Adaptive Cross Approximation Algorithm Research Articles

Fast computation of monostatic radar cross section using compressive sensing and ACA-accelerated block LU factorization method

Compressive sensing (CS) technique in conjunction with adaptive cross approximation (ACA) algorithm is applied to calculate the monostatic radar cross section with many required sampling angles based on block LU factorization method. CS technique is used to construct a new excitation matrix and reduce the number of right-hand side. ACA algorithm is applied to all steps of the solution including impedance matrix filling, block LU solve, and excitation matrix compression to accelerate the computation process and reduce the memory consumption. Finally, the real-induced currents can be recovered by the orthogonal matching pursuit algorithm or compressive sampling matching pursuit algorithm, which have relatively low computational complexity than computation by the traditional method of moments (MoM). Numerical results are presented to validate the efficiency and accuracy of this method through comparison with the traditional MoM and other rigorous solutions.

An algorithm for low-rank approximation of bivariate functions using splines

We present an algorithm for the approximation of bivariate functions by “low-rank splines”, that is, sums of outer products of univariate splines. Our approach is motivated by the Adaptive Cross Approximation (ACA) algorithm for low-rank matrix approximation as well as the use of low-rank function approximation in the recent extension of the chebfun package to two dimensions. The resulting approximants lie in tensor product spline spaces, but typically require the storage of far fewer coefficients than tensor product interpolants. We analyze the complexity and show that our proposed algorithm can be efficiently implemented in terms of the cross approximation algorithm for matrices using either full or row pivoting.We present several numerical examples which show that the performance of the algorithm is reasonably close to the best low-rank approximation using truncated singular value decomposition and leads to dramatic savings compared to full tensor product spline interpolation.The presented algorithm has interesting applications in isogeometric analysis as a data compression scheme, as an efficient representation format for geometries, and in view of possible solution methods which operate on tensor approximations.

Fast Direct Solution of Electromagnetic Scattering from Left-Handed Materials Coated Target over Wide Angle

When solving the electromagnetic scattering problems over wide angle, the traditional method of moments (MoM) needs to repeat the solving process of dense systems of linear equations using the iteration method at each incident angle, which proved to be quite inefficient. To circumvent this problem, a fast numerical method based on block LDLT factorization accelerated by adaptive cross approximation (ACA) algorithm is presented to analyze the electromagnetic scattering of left-handed materials (LHM) coated target. The ACA algorithm is applied to impedance matrix filling and all steps of block LDLT factorization process, which can accelerate the computation process and reduce the memory consumption. The numerical results proved that the proposed method is efficient in calculating monostatic RCS of LHM coated target with many required sampling angles. Compared with the traditional MoM, computation time and memory consumption are reduced effectively.

EXTRACTING SURFACE MACRO BASIS FUNCTIONS FROM LOW-RANK SCATTERING OPERATORS WITH THE ACA ALGORITHM

The Adaptive Cross Approximation (ACA) algorithm has been used to compress the rank- deficient sub-blocks of the matrices that arise in the numerical solution of integral equations (IEs) with the Method of Moments. In the context of the linear embedding via Green's operator (LEGO) method — a domain decomposition technique based on IEs — an electromagnetic problem is modelled by combining bricks in turn described by scattering operators which, in many situations, are singular. As a result, macro basis functions defined on the boundary of a brick can be generated by applying the ACA to a scattering operator. Said functions allow compressing the weak form of the LEGO functional equations which then use up less computer memory and are faster to invert.

A Hybrid Approach for Fast Computation of Multiple Incident Angles Electromagnetic Scattering Problems with Compressive Sensing and Adaptive Cross Approximation

A hybrid compressive approach for fast computation of the electromagnetic scattering problems with multiple incident angles is proposed. The compressive sensing (CS) technique is firstly introduced to the method of moment (MoM) to reduce the number of the right-hand sides (RHS), but since the resulting excitation matrix contains linear dependency and has low-rank characteristics, the adaptive cross approximation (ACA) algorithm is used to recompress such excitation matrix, keeping only the necessary physical information. The hybrid compressive approach can reduce the number of the RHS to lower level. In fact, the ratio of the number of the RHS between the compressed excitation matrix and the original excitation matrix in the conventional MoM is close to one to fifteen. Numerical results are presented to validate the efficiency and accuracy of this method, which turns out to be highly efficient and accurate for the solution to multiple incident angles electromagnetic scattering problems.

A fast block low-rank dense solver with applications to finite-element matrices

This article presents a fast solver for the dense “frontal” matrices that arise from the multifrontal sparse elimination process of 3D elliptic PDEs. The solver relies on the fact that these matrices can be efficiently represented as a hierarchically off-diagonal low-rank (HODLR) matrix. To construct the low-rank approximation of the off-diagonal blocks, we propose a new pseudo-skeleton scheme, the boundary distance low-rank approximation, that picks rows and columns based on the location of their corresponding vertices in the sparse matrix graph. We compare this new low-rank approximation method to the adaptive cross approximation (ACA) algorithm and show that it achieves better speedup specially for unstructured meshes. Using the HODLR direct solver as a preconditioner (with a low tolerance) to the GMRES iterative scheme, we can reach machine accuracy much faster than a conventional LU solver. Numerical benchmarks are provided for frontal matrices arising from 3D finite element problems corresponding to a wide range of applications.

Multilevel Fast Adaptive Cross-Approximation Algorithm With Characteristic Basis Functions

This paper presents a multilevel fast adaptive cross-approximation (MLFACA) algorithm for accelerated iterative solution of the method of moments (MoM) matrix equation for electrically large targets. The MLFACA compresses the impedance submatrices between well-separated blocks into products of sparse matrices, constructed with the aid of the fast adaptive cross-sampling (FACS) scheme and the butterfly algorithm. As a result, the MLFACA can reduce both the computational time and the storage of the MoM to ${\mathbf{O}}({\mathbf{N}}\;{\mathbf{log}}^{\mathbf{2}}{\text{N}})$ , where ${\mathbf{N}}$ is the number of the Rao–Wilton–Glisson (RWG) basis functions in the analyzed target. Meanwhile, the MLFACA maintains the adaptive and kernel-independent properties. Furthermore, the characteristic basis function method (CBFM) is employed to decrease the size of the outer matrices of the MLFACA to further reduce the storage and iteration time. Numerical results are presented to demonstrate the advantages of the proposed method, including a successful solution of a scattering problem involving 10 861 668 RWG basis functions.

Parallel Marching-on-in-Degree Solver of Time-Domain Combined Field Integral Equation for Bodies of Revolution Accelerated by MLACA

Time-domain combined field integral equation (TD-CFIE) for bodies of revolution (BORs) is solved by marching-on-in-degree (MOD) method. A multilevel partitioning is adopted to group the spatial basis functions along the longitudinal dimension. The interactions of the adjacent groups are calculated directly in the traditional manner, while the impedance matrices associated with the well-separated groups at each level are computed by the multilevel adaptive cross approximation (MLACA) algorithm. The hybrid MPI and OpenMP parallel programming technique is utilized to further accelerate the solving process on a shared-memory computer system. Two numerical results demonstrate that the proposed method can greatly reduce the memory requirement and CPU time, thus enhance the capability of MOD method.

Stochastic Estimation of the Frobenius Norm in the ACA Convergence Criterion

The adaptive cross approximation (ACA) algorithm has been used in many fast Integral Equation solvers for electromagnetic Radiation and Scattering problems. It efficiently computes a low rank approximation to the interaction matrix between mutually distant parts of a scattering object. The ACA is an iterative algorithm that needs an accurate and efficient convergence criterion. The evaluation of this criterion may consume a considerable part of the computational resources. This communication presents an efficient new way to evaluate the convergence criterion, using a stochastic approach.

Multilevel Complex Source Beam Method for Electromagnetic Scattering Problems

This letter proposes a multilevel complex source beam (MLCSB) method for electromagnetic (EM) scattering problems. The multilevel structure is obtained with the octree grouping technique. Each basis function in the nonempty group is expanded with CSBs that tightly surround the group. The interactions between far groups are expressed with interactions of these CSBs. To accelerate the computation, the symmetry and shift invariance of the Green’s function is employed. The adaptive cross approximation (ACA) algorithm is then applied to accelerate the computation of interactions of CSBs. Due to the directionality of the radiation pattern of the CSBs, the interactions of CSBs are truncated with an angle threshold. Numerical examples demonstrate the efficiency of this method.

Hybrid Accelerated Method via Volume-Surface Integral Equation for Efficient Calculation of Monostatic RCS

A novel efficient hybrid accelerated method is proposed for the fast analysis of the monostatic electromagnetic scattering problems arising from volume-surface integral equations (VSIE) formulation. In the first step, by utilizing the low rank property, several largest eigenvalues and corresponding eigenvectors of the multiple right hand sides can be computed and saved efficiently by adaptive cross approximation (ACA) algorithm. The iterative solution of linear equations is required at these principle eigenvectors. Compared with solving linear equations at each angle repeatedly, the proposed method is able to greatly reduce the number of equations. In the second step, a disturbed symmetric successive over-relaxation (D-SSOR) preconditioner is constructed to speed up the convergence rate of iterative methods. Numerical results demonstrate that the present method can reduce the computational time significantly for monostatic VSIE calculation with high accuracy.

Simulation of Complex Multiscale Objects in Half Space With Calderón Preconditioner and Adaptive Cross Approximation

Simulation of electromagnetic scattering by complex multiscale objects in a half space is addressed in this communication. The objects under investigation can be situated above, embedded in, or even penetrating the interface of the half space. In this analysis, the electric field integral equation is formulated with the kernel of a half-space Green's function. The Calderón preconditioner is employed to improve the convergence of the Krylov-subspace iterative solvers. A new preconditioning operator is proposed to substantially reduce the construction cost, yet maintain the effectiveness. Moreover, a novel multilevel implementation of the adaptive cross approximation (ACA) algorithm is further adopted to make the method capable for problems with moderate size. The proposed scheme is examined by several multiscale problems, which constitute the most common scenarios in the near-surface applications.

A vectorized multiscale compressed decomposition‐based solver for partial element equivalent circuit method

SummaryThe high level of integration has made the analysis and design of integrated circuits and packages increasingly challenging. Hence, there exists an urgent need to reduce the computational complexity of existing numerical methods. The integral equation‐based method known as the partial element equivalent circuit (PEEC) method naturally generates an equivalent circuit that can be analyzed in both the time and frequency domains. The enforcement of Kirchhoff laws to the equivalent circuit may easily result into a very large set of equations whose solution can be extremely time‐consuming.In this paper, we propose a vectorized version, over the frequency sweep, of the adaptive cross approximation algorithm. Furthermore, the multiscale block decomposition is applied to the PEEC method, powered by a vectorization strategy and an efficient management of the random access memory. It is found that the proposed use of vectorization and compression techniques in the framework of the multiscale block decomposition results in a significant computational speedup of the frequency‐domain analysis of PEEC models. The efficiency and accuracy of the proposed method are demonstrated through its application to two pertinent problems. Copyright © 2014 John Wiley & Sons, Ltd.

On the Convergence of the ACA Algorithm for Radiation and Scattering Problems

The adaptive cross approximation (ACA) algorithm, when used to accelerate the numerical solution of integral equations for radiation and scattering problems, sometimes suffers from inaccuracies. These inaccuracies occur when the ACA convergence criterion, which is based on an approximation of the residual relative error, is prematurely satisfied. This paper identifies the two sources of this problem and proposes adaptations of the algorithm that remedy them.

Fast Analysis of Electromagnetic Scattering From Three-Dimensional Objects Straddling the Interface of a Half Space

The electromagnetic scattering from 3-D perfectly electrically conducting (PEC) objects straddling the interface of a half-space is analyzed in this letter. The adaptive cross approximation (ACA) algorithm is adopted to enhance the computational efficiency. As the object is naturally separated into two parts by the interface of the half-space, two different mesh densities are required due to the contrast of the two background materials. Therefore, two multilevel tree structures are set up individually in the two regions. When the source and field points are in the same region, which corresponds to the primary terms and reflection terms, the implementation is similar to the free-space case. However, when considering the mutual coupling of the two regions, which corresponds to the transmitted terms, difficulties arise since the two trees are independent and the numbers of levels are, in general, different. In this case, a z-dependent grouping strategy is proposed to redefine the well-separated interactions and match the abrupt changes at the interface. Three clustering strategies are proposed and discussed carefully to account for the mutual coupling in the two different regions of the half-space. The combined field integral equation (CFIE) formulation is further applied to improve the convergence of the system. Several numerical examples are demonstrated to validate the proposed schemes.

Angular glint calculation and analysis of radar targets via adaptive cross approximation algorithm

Angular glint is a significant electromagnetic (EM) scattering signature of extended radar targets. Based on the adaptive cross approximation (ACA) algorithm accelerated method of moments (MoM) and the plane incident wave assumption, the narrowband, wideband and newly developed high-resolution range profile (HRRP) based angular glint calculation formulations are derived and applied to arbitrarily shaped three-dimensional (3D) perfectly electrically conducting (PEC) objects. In addition, the near-field angular glint is emphasized, which is of great importance for radar-seeker applications. Furthermore, with the HRRP based angular glint, an approach to rigorously determine range resolution cells which own relatively smaller angular glint is provided. Numerical results are presented with new findings to demonstrate the usefulness of the developed formulations.

Efficient Analysis of EM Scattering From Bodies of Revolution via the ACA

In this communication, the adaptive cross approximation (ACA) algorithm is utilized to analyze the scattering from arbitrary metallic bodies of revolution (BoRs) that is formulated by the electric field integral equation approach. For a given mode, a multilevel partitioning is used to group the basis functions of the BoRs along the longitudinal dimension. The interactions of the adjacent groups are calculated directly by the method of moments (MoM), and the interactions of well-separated groups are compressed by the ACA algorithm. The memory requirement and CPU time consumption are reduced drastically.

Fast Adaptive Cross-Sampling Scheme for the Sparsified Adaptive Cross Approximation

A fast adaptive cross-sampling (FACS) scheme for the sparsified adaptive cross approximation (SPACA) algorithm is proposed to improve the conventional uniform spatial sampling. The FACS adaptively samples each well-separated block pair in an iterative manner to reach a given sampling error. At each iteration, the FACS first selects a set of initial samples with uniform spatial distribution for each block, and then uses the adaptive cross approximation (ACA) to find the important samples from the initial samples. Compared to the uniform spatial sampling, the FACS is easier to control the sampling error and needs fewer samples for the same sampling error. By reducing the number of samples, the FACS can enhance the efficiency of the SPACA. Numerical results are shown to demonstrate the merits of the proposed scheme.

Improved multilevel physical optics algorithm for fast computation of monostatic radar cross section

This paper proposes an acceleration technique to fast evaluate the monostatic radar cross section (RCS) with the multilevel physical optics (MLPO) algorithm. The proposed method combines the adaptive cross approximation (ACA) algorithm with the MLPO to fast evaluate two-dimensional monostatic RCS (two-dimensional monostatic RCS over a range of elevation angles θ and azimuths angles φ at a fixed frequency f, or a range of frequencies f and azimuths angles φ at a fixed elevation angle θ) responses. Owing to the phase compensation in the MLPO, the matrix corresponding to the compensated back-scattered field of each group is highly rank-deficient, which is compressed with ACA in a multilevel fashion therefore a lot of central processing unit time will be saved. The rank-deficiency, accuracy and computational complexity of the proposed method have been studied through a couple of numerical examples, which illustrate the effectiveness of the proposed approach.

The modified multilevel compressed block decomposition algorithms for analyzing the scattering of objects in half space

The convergence rate of iterative methods can vary in an unpredictable way. It is related to the matrix condition number, which is notoriously bad for the electric field integral equation in the large-scale electromagnetic problems. Therefore, an efficient direct solution—a multilevel compressed block decomposition (MLCBD) algorithm based on the adaptive cross-approximation algorithm—is applied to overcome this problem; it is very efficient for the monostatic problems. Simulation results of the objects up and below ground in half space demonstrate that the proposed MLCBD method is efficient for analyzing electromagnetic problems. Copyright © 2013 John Wiley & Sons, Ltd.