A Truncated Singular Value Decomposition Enhanced Nested Complex Source Beam Method

This work presents a novel matrix compression algorithm to improve the computational efficiency of the nested complex source beam (NCSB) method. The algorithm is based on the application of the truncated singular value decomposition (TSVD) to the multilevel aggregation, translation, and disaggregation operations inNCSB. In our implementation, the aggregation/disaggregation matrices are solved by the truncated far-fieldmatching, which is based on the directional far-field radiation property of the complex source beams (CSBs). Furthermore, the translation matrices are obtained according to the beam width of CSBs. Due to the high directivity of the radiation patterns of CSBs, all the far-field related interaction matrices are low-ranked. Therefore, TSVD can be employed and a new set of equivalent sources can be constructed by a linear combination of the original CSBs. It is proved that the radiation power of the new sources is proportional to the square of the corresponding singular values. This provides a theoretical guideline to drop the insignificant singular vectors in the calculation. In doing so, the efficiency of the original NCSB method can be much improved while a reasonably good accuracy is maintained. Several numerical tests are conducted to validate the proposed method.


Introduction
Integral equation (IE) method has been intensively studied in the analysis of electromagnetic (EM) radiation and scattering in recent years.For a numerical solution, the IE is usually discretized into a matrix equation by the method of moments (MoM) [1].In order to conduct large-scale simulations, a variety of numerical techniques have been developed based on MoM.These techniques are carried out efficiently either by reduction of the far-field interaction exploiting the physical or mathematical properties of the IE or by using well-designed basis/test functions to reduce the total number of unknowns.
In the first category, the multilevel fast multipole algorithm (MLFMA) [2] is widely used, where the far-field interactions can be accelerated by aggregation, translation, and disaggregation operations.Besides it, the low rank property of the MoM far-field submatrices leads to a series of matrix decomposition methods, such as the multilevel matrix decomposition algorithm (MLMDA) [3] and the adaptive cross approximation (ACA) [4,5].For the second category, the higher order basis functions (HOBF) [6], phase extracted basis functions (PE) [7,8], characteristic basis function (CBF) [9,10], or body of revolution (BoR) [11] MoM are proposed to reduce the number of unknowns in a given problem.
Following the first category, a complex source beammethod of moments (CSB-MoM) is recently proposed to accelerate the far-field interactions of MoM [12].In this method, the object is divided into groups and complex source beams (CSBs) [13] are used to expand the fields of the basis functions residing in each group [14,15].Hence, the far-field interactions of these basis/test functions can be accounted for by their equivalent CSB expansions.Due to the directional nature of CSBs [16], the interactions usually involve only a small portion of the total CSBs.A multilevel version of this method is developed in [17] to further improve the computational efficiency.The branch cut issue of CSBs, which might degrade the accuracy in the evaluation of group interactions, can be avoided by a proper choice of the CSB parameters.
Unfortunately, in this method, the CSB expansions at any level are calculated directly from the basis functions contained in the finest level.This operation is very computationally expensive.Hence, the application of this method to electrically large problems is prohibited.To overcome this difficulty, a nested complex source beam (NCSB) method is proposed by utilizing an equivalent relationship between adjacent levels [18].This relationship is built by treating CSBs in the child group as new sources and applying the far-field matching to get their CSB expansions in the parent group.In doing this, the computational complexity of NCSB can be reduced to ( log ), where  is the number of unknowns.However, it should be noted that the CSBs involved in the translation process have to be determined empirically.Moreover, the aggregation and disaggregation matrices are still in a dense format.
To fully exploit the directional property of CSBs, a truncated singular value decomposition (TSVD) method is applied to compress the aggregation, translation, and disaggregation matrices of NCSB in this paper.After SVD, a set of equivalent sources can be obtained by a linear combination of the original CSBs.A theoretical proof reveals that the radiation power of the new sources is proportional to the square of the corresponding singular values.This provides a guideline for the truncation in the calculation.Thereby, the proposed method not only leads to a significant improvement of the computational efficiency but also provides a flexible compromise between accuracy and computational cost.

Formulations
Given a 3D perfectly electrical conducting (PEC) body defined by its surface, a MoM matrix equation can be obtained as where I is the unknown vector containing the expansion coefficients of the current, V is the excitation vector, and Z is the MoM impedance matrix.After grouping, this dense matrix can be separated into the near-field interaction part Z  and the far-field interaction counterpart Z  .
In CSB-MoM, the far-field interactions between different groups are carried out by a series of CSBs launched on a complex equivalence surface enclosing each group [14].The far-field part of the matrix-vector product (MVP) in CSB-MoM can be represented as where  and   denote the observation and source groups and  indicates the finest level (single level in CSB-MoM).

W 𝑝 󸀠
and Y   are, respectively, the expansion matrix and local expansion matrix for both  and  components.The farfield matching technique can be used for constructing W     and Y   in electric field integral equation (EFIE), magnetic field integral equation (MFIE) [19], and PMCHWT integral equation [20].T   ,  is the translation matrix, of which the elements are expressed as where p and p   denote the unit vectors of th and   th CSB, respectively.The complex position vectors r , and r   ,  are the launch points of CSBs.(r   | r     ) is the dyadic Green function with complex arguments.
By using this representation, the CSB expansion coefficients for the source group   can be expanded from the surface currents as On the other hand, the CSB expansion coefficients R ,  for the receiving group  are translated from the transmitting group   : As has been mentioned previously, the directional property of CSBs can be used to reduce the computational cost in the translation procedure.The translation window can be set based on numerical experiments for different group sizes and CSB parameters, as is done in [18].

NCSB Formulations.
Motivated by the idea of MLFMA [2], CSB-MoM can be naturally extended to a multilevel version if a similar octree data structure is adopted.However, the multilevel CSB or MLCSB proposed in [17] calculates the CSB expansion coefficients for each level directly from the basis functions in the finest level.The lack of a proper mechanism of interlevel aggregation and disaggregation prohibits this method from analyzing large-unknown problems.
To end this, the NCSB is proposed in [18] to introduce the aggregation and disaggregation operations between every two adjacent levels through a proper far-field matching.
In the following, we obtain the aggregation matrix in a slightly different way by further utilizing the directional property of the radiating far field of CSBs.Moreover, the symmetry is fully exploited so that only one aggregation matrix is needed for all the eight child groups of one parent group.First, a linear system is set up to build the equivalent relationship of CSBs between two adjacent levels: where Z  is the matching matrix that connects the equivalent CSB sources with the far fields F +1, .The expression of the matching matrix is the same as the matrix used for CSB expansion with electric current type equivalent sources in CSB-MoM [12,14].Different from CSB-MoM of single level, the equivalent CSB sources in Z  are launched in level , and the far fields here are radiated by CSBs in child level  + 1 with both  and  components: where the elements in each block matrix are It is noted that, for a parent group in the octree, there are eight different child groups at most.Hence, eight corresponding aggregation matrices are required in the calculation.However, by fully exploiting the symmetry property, we discover that only one aggregation matrix is needed and others can be easily deduced from it.In the following, the subscript  will be omitted for simplicity.
In our method, the radiating far fields on the righthand-sides of ( 6) can be fast calculated by truncating CSB directional fields within the paraxial regions.Once Z  and F +1,c are assembled, the aggregation matrix for level can be numerically solved.Here   and  +1 are the numbers of CSBs in levels  and  + 1.In this work, we apply the least square method to improve the stability of the solution.In (9), the element [ Ã

𝑙
]    +1 in the block aggregation matrix is the mapping coefficient between the  +1 th CSB in the child group and the   th CSB in its parent group.By using this relationship, the disaggregation matrix can be easily obtained from the transpose of the aggregation matrix.The CSB expansion coefficients of the parent group in level  can be obtained efficiently from its child groups in level  + 1 with the aggregation matrix: Similar to MLFMA, the CSB expansion coefficients of a receiving group in level  + 1 is obtained from both of the translation in the same level and the disaggregation from its parent level .Therefore, we have As has been elaborated in the above, the NCSB algorithm can be constructed in much the same way as MLFMA.However, all the operation matrices involved in NCSB are dense matrices, in contrast with the diagonal matrices in MLFMA.Improvement can be made by utilizing the unique property of CSBs.Specifically, all the CSBs are directional; that is to say, a CSB can only directionally interact with another CSB in the far-field region.Hence, the dense aggregation/disaggregation and translation matrices are usually low-ranked.To take advantage of the directional property of CSBs, TSVD will be applied to compress these matrices in the following sections.

TSVD in Aggregation/Disaggregation Process.
For the aggregation matrix in (9), the SVD factorization can be applied as where The physical meaning of the above SVD can be interpreted as follows.One set of the child CSBs is coupled to another set of parent CSBs, with the coupling strength governed by the corresponding singular value.Hence, after SVD, the aggregation procedure is converted from beam-tobeam coupling into mode-to-mode coupling.Furthermore, we can prove that the radiated power by each child CSB mode is proportional to the square of the singular value (see Appendix).This power-related mode provides a helpful guideline for extracting the most significant part in the aggregation matrix.
If   is the effective rank of A  for a prescribed threshold , the SVD decomposition in (12) can be truncated and approximated as where Σ  is formed by the first (largest)   singular values in Σ   satisfying    +1 <  1 .Ũ  and Ṽ  denote the corresponding submatrices consisting of the first   columns of U   and V   , respectively.Hence, Ṽ  contains all the child CSB modes having radiated power higher than  2  2 1 .According to the prescribed accuracy, this reduced set of CSBs in child group suffices to represent the far field of the parent group.Since disaggregation is implemented as the transpose of aggregation, the TSVD procedure of disaggregation matrix can be implemented by fully utilizing Ũ  and Ṽ  in (13).

TSVD in Translation Process.
As the CSB decays rapidly in the directions orthogonal to the beam propagation axis, in the translation process, the interactions between farfield groups are carried out by CSBs whose beam axis are alined with the line connecting the group centers.These CSBs can be easily selected by setting a conical truncation window.To avoid amount of experiments needed to determine the minimal conical angle, in this paper, we first set the truncation angle when the field of a CSB decays to −20 dB of the aforementioned prescribed accuracy times the maximum value.According to this criterion, the CSB selection matrices L  ,  and L ,  are introduced to pick up where Ũ,  ,  , Σ,  ,  , and Ṽ,  ,  are defined similarly as in (13), except that the threshold is set as  2 instead of  to account for the effects from both of the transmitting and receiving groups.
The proposed TSVD procedure is effective by providing new set of CSB modes based on the singular vectors.The significance in the total radiated power is governed by the descending singular values.The SVD process fully exploits the spatial window property of CSBs, which is independent of the geometry, current distribution, and types of IE.By defining an error threshold, redundant modes which contribute negligibly to the radiated power are excluded.Therefore, the aggregation, translation, and disaggregation matrices can be represented in a very compact form.Furthermore, the balance between accuracy and computational efficiency can be easily controlled by adjusting the truncation threshold.

Numerical Results
In this section, several numerical tests are conducted to demonstrate the efficiency and validity of the proposed method.
Firstly, the low rank property of the aggregation matrices of NCSB method is studied.Figure 1 shows the normalized singular values of the aggregation matrices for level 2 (with electrical size of 8 and aggregating from level 3) and level 3 (with electrical size of 4 and aggregating from level 4).The numbers of CSBs for levels 2, 3, and 4 are 5738, 1758, and 682, as listed in Table 1.Considering both  and  components of CSBs, the dimensions of the aggregation matrices for level 2 and level 3 are 11476 × 3516 and 3516 × 1364, respectively.As shown in Figure 1, the singular values for both matrices decrease rapidly, indicating a quick decay of the radiated power from individual child CBS mode.Due to this fact, the aggregation matrices are truncated by thresholds of  = 10 −2 , 10 −3 , and 10 −4 for comparison.Figure 2 depicts the relative errors of the radiated far field (an effective measure of the aggregation error) with respect to the observation angles.In particular, this relative error is defined as and here E() is the reference field, E agg () is the field obtained via aggregations by TSVD, and  is the observation angle which is set on a azimuth circle.In Figure 2, the reference field is radiated directly by the CSBs of a group with the size of 2 in the fourth level.Then two aggregation steps with TSVD are performed to this group until the second level.Finally, the field E agg () is calculated by the CSBs in the second level.Figure 2 shows that the relative aggregation error can be effectively controlled by the TSVD threshold .

Conclusion
In this paper, we have proposed an efficient implementation of NCSB.Different from previous work, the aggregation process is first constructed by the truncated far-field matching.
The dimension of the translation matrix is then reduced based on the beam width of the CSBs.Finally, TSVD is used to compress the aggregation, translation, and disaggregation matrices by fully exploiting the directional property of CSBs.
It is shown that the radiated power of the new sources from SVD is proportional to the square of the corresponding singular values.This power-dependent mode provides a theoretical guideline for extracting the most significant CSB contributions in the calculation.Therefore, the desired balance between accuracy and computational efficiency can be easily controlled by adjusting the truncation threshold.

Figure 1 :
Figure 1: Normalized singular values of the aggregation matrix.

Figure 5 :
Figure 5: Bistatic RCS of a PEC sphere with a diameter 32, by NCSB-TSVD and Mie-series solution.
[U   ] 2  ×2  and [V   ] 2 +1 ×2 +1 are unitary matrices and Σ   = diag{ 1 ,  2 , . . .,  2 +1 } 2  ×2 +1 is a diagonal matrix whose elements are the nonnegative real singular values listed in a descending order.The columns of V   represent a new set of orthonormal child CSB modes in level  + 1, which are obtained as linear combinations of the original CSBs.Matrix U   defines a complete set of orthonormal parent CSB modes associated with the child modes.Diagonal matrix Σ   maps the child modes to the corresponding parent modes with weighting of the singular values.

Table 1 :
Computational statistics.×= [L ,  ]   ×  [ T  ,  ]   ×  [L  ,  ]   ×  , ,  ) is 1,if the corresponding CSB is within the conical truncation window; otherwise they are set to 0.After this truncation based on the beam width of CSBs, the dimension of the translation matrix T  ,  is reduced from   ×   to   ×   .However, it is still rank-deficient.The TSVD procedure is then applied to this reduced translation matrix to gain further compression: ,