sources. The matrix redundancy can be reduced by exploiting at least three different ways.

[15] 1. A (linear) transformation that “reveals” the relevant part of the matrix can be employed, enhancing the dynamics of the entries, clipping lower entries, and storing the (largely) sparse resulting matrix; this is typical of block QR [Burkholder and Lee, 2004; Kapur and Long, 1998] and wavelet-transform approaches [Canning, 1993; Golik, 2000; Pirinoli et al., 2001; Su and Sarkar, 2002].

[16] 2. A sort of “interpolatory” scheme can be employed in which one only stores information required to evaluate, with a given precision, the matrix elements, which are actually computed only when needed [Bleszynski et al., 2003; Chew, 1993].

[17] 3. Nonredundant, entire domain functions can be used to represent the unknown currents and to minimize the number of unknowns and thus the dimension of the matrix [Bucci and Di Massa, 1995; Harrington and Mautz, 1971, 1985; Harrington et al., 1972; Kwon et al., 2001; Prakash and Mittra, 2003; Sun et al., 2003; Tiberi et al., 2003]. In recent years, another class of methods has emerged which groups basis functions (typically of low polynomial order) into “aggregate functions” defined over portions of the structure that are significantly larger than one cell of the initial mesh [Bleszynski et al., 1996; Matekovits et al., 2001; Suter and Mosig, 2000].

[18] Concerning the second point, it must be noted that the exploitation of entire domain functions, as the characteristic modes [Harrington and Mautz, 1971], without a suitable approach (like the multigrid one), involves an overhead in the matrix elements computation. However, entire domain functions ensure a significant saving of both memory and computer time in the solution of the linear system. Further advantages could be obtained when their theoretical expression is available or when the scattering systems is made by a large number of similar elements [Bucci and Di Massa, 1995]. From the point of view of matrix filling, in the case of metallic scatterers in free space, a technique that has regained attention in recent years appears particularly attractive: the method of auxiliary sources [Shubitidze et al., 2004; Zaridze and Karkasbadze, 1986; Leviatan and Boag, 1987; Leviatan, 1990; Boag and Mittra, 1994; Zaridze et al., 1998; Kaklamani and Anastassiu, 2002].

[19] To reduce the computational cost in the solution of the linear system (1) (the third point), iterative methods are usually adopted that require the repeated evaluation of a matrix-vector multiplication; the core of such methods is made by algorithms for the matrix-vector multiplication that reduce the order of the number of elementary operations from N2, N being the matrix dimension, to a smaller power of N (typically 1.5) or even, in the case of the multilevel fast multipole algorithms (MLFMA), to N log(N) [Chew, 1993; Chew et al., 1997; Lu and Chew, 1994; Epton and Dembart, 1995].

[20] In light of the above discussion, it is clear that, ideally, a technique should have the minimal size compatible with the required precision and should require the minimal computational cost; that is, it should be nonredundant and fast. Thus we are led quite naturally to the following two fundamental questions: (1) What is the minimal rank required to substitute, with a given error, the involved operators with finite dimensional ones? and (2) for a problem of given size, what is the minimal computational cost required to tackle steps 1 and 2, and how can we develop algorithms approaching this limit? These points will be considered in sections 3 and 4.

3. Assessing the Size of the Problem

[21] The fundamental property which allows to answer question 1 is the compactness [Gohberg and Krein, 1969] of the operator G0. Compact operators can be uniformly approximated by finite rank ones, with arbitrarily small error. The best approximation of rank n, that is, that providing the minimal error (in the uniform operator norm), G0n, say, is obtained by truncating to n terms the singular value (or Schmidt) decomposition (SVD) [Gohberg and Krein, 1969] of G0:

equation image

where σi > 0 are the singular values, and ∣eio >, < jio∣ denote the corresponding left and right singular vectors in the Dirac bra-ket notation. The σi form a nonincreasing sequence going to zero as n → ∞, and the singular vectors provide an orthonormal basis in the output and input spaces, respectively. Moreover, we have

equation image

where ∥.∥ denotes the operator norm.

[22] Clearly, relation (4) answers question 1 for step 2, provided we are able to evaluate the singular values, which can be done in closed form only for those geometries allowing separation of the Maxwell equations. However, by exploiting the properties of the kernel of G0, it can be shown [Bucci and Franceschetti, 1989] that for large sources (provided the observation domains are some wavelengths far from them) the singular values have a step-like behavior: They decrease slowly until a critical value of n is reached, then they go to zero very rapidly (see Figure 1). This implies that for reasonable approximation errors, the problem size, N0, is practically independent of the required precision. N0 can be explicitly estimated, once the geometries of the sources and of the domain of interest are given. In the case in which the region of interest surrounds the sources, it turns out [Bucci et al., 1998] that

equation image

where ∑ is the surface bounding the support of the sources. Moreover, simple and nonredundant sampling representations (depending only on the source geometry) for the radiated field can be developed [Bucci et al., 1998].

Figure 1.

Behavior of the singular values of the radiation operator.

[23] Assessing the size of step 1, which requires the inversion of the operator G, is a more difficult task. As a matter of fact, the properties of the operator G depend on the problem formulation (EFIE, MFIE, EMFIE, etc.) and dimensionality, as well as on the antenna structure. Under MFIE and EMFIE formulations, G is continuous and, if internal resonances are avoided, continuously invertible. On the other side, under the more usual EFIE formulation, the inverse is noncontinuous, that is, unbounded. Accordingly, no finite dimensional approximation of G can lead to a uniformly bounded error in the determination of the induced currents J, and increasing the size of the finite dimensional approximation to G makes step 1 ill conditioned.

[24] The above conclusions, which are usually scarcely pointed out, seem to prevent the possibility of a meaningful definition of the size of step 1. However, we are not interested in the evaluation of J for arbitrary incident fields, but only for fields radiated by (uniformly bounded) sources external to the antenna support. Because of, again, the compactness of the radiation operator, the set of all such fields is compact, hence can be uniformly approximated, with arbitrarily small error, by elements of a finite dimensional subspace. If the primary sources are some wavelength far from the antenna or scattering system, the dimension of the subspace is essentially equal to N0, as given by (5).

[25] Accordingly, the operator G can be substituted with GN0 = PN0G, PN0 being the orthogonal projector [Taylor, 1958] onto the subspace of the incident fields, and step 1 can be settled in the null space [Taylor, 1958] of (1 − PN0)G, which is of dimension N0. Thus the size of step 1, and hence of the whole problem, is N0. This checks with the fact, first explicitly noted by Bucci and Di Massa [1995], that the set of the first N0 characteristic fields (i.e., the fields radiated by the characteristic currents) provides an accurate representation of the scattered field and shows that problem 1 can be settled in the space spanned by the first N0 characteristic modes.

4. Reducing the Computational Cost

[26] Let us now come to question 2. From an abstract point of view, the answer is trivial: As the operators G and G0 are of size N0, in principle the minimal computational cost of both steps 1 and 2 is of O(N0).

[27] This could be achieved, for instance, if the SVD of the corresponding optimal finite rank approximations were known in advance and the input and output quantities were expressed in terms of left and right singular vectors. In practice, this never happens, and the involved matrices are nondiagonal and, if nonredundant, full. Accordingly, apart from the filling cost and assuming nonredundancy, the computational cost of performing the heaviest step 1 by matrix inversion is of O(N03), whereas that of step 2 is of order N0M, M being the number of required field values.

[28] In order to see if the ideal limit of an O(N0) computational cost can be approached and to enumerate possible ways to reach this goal, let us first consider the simplest step 2, assuming that we are interested in evaluating the complete far-field pattern. This requires a value of M much larger than N0, which is essentially equal to the number of samples at the Nyquist rate [Bucci et al., 1998] required to represent the radiated field. By using optimal “traveling” sampling algorithms [Bucci and D'Elia, 1996], each required value can be computed starting from a limited number of near-field samples, taken at rate slightly larger than the Nyquist one. Thus, once a number of samples slightly larger than N0 has been computed, the evaluation of the required field values has a computational cost of O(M), which is, obviously, the best we can achieve. If we adopt a multilevel fast multipole algorithm or that recently proposed by Boag and Letreu [2003], the evaluation of the required samples via the radiation integral has a computational cost of O(N0log2(N0)). Accordingly, for step 2, the ideal limit of an O(N0) + O(M) cost can be closely approached.

[29] Coming to step 1, its cost can be reduced to O(N02) if an iterative approach is adopted, provided that the convergence is size-independent. By the way, this raises another question, that is, the speed of convergence of these iterative algorithms, which depends on the spectrum of the matrix in a nonobvious manner, and the issue of keeping the iteration count at bay is a fundamental one, since it tends to increase with the size of the problem.

[30] To see if the cost can be further reduced, approaching the ideal limit O(N0), let us note that (1) each step of an iterative algorithm involves the product of the system matrix by the vector of the unknowns, and (2) matrix entries have an individual physical meaning in terms of field radiated by a basis function source and “received” by a test function. Accordingly, if we adopt impulsive basis and test functions, each step of an iterative algorithm amounts to evaluating the field radiated by a (large) number of sources at a (large) number of points. Moreover, because the Green dyadic can be directly evaluated (or read from a lookup table), precalculation and storage of the system matrix is not required.

[31] As said before, in their FFT-like multilevel version, fast multipole algorithms allow one to perform such evaluation with a computational cost of O(Nlog2(N)), N being the number of source and test points. Unfortunately, a straightforward, direct discretization of the operator G requires a large number of points per wavelength to ensure a reasonable accuracy, leading to strongly redundant matrices, of dimension NN0. This can nullify the advantage of MLFMA as compared to direct multiplication by a nonredundant matrix or even a direct solution of (1), provided that the cost for evaluating the basis functions (characteristic modes, synthetic functions [Matekovits et al., 2001], etc.) and for matrix filling is not large. This happens, for instance, in the case of systems made of many similar, relatively small parts.

[32] Accordingly, the question is: It is possible to devise an approach exploiting a number of sources and test points not significantly larger than N0, yet ensuring the required precision? To provide a general answer to this question is clearly a difficult task. However, in the case of large, smooth metallic scatterers, a possible, positive answer can be provided by the method of auxiliary sources (MAS), as shown in section 5.

5. Toward an Effective Technique

[33] In the MAS, the induced current J is replaced by a set of “equivalent” auxiliary point sources, placed beneath the conductor surface, thus eliminating the singularities of the scattering integral equation. The MAS was mathematically justified by Kupradze [1967], who proved that the MAS solution approaches the exact one as the number of auxiliary sources goes to infinity. A simpler justification, more apt to our purpose, stems out from the following two facts, both consequences of the properties of the free space Green diadics: (1) If the field generated by the auxiliary sources fits, with a given precision, the scattered field on the scatterer's surface S, it represents with the same (or higher) precision the scattered field at its exterior, and (2) the set of the fields radiated by all finite families of point sources inside S is dense in the set of all fields on S.

[34] It follows that any scattered field can be approximated with arbitrary precision, on the scatterer surface and at its exterior, by the field radiated by a finite number of auxiliary sources. Accordingly, the crucial questions for an effective exploitation of the MAS are: (1) Can we represent the scattered field with a nonredundant number of auxiliary sources? and (2) can we determine the amplitudes of these sources by point matching the incident field? Both questions have a positive answer in the case of smooth metallic scatterers in free space.

[35] As a matter of fact, for such scatterers, if the arc length is adopted to parameterize the coordinate curves on the scatterer's surface S (see Figure 2), the incident field due to (not too near) arbitrary external sources is essentially band limited to β, the free space propagation constant [Bucci et al., 1998]. This implies that the incident field is determined with high accuracy by its Nyquist samples, with an intersample distance slightly smaller than λ/2, λ being the wavelength. In the same way, if the distance of the auxiliary sources from S is small compared with its size, but not less than some few wavelengths, their field on S is also essentially band limited to β, so that it is determined by its samples at the Nyquist lattice.

Figure 2.

Scattering from a smooth conductor.

[36] Accordingly, if the tangential field of the auxiliary sources fits the opposite of the incident tangential field at the sampling points, they provide the scattered field everywhere. This means that a number of auxiliary sources slightly larger then N0 allows an accurate representation of the scattered field, and the amplitudes of the sources can be determined by fitting the incident field at a proper lattice of points on the scatterer surface.

[37] Thus we conclude that in the case of (large) smooth metallic scatterers, it must be possible to develop a nonredundant MAS approach. It must be stressed that to the best of the author's knowledge, this is the first proof of the possibility of achieving nonredundancy in the MAS.

[38] According to the discussion of section 4, this means that if we couple the MAS with some kind of MLFMA, as, for instance, those described by Engheta et al. [1992] and Song et al. [1997] can allow the development of suboptimal numerical techniques. While the above conclusion shows that in principle the ideal limit can be approached in a practically relevant case, a full exploitation of this possibility requires, at least, the following.

[39] 1. Quantitative criteria should be determined, allowing assessment, for sufficiently general scattering systems, of the minimum number of sampling points (hence of auxiliary sources) required to ensure a given accuracy.

[40] 2. A rule for positioning the sources inside the scatterer's surface S should be determined. As is well known, this is a crucial point. In fact, as the auxiliary sources (ASs) approach S, the bandwidth needed to ensure the required precision of the corresponding field over S can become significantly larger than β, which causes an aliasing error when applying the point-matching procedure. On the other side, because of the behavior of the singular values of the radiation operator (see Figure 1), the matrix G becomes rapidly more and more ill conditioned as the sources move away from S and/or their number increases.

[41] Notwithstanding their relevance, these points have been scarcely dealt with in the MAS literature. In particular, the first one has not been considered at all, and a largely redundant number of ASs is typically adopted. The second point has been considered in a satisfactorily way only very recently [Anastassiu et al., 2004], with reference to the particular 2-D case of a circular cylinder under plane wave illumination.

[42] It is clear that providing a general and satisfactory answer to the above questions is by no means an easy task. To enlighten the way, which could lead to achieving the goal, let us consider the case of a perfectly conducting cylinder illuminated by a line source (see Figure 3) whose analytical solution is well known. Because of the involved symmetry, the auxiliary sources are uniformly placed on a circle of radius a0 < a.

Figure 3.

Geometry of the scattering from a conducting cylinder.

[43] Figures 4a and 4b refer to the case of a cylinder of radius a = 5λ, illuminated by a plane wave and by a source 2λ far from its surface, respectively. They show, for decreasing values of the ratio a0/a, the behavior of the relative error (in the mean square norm) we make by evaluating the scattered far field with the MAS, as a function of the ratio (nn0)/n0, n being the number of sampling points (and auxiliary sources) and n0 = 2βa, corresponding to an intersample distance equal to λ/2. As can be seen, the error rapidly saturates as the sources move inside the scatterer, and, for smaller values of a0/a, its behavior becomes erratic after a certain oversampling ratio.

Figure 4.

Relative error in scattered far-field evaluation via the MAS (a = 5λ): (a) plane wave incidence and (b) rs = 7.

[44] The saturation is caused by the fact that as the ASs shrink, the band limitation error of their field over S decreases, becoming negligible in comparison with that of the incident field. Accordingly, the saturation curve should reflect the behavior of this last error, which has been shown [Bucci and Franceschetti, 1989] to be (asymptotically) proportional to

equation image

where the constant C depends on the shape of the domain of interest.

[45] Fitting the data of Figures 4a and 4b, we get C = (3/2)3/2 and the dotted curves reported in the same figures. As can be seen, the agreement is excellent, irrespective of the source position. This would answer the first question, if it were not for the presence of the erratic part in the error behavior.

[46] As pointed out by Anastassiu et al. [2004], this is a consequence of the ill conditioning of the matrix G, which makes the norm of G−1 larger and larger as the sources move away from the surface and/or the oversampling ratio increases. However, the norm of the matrix G0G−1stays bounded, so that the ill conditioning of G does not have repercussions on the evaluation of the scattered field, until the condition number becomes so large that round-off errors become significant. This explains the sudden transition between regular and erratic behavior.

[47] For very low error levels, the ill posedness of G does not allow one to reach the limiting curve (6), and the maximum allowable condition number fixes the (redundant) number of required ASs, as well as the corresponding optimal value of a0 [Anastassiu et al., 2004]. On the other side, for reasonable values of the error (larger than −120 dB, say) it is the limiting curve (6) that determines the (not redundant) number of ASs, and we need some rule to choose a0. According to the previous considerations, the value of a0 must be such that it ensures a sufficiently low band limitation error for the field radiated by the ASs (laying on the circle of radius a0) on the circle of radius a. Now, according to (6), this error depends on the quantity βa0(n/(2βa0) − 1)3/2, which in the most critical case n = n0 reduces to βa0(a/a0 − 1)3/2 = 2π/λ(a/a0 − 1)3/2. Accordingly, in order to ensure a constant error, we must have

equation image

being, for large scatterers, a/a0 ≈ 1. The value of the constant A in (7) can be determined from the curves of Figure 4, which show that for a/λ = 5 the limit curve is practically reached for a0/a = 0.6. Substituting in (7), we get A = 0.5.

[48] As a check, Figure 5 shows the behavior of the MAS error under plane wave incidence when choosing a0/a according to (7), compared with the limit behavior (6), for values of a/λ ranging from 1 to 64. As can be seen, the agreement is excellent, even for values of a/λ comparable to 1. For large sizes, an oversampling of the order of 10–20% ensures a negligible error.

Figure 5.

Relative error in scattered far-field evaluation via the MAS under plane wave incidence.

[49] It must be stressed that (6) and (7) have been obtained by exploiting general results concerning the properties of scattered fields, that is, independently of the availability, in the case at hand, of an analytical solution. Accordingly, they are not so much relevant by themselves, but because they show that a clever exploitation of such properties as the field effective bandwidth can possibly lead to answering of our two fundamental questions in the general case. In particular, the availability of explicit and general expressions for the band limitation error [Bucci and Franceschetti, 1987] could be exploited to devise a criterion to find the optimal positions of the sampling points, whereas the “natural” coordinate system involved in the optimal sampling representations [Bucci et al., 1998], which is strictly tied to the scatterer's geometry, could allow one to face the problem of determining the optimal positions of the ASs. The same tool could be possibly used to address the case of nonsmooth scatterers, at least in the case of regular surfaces (i.e., surfaces composed of smooth pieces, with a finite number of corners and edges).