Sparse representation of plane wave response matrices for convex targets using local solution modes with band-limited excitations

Authors


Abstract

[1] A procedure is outlined for determining compressed representations of the plane wave response matrix (P matrix) for transverse magnetic scattering with respect to the z axis from convex cylinders. The method is based on the determination of band-limited spectral modes that excite spatially localized solutions to the wave equation and satisfy global boundary conditions. Numerical examples indicate that the proposed method provides a representation of the P matrix with reduced computational complexity.

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]

equation image

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 Ei 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(N2) 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(N3) matrix inversion techniques to determine Z−1. Section 7 indicates directions for future research that are expected to help overcome this O(N3) 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.

2. Convex Targets in Two Dimensions

[8] It is expected that the local-global solution (LOGOS) based compression method discussed below will be useful in a variety of two- and three-dimensional electromagnetic interaction problems. However, in the remainder of this paper, we restrict our description and application of the method to boundary integral equation formulations of transverse magnetic scattering with respect to the z axis (TMz) from convex, perfect electrically conducting rectangular cylinders. The scattering problems are formulated using electric (EFIE), magnetic (MFIE), and combined (CFIE) field integral equations [Peterson et al., 1998].

[9] In all of the examples we consider, finite projections of the continuous integral equations are obtained using a point-matching moment method discretization with N basis and N testing functions [Peterson et al., 1998]. This reduces the EFIE to a matrix equation of the form [Peterson et al., 1998]

equation image

where the vector Ezi contains samples of the z directed incident field on the surface of a target, and Jz is the vector of unknown coefficients for the z directed surface current.

[10] For the MFIE, a point-matching discretization yields a matrix equation of the following form [Peterson et al., 1998]:

equation image

where I is the N × N identity matrix, and Jzi is determined from the incident magnetic field. The CFIE is obtained by combining (1) and (3) [Peterson et al., 1998]:

equation image

where η is the characteristic impedance of the background medium, and 0 ≤ α ≤ 1. For convenience, we rewrite equation (4) as

equation image

The definitions of the matrix Cα and the vector Fαi are evident from a comparison of equation (4) with equation (5). The desired solution can be represented as

equation image

3. P Matrix

[11] As mentioned in section 1, in lieu of considering compression methods for Cα−1, we consider the related problem of compressing the plane wave response matrix (P matrix), which defines the currents excited on the surface of a target by a spectrum of incident plane waves. While the P matrix itself provides adequate information for a number of important applications, we expect that it might also be useful in developing more general representations for Cα−1.

[12] The P matrix is obtained by assuming that the incident field can be represented as a sum of propagating plane waves:

equation image

where ρ = equation imagex + equation imagey, k = k(equation image cos equation image + equation image sin ϕ), k = 2π/λ, and fi(ϕ) is the plane wave spectrum of the incident field. For the TMz polarization, the magnetic field satisfies the following relation at every point on the surface of a target:

equation image

where equation image is the outward unit normal at a point on the surface of the target. In the following we assume that the coordinate origin used in equations (7) and (8) is located at the center of the convex target.

[13] In the following we denote the respective discrete forms of equations (7) and (8) as

equation image

where D and Dn are discrete representations of the continuous integral operator appearing in these equations. In the numerical examples considered below, the vectors Ezi, Jzi and fi are obtained as point samples of their continuous counterparts. Similarly, the elements of D and Dn are point samples of ejk · ρ and (equation image · equation image)ejk · ρ multiplied by 1/Nϕ, where Nϕ is the number of uniformly spaced samples used to discretize equations (7) and (8). Accurate representation of these operators requires that Nϕ scales approximately linearly with the electrical size of the target (cf. equation (29) below).

[14] With the definition

equation image

we express equation (6) as

equation image

where

equation image

specifies the coefficients of the surface current approximation excited by a weighted sum of incident plane waves. For this reason, Pα is referred to as the plane wave response matrix (P matrix). If an exact discretization is used, the P matrix will be independent of α. We have retained the subscript on Pα in equation (12) because, in practice, the numerical approximation obtained for the P matrix will depend on the formulation used. We also note that while “P matrix” is an unfamiliar term, the associated mathematical operation (a mapping from incident plane waves to surface currents) is a common component in the numerical solution of a number of practical problems.

[15] Figure 1 illustrates the rectangular target, and Figure 2 illustrates the plane wave response matrix for TMz scattering from the rectangular target when a = 10λ and b = 3λ. The P matrix was obtained using a CFIE formulation with α = 1/2. Each column of the matrix in Figure 2 provides an approximation of the electric current excited on the surface of the target for incident plane waves originating from ϕ = 0 (far left) to ϕ = 2π (1 − 1/Nϕ) (far right). The P matrix is N × Nϕ where Nϕ = O(N) for electrically large targets.

Figure 1.

Rectangular target geometry.

Figure 2.

Gray scale image of absolute value of elements of P matrix for TMz scattering from a 10λ × 3λ perfect electrically conducting rectangle. A logarithm scaling is used. Black indicates matrix elements greater than 10, and white indicates matrix elements smaller than 0.01. N = 312 uniformly spaced samples were used to discretize the target. Nϕ = 174 angles were used to sample the incident plane wave spectrum.

4. Spatial Groups

[16] The method discussed below relies on a multilevel spatial decomposition of points on the surface of a target. Because we have restricted our attention to TMz electromagnetic scattering from convex cylinders, the target points can be uniquely ordered in the polar variable ϕ. The desired multilevel tree is obtained by recursively bisecting these points into nested groups. An illustration of the resulting multilevel spatial tree is provided in Figure 3.

Figure 3.

Illustration of multilevel binary tree used to define spatial groups on surface of convex target.

[17] The number of levels in the multilevel binary tree will be denoted by L. The individual levels are indexed by l, l = 1, …, L. The level l = 1 is the root level of the tree. The root level consists of a single group containing all spatial samples. The number of groups at the lth level of the tree is M(l) = 2l − 1, and the number of spatial samples in each group is approximately m(l) = N/M(l). The total number of levels, L, is chosen so that the level L groups have a maximum dimension of approximately one wavelength.

[18] The index i(l) is used to enumerate the groups at each level, i(l) = 1, …, 2l − 1. In the remainder of this paper we will denote a matrix associated with a given group, i(l), at specific level, l, using notation of the form Xi(l). The subscript i(l) is used to indicate both the level and the group number at that level with which the matrix is associated.

[19] The two level l groups contained by a given group at level (l − 1) are referred to as the level l children of that level-(l − 1) parent. Groups at level L have no children, and the root group has no parent. For p > l, the level p groups contained by a level l group, i(l), are referred to as the level p descendants of i(l). This set is denoted d[i(l),p]. The set d[i(l),l] is empty.

5. P Matrix Compression

[20] For a given tolerance ɛ, the band-limited, LOGOS-based compression method determines an order ɛ compressed representation of the P matrix using a basis of band-limited plane wave modes which excite currents in a sequence of successively larger spatial regions. Section 5.1 describes the LOGOS procedure used to determine incident plane wave modes which excite spatially localized current solutions that satisfy the global boundary conditions. A more efficient version of the LOGOS procedure based on band-limited plane wave modes is outlined in section 5.2. The resulting multilevel compression scheme for the P matrix is summarized in section 5.3.

5.1. LOGOS Modes

[21] Let the cross-sectional contour of a convex cylinder be divided into two regions based on the ith group at level l (group i(l)). Let region 1 consist of points on the surface of a target inside group i(l), and let region 2 consist of the remaining points. Define the matrix Dα,1 as that portion of the plane wave transform matrix (equation (10)) which specifies the currents incident on region 1, and let Dα,2 denote the remainder of Dα. Similarly, let the matrices Cα,11 and Cα,21 denote submatrices of Cα which specify the fields produced in region 1 and region 2, respectively, because of current sources in region 1. With these definitions, equation (5) can be written

equation image

where we have used Fi = Dαfi, and J1 (J2) is the part of the total current vector J that is located in region 1 (region 2). In equation (13), note that while the spatial variables have been split into distinct regions, the spectral variables (fi) are not split into distinct regions or groups.

[22] A region 1 LOGOS mode is a source/excitation pairing, (Ji, fii), in which the current is localized to region 1 and the pair satisfy the global boundary conditions. (The subscript i on Ji and fii is an integer used to index the LOGOS modes.) The remainder of this section outlines a numerical procedure to determine all LOGOS modes localized to a given spatial region which also satisfy the global boundary conditions to O(ɛ).

[23] To identify all solution modes localized to region 1, we first impose a local boundary condition,

equation image

The associated global boundary condition is

equation image

Combining equations (14) and (15) provides a single local-global condition on the desired modes,

equation image

The fii which satisfy equation (16) provide local solutions in region 1 which satisfy the global radiation condition from region 1 to region 2. Since we are not interested in the trivial solution fii = 0, we augment equation (16) with the condition

equation image

[24] The operators appearing in equations (16) and (17) are finite dimensional matrices, and there are generally no fii which exactly satisfy these simultaneous conditions. To find fii which approximately satisfy equation (16) subject to equation (17), we impose an approximate version of equation (16):

equation image

where

equation image

The symbol ∥·∥ is used to indicate the 2 norm of a vector.

[25] The fi satisfying equation (18) can be determined from the singular value decomposition (SVD) [Golub and Van Loan, 1996] of the augmented system

equation image

which is an N × Nϕ matrix. The SVD strategy described below simultaneously imposes equations (18) and (19) by first computing modes in the domain of X which contribute significantly to the total incident field on the surface of the target (cf. equation (22)). Next, we identify the modes in the domain of this truncated representation of X which contribute the fraction (1 − ɛ) or more of their energy to the rows of equation (20) associated with region 1.

[26] To begin this procedure, let usvH denote the order ɛ expansion obtained by truncating the exact SVD of X to retain only those singular modes which satisfy

equation image

where the si indicate the singular values of X:

equation image

Observe that the representation usvH in equation (22) includes all singular modes which significantly contribute the field incident on the target relative to the tolerance ɛ.

[27] A second set of SVDs can now be used to determine modes in the domain of XusvH which also satisfy equation (18). Beginning with equation (22), we observe that the matrix vs−1 provides a linear transform of X composed of vectors ordered by the degree to which they contribute to the incident field on the surface of the target. Following F. X. Canning (Compression of interaction data using directional sources and testers, 2003, patent application 20040010400), we seek an additional transform which sorts these modes by the degree to which their energy is localized within region 1. Let û denote the rows of u corresponding to observers located within region 1. The desired localizing transform is obtained by performing an SVD of û,

equation image

Let Si denote the diagonal elements of S. Because û is obtained by truncating the unitary matrix u, the singular values Si are bounded from above by unity. Those singular values which are close to 1 correspond to modes which focus a majority of their energy within region 1. We refer to those columns of V having singular values which satisfy

equation image

as modes which are localized within region 1 to order ɛ. (As discussed in section 5.3, the numerical examples presented below will use a localization tolerance of ɛ3 instead of ɛ2 in equation (24).)

[28] Let equation image indicate the subset of V with singular values Si satisfying equation (24). The columns of the matrix vs−1equation image provide a set of plane wave modes which satisfy equation (18) in region 2 to order equation image. Let Ki(l) denote the orthonormal basis which spans vs−1equation image (obtained, for example, from a QR decomposition),

equation image

The modes Ki(l) excite local currents which satisfy global boundary conditions. Because they excite local currents which satisfy global boundary conditions, we refer to the columns of Ki(l) as local-global solution (LOGOS) modes.

[29] The nonzero entries of the product

equation image

are, to order ɛ, located in rows corresponding to observation points in region 1 (i.e., group i(l)). In fact, according to equation (24), for a given column of Bi(l), the norm of the rows in that column associated with region 1 is O−2) larger than the norm of the remaining rows of the column. However, it follows from equation (22) that the condition number of Λi(l) is O−1). This is the reason that the truncation parameter used in equations (18) and (24) is the square of the desired tolerance.

[30] Our purpose in the remaining sections will be to use the spatial localization provided by the modes Ki(l) to determine an efficient order ɛ representation of the P matrix. In particular, the contribution of the localizing modes Ki(l) to the P matrix can be represented as

equation image

Ki(l) is evidently sparse relative to Pi(l) because of the spatial localization provided by the modes Ki(l). However, the matrix Ki(l) is not significantly sparser than Pi(l), since the number of plane wave modes (Nϕ) required to represent the P matrix is proportional to the electrical size of a target. In section 5.2 we consider the use of band-limited functions to control the sparsity of Ki(l).

5.2. Band-Limited LOGOS

[31] At a fixed frequency, the electromagnetic bandwidth (BW) associated with a region of space having a maximum physical dimension d scales approximately as d/λ [Bucci, 1989]:

equation image

The number of spectral samples (Nϕ) required to accurately represent these degrees of freedom to spatially separated observers is approximately [Coifman et al., 1993]

equation image

[32] Consider the problem of determining the localizing transform Ki(l) when the electrical size of group i(l) (region 1) is small relative to the remainder of the target (region 2). If d1 and d2 indicate the maximum dimensions of regions 1 and 2, then d2d1. The number of plane wave modes required to represent all incident current distributions generated in region 1 by distant sources is approximately Nϕ(d1). This is the number of columns required to accurately represent the domain of Dα,1 in equation (20).

[33] The number of plane wave modes required to represent all incident current distributions in region 2 is Nϕ(d2). Thus the number of columns required to represent Dα,1 and Dα,2 in equation (20) is much larger than the number of columns required to accurately represent Dα,1. (Recall that Dα,1 and Dα,2 are submatrices of Dα.)

[34] However, if we restrict our consideration to surface current solutions which are localized within region 1, then the incident currents in region 2 must cancel the fields radiated from region 1 to region 2. Since the number of degrees of freedom (DOF) in region 1 is limited to Nϕ(d1), it follows that the number of DOF required to represent the fields incident on region 2 is also on the order of Nϕ(d1). This suggests that when the current solution is localized to region 1 and d2d1, the number of angular modes required to represent Dα,2 in equation (20) is similar to the number required for Dα,1.

[35] Having established that the number of DOF required to represent the plane wave transforms Dα,1 and Dα,2 is roughly Nϕ(d1), appropriate discrete forms of the resulting, subsampled operators remain to be determined. It is well known that if the coordinate origin is located at the center of region 1, then the rows of Dα,1 have a bandwidth on the order of kd1 [Chew et al., 2001]. In this case, Dα,1 can be accurately represented by subsampling with Nϕ(d1) points. Let Dα,1 indicate the resulting matrix:

equation image

In equation (30), ρ1 indicates the center of region 1, and E(ρ1) is the diagonal operator which shifts the coordinate origin to the center of region 1,

equation image

The matrix S(Nϕ(d2), Nϕ(d1)) in equation (30) symbolizes the spectral subsampling operation applied to the rows of the product Dα,1E(ρ1). The argument Nϕ(d2) (Nϕ(d1)) indicates the number of rows (columns) in the matrix S. If d2d1, then Nϕ(d2) ≫ Nϕ(d1) (cf. equation (29)). In this case, S is a tall, thin rectangular matrix. One way to implement the indicated subsampling operation is using a fast Fourier transform (FFT) spectral truncation and subsequent inverse FFT procedure along each (band-limited) row of the product matrix Dα,1E(ρ1). Fortunately, the subsampling can be accomplished more simply by directly sampling the rows of Dα,2E(ρ1) at the desired Nϕ(d1) points. The latter scheme is used in the following examples.

[36] As suggested above, the number of angles used to represent Dα,2 can also be reduced to approximately Nϕ(d1) points. However, because the rows of the product Dα,2E(ρ1) are not band-limited functions, the matrix equation imageα,2 cannot be obtained by subsampling the rows of Dα,2E(ρ1) at Nϕ(d1) points. To avoid aliasing, we must apply a low-pass filter prior to subsampling:

equation image

In equation (32), F12 is an ideal low-pass filter which eliminates all angular modes aliased by the subsampling operation. The subsampling operator is the same as that indicated in equation (30), and it can again be implemented using an FFT-based filtering operation. However, a more efficient alternative is possible.

[37] Although equation (32) provides a reduced representation of Dα,2, its direct implementation requires that we first sample Dα,2 at Nϕ(d2) points prior to filtering and subsampling. A more efficient implementation is obtained by expanding the plane wave kernel of Dα,2E(ρ1) in a Fourier series and retaining only those modes which are in the passband of F12. In this case, the ideal filter F12 effectively truncates the Fourier series, thereby facilitating numerical evaluation of the remaining terms at Nϕ(d1) points on the interval [0,2π].

[38] The Fourier expansion of the plane wave kernel is [Harrington, 2001]

equation image

where δ1 = (ρρ1) is the vector from the center of region 1 to the point ρ. Since the left side of equation (33) is the kernel of Dα,2E(ρ1) in equation (32), the elements of the filtered operator, equation imageα,2, are obtained by truncating the sum on the right side of equation (33) to include only those Fourier modes in the passband of F12. Finally, we note that if m terms are retained in the expansion, the cost to evaluate the left side of equation (33) at m uniformly sampled points in ϕ using the FFT is O(mlog m).

[39] With equations (30) and (32), the filtered form of equation (20) is

equation image

which has dimension N × Nϕ(d1) and has a domain coordinate origin located at the center of region 1. This is the form of the localizing condition which will be used in the numerical examples to follow. In section 5.2, we summarize a simple multilevel algorithm which uses the band-limited, localizing modes obtained from equation (34) to compress the P matrix.

5.3. Multilevel Implementation

[40] Let K(L) denote the collection of all band-limited plane wave modes which excite currents localized to one of the M(L) spatial groups at level L. These modes are obtained by applying the LOGOS algorithm discussed above to the band-limited condition (34). We represent this procedure as

equation image

where equation imagei(L) is equation (34) for a given group at level L. As indicated by equation (27), we obtain a projection of the P matrix onto this basis as

equation image

where Λ(L) = PK(L).

[41] We would like to project the error, PP(L), onto a set of modes localized at the parent level, l = L − 1. However, a direct application of equation (34) at the parent level will yield a basis which contains modes already represented by K(L). This can be avoided by first removing all level L modes from the domain of equation imagei(L − 1). If Kd[i(L − 1),L] denotes the previously determined (via equation (35)) plane wave modes which localize currents within a level L descendant of group i(L − 1), then the reduced local-global condition for group i(L − 1) is

equation image

With the definition Kd[i(l),l] = 0, the multilevel generalization of equation (37) is

equation image

As with equation (35), K(l) is determined from equation imagei(l) using the LOGOS procedure,

equation image

[42] Having determined the K(l) via equation (39), the approximation to the P matrix is

equation image

where Λ(l) is the level l matrix of footprints:

equation image

The expansion (40) can be equivalently represented as

equation image

where Λ is the matrix obtained by concatenating all of the Λ(l). Similarly, the columns of K are obtained by concatenating the K(l).

[43] It is important to recognize that the multilevel representation defined by equations (39)–(42) does not directly enforce orthogonality on either K or K(l). (The Ki(l) are orthonormal because of equation (25).) This lack of orthogonality on the right side of equation (38) has been observed to reduce the efficiency of the method in some cases. However, as shown in section 6, it is possible to indirectly obtain an orthonormal basis (to order ɛ) by increasing the localizing tolerance on the right side of equation (24) from ɛ2 to ɛ3.

[44] The representation of the P matrix provided by equation (42) is efficient for convex targets because both Λ and K have sparse representations. In the former case, the only nonzero elements retained in each column of Λ are within the single illuminated spatial region to which a given mode of K is directed. Although K is a full matrix, the columns of K are band-limited functions and can be efficiently stored and manipulated in the spectral domain.

[45] To compute the order ɛ approximation of the current excited by an incident plane wave spectrum finc via equation (42), we use a plane wave translation and filtering procedure similar to that used in the disaggregation step of the multilevel fast multipole algorithm [Chew et al., 2001]. That is, the plane wave spectrum incident on the root group (l = 1) is first projected onto the spectral modes defined at the root level, K(1), with the result providing the complex weights for the level 1 current modes, Λ(1). The weighted current modes are subsequently accumulated in the global surface current vector. Proceeding to the next (finer) level, the level 2 spectrum for each child of the root group is computed from the incident spectrum via spatial translation and low-pass filtering. Each level 2 spectrum is then projected onto the local spectral modes, Ki(2). This result provides the complex weights for the localized current modes, Λi(2), which are again accumulated in the global surface current vector. This procedure is repeated for all groups at all levels in the multilevel tree, proceeding from the coarsest level to the finest level.

6. Numerical Examples

[46] Consider the problem of TMz scattering from the cylinder illustrated in Figure 1 when a = b. Figure 4 illustrates the Λ and K matrices of equation (42) when a = 50λ. A seven level (L = 7) binary tree was used to form the nested spatial groups. The number of groups at the finest level was M(L) = 64. The P matrix was formed using a CFIE formulation with α = 0.5 and a sampling density of twelve points per wavelength (N = 2400). A tolerance of ɛ = 10−2 was used in the LOGOS procedure. The actual normalized RMS error in the difference PαΛK was 2.2 × 10−3. The implementation of the LOGOS procedure for this example used a localization tolerance of ɛ3 instead of ɛ2 in equation (24) (cf. section 5.3).

Figure 4.

(top) Matrix Λ and (bottom) FFT of columns of K in local coordinates for scattering from square cylinder with a = b = 50λ. Nonzero entries are black.

[47] The total number of nonzero entries used to define Λ and K for the case considered in Figure 4 is 2.44 × 105. The number of nonzero entries in the P matrix is 2.22 × 106. The number of angular modes used for both Λ and K is 429. The number of singular modes which must be retained to obtain an SVD-based expansion of Pα with a similar relative RMS error (1.3 × 10−3) is 419. Thus, the number of modes used by the expansion (42) is close to optimal in this case.

[48] Figure 5 displays the increase in the complexity of the representation (42) as a function of N for the square cylinders considered in Figure 4 and also for rectangular cylinders having fixed heights of b = 10λ. In all cases, the contours are sampled with twelve points per wavelength. The complexity of the sparse representation is defined as the number of nonzero complex numbers required to store Λ and K. Similarly, the complexity of the P matrix is defined as the number of nonzero entries in Pα. For all cases considered, the complexity of the LOGOS-based representation increases at a rate which is less than O(N1.5), while the complexity of Pα increases approximately as O(N2). The formulation and tolerances used for the cases reported in Figure 5 are the same as those used in Figure 4.

Figure 5.

Total number of nonzero entries (nnz) in P matrix (squares) and in band-limited representation of P matrix (circles) as function of N for a square cylinder (open symbols) and a rectangular cylinder for which b = 10Λ (solid symbols). The dash-dotted line is the equation nnz = 0.8 N2. The dashed line is nnz = N1.5.

7. Summary and Discussion

[49] It has been demonstrated that a multilevel basis of band-limited LOGOS modes provides a sparse transformation (i.e., K) that yields a compressed representation of the plane wave response matrix for electrically large, convex objects. The complexity of the compressed representation has been observed to scale at a rate less than O(N1.5). To the best of our knowledge, this is the first demonstration of the existence of a sparse, error-controlled representation of the spectral solution operator (P matrix) having a complexity less than O(N2). Furthermore, because of the spatially localized nature of the current basis functions that comprise Λ, the compressed representation of the P matrix reported above leads naturally [Bucci, 1989] to a compressed representation of the plane wave scattering matrix wherein band-limited functions are used to represent both the incident fields (the fi above) and scattered fields. That the scattered fields have a band-limited representation follows directly from the spatially localized nature of the currents in Λ.

[50] The strategy used to compress the P matrix in this paper is based on two related concepts. First, it is possible to determine spectral modes which excite spatially localized currents that satisfy global boundary conditions. Second, the number of spectral DOF required to define these localizing modes is proportional to the electrical size of the localization region. Numerical examples indicate that the complexity of the resulting band-limited representation of the P matrix increases at a rate smaller than O(N1.5) for convex rectangular targets. Similar results have been observed for convex elliptical targets in two dimensions (in this case, the complexity of the compressed P matrix scales more slowly than observed above for rectangular cylinders). After appropriate modifications, it is expected that the results presented here can be extended to compress the P matrix for three-dimensional convex targets.

[51] The computational complexities associated with the compression method reported here can be further reduced. One way this might be achieved is by allowing spatial groups to overlap, which would allow more of the LOGOS modes to be captured at finer levels of the spatial tree. (This is expected to be particularly beneficial in determining LOGOS modes near the corners of the rectangular cylinders.) Additional computational savings might be obtained by imposing an orthogonalization condition on the localizing basis K. This would in turn allow us to use the localization tolerance of ɛ2 indicated by equation (24) instead of the more restrictive value of ɛ3 used in the numerical examples presented above.

[52] Although the band-limited LOGOS representation provides a relatively efficient representation of the P matrix, the CPU and memory resources required to determine this representation using the algorithms reported above scale approximately as N3 and N2, respectively. These costs are due to the SVD of equation image required by the LOGOS algorithm. (At the coarsest levels of the spatial tree, the number of rows and columns in equation image each scale linearly with N.) It is expected that these complexities can be significantly reduced.

[53] To indicate one way in which the cost to determine the band-limited representation of Pα might be reduced, we observe that the operational cost to determine the modes which localize currents at the finest levels of the spatial tree asymptotically scales as O(N2). This follows from the facts that (1) equation imagei(l) is O(N) × O(1), so its SVD is an O(N) operation, and (2) there are O(N) groups at the finest level of the tree. It may be possible to reduce the cost to find the SVD of equation imagei(l) at coarser levels by removing all previously determined angular modes from the domain of equation imagei(l). (Equation (38) only specifies the removal of angular modes associated with descendants of group i(l).) This would reduce the rank of equation imagei(l) and might facilitate the application of more efficient SVD algorithms designed for low-rank systems.

[54] It may also be possible to reduce the O(N2) cost currently required to determine the K(l) at fine levels by reducing the number of constraints used to enforce the global boundary condition for small groups. One way this reduction might be accomplished is by sparsely imposing constraints on the scattered fields in regions of the simulation domain sufficiently separated from the region of space to which the fine-level LOGOS modes are localized. Another possibility is to control the angular radiation patterns of the localized current modes. These possibilities are topics of ongoing investigations.

[55] The numerical examples considered in this paper have been limited to convex targets. To be useful in a variety of practical situations, it will be necessary to extend the results reported above to more general scattering configurations. We are presently considering two ways in which this might be done. Adams et al. [2005b] have demonstrated that if combined with a sparse representation of the pseudoinverse of the plane wave transform (Dα above), the P matrix provides a sparse representation for some interior (e.g., cavity) scattering problems excited by finite sources. (Sparse representations of the pseudoinverse of Dα have been discussed elsewhere [Adams et al., 2006b].)

[56] Second, it is expected that the extension of the algorithms and concepts discussed in this paper to nonconvex targets will require the incorporation of individual LOGOS modes that are localized to multiple, distinct spatial regions. Such solution modes might be interpreted as multiple-bounce LOGOS modes. In this context, the results of the present paper can be viewed as the application of single-bounce LOGOS modes to the determination of sparse representations of the spectral solution operator (P-matrix) associated with convex targets.

Acknowledgments

[57] The authors gratefully acknowledge reviewer comments that helped improve the discussion of this paper. The work was supported by the Office of Naval Research under contract N00014-04-1-0485 and by the Center for Computational Sciences at the University of Kentucky.

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