José M. Tamayo

Multiscale Compressed Block Decomposition for Fast Direct Solution of Method of Moments Linear System

The multiscale compressed block decomposition algorithm (MS-CBD) is presented for highly accelerated direct (non iterative) solution of electromagnetic scattering and radiation problems with the method of moments (MoM). The algorithm is demonstrated to exhibit N2 computational complexity and storage requirements scaling with N1.5, for electrically large objects. Several numerical examples illustrate the efficiency of the method, in particular for problems with multiple excitation vectors. The largest problem presented in this paper is the monostatic RCS of the NASA almond at 50 GHz, for one thousand incidence angles, discretized using 442,089 RWG basis functions. Being entirely algebraic, MS-CBD is independent of the Greens function of the problem.

Fast Iterative Solution of Integral Equations With Method of Moments and Matrix Decomposition Algorithm – Singular Value Decomposition

The multilevel matrix decomposition algorithm (MLMDA) was originally developed by Michielsen and Boag for 2D TMz scattering problems and later implemented in 3D by Rius et al. The 3D MLMDA was particularly efficient and accurate for piece-wise planar objects such as printed antennas. However, for arbitrary 3D problems it was not as efficient as the multilevel fast multipole algorithm (MLFMA) and the matrix compression error was too large for practical applications. This paper will introduce some improvements in 3D MLMDA, like new placement of equivalent functions and SVD postcompression. The first is crucial to have a matrix compression error that converges to zero as the compressed matrix size increases. As a result, the new MDA-SVD algorithm is comparable with the MLFMA and the adaptive cross approximation (ACA) in terms of computation time and memory requirements. Remarkably, in high-accuracy computations the MDA-SVD approach obtains a matrix compression error one order of magnitude smaller than ACA or MLFMA in less computation time. Like the ACA, the MDA-SVD algorithm can be implemented on top of an existing MoM code with most commonly used Greens functions, but the MDA-SVD is much more efficient in the analysis of planar or piece-wise planar objects, like printed antennas.

Fast Direct Solution of Method of Moments Linear System

A fast direct (non iterative) solution method for the method of moments (MoM) in electromagnetics is proposed. The method uses the well known matrix decomposition method (MDA) and singular value decomposition (SVD) to achieve block-wise compression of the MoM impedance matrix, followed by a block-wise LU factorization that preserves the initial compression. A number of examples are presented involving problems ranging from ten to seventy thousand unknowns.

Multilevel Adaptive Cross Approximation (MLACA)

The Multilevel Adaptive Cross Approximation (MLACA) is proposed as a fast method to accelerate the matrix-vector products in the iterative solution of the linear system that results from the discretization of electromagnetic Integral Equations (IE) with the Method of Moments (MoM). The single level ACA, already described in the literature, is extended with a multilevel recursive algorithm in order to improve the asymptotical complexity of both the computational cost and the memory requirements. The main qualities of ACA are maintained: it is purely algebraic and independent of the integral equation kernel Greens function as long as it produces pseudo-rank-deficient matrix blocks. The algorithm is presented in such a way that it can be easily implemented on top of an existing MoM code with most commonly used Greens functions.

Fast and Accurate Computation of Hypersingular Integrals in Galerkin Surface Integral Equation Formulations via the Direct Evaluation Method

Hypersingular 4-D integrals, arising in the Galerkin discretization of surface integral equation formulations, are computed by means of the direct evaluation method. The proposed scheme extends the basic idea of the singularity cancellation methods, usually employed for the regularization of the singular integral kernel, by utilizing a series of coordinate transformations combined with a reordering of the integrations. The overall algebraic manipulation results in smooth 2-D integrals that can be easily evaluated via standard quadrature rules. Finally, the reduction of the dimensionality of the original integrals together with the smooth behavior of the associated integrands lead up to unmatched accuracy and efficiency.

Compressed Block-Decomposition Algorithm for Fast Capacitance Extraction

A novel algorithm, the compressed block decomposition, is presented for highly accelerated direct (noniterative) method-of-moment capacitance extraction. The algorithm is based on a blockwise subdivision of the method-of-moment potential coefficient matrix. Matrix subblocks corresponding to distant subregions of the problem geometry are not calculated directly but approximated in a compressed form. Subsequently, the matrix is decomposed using an algorithm that preserves the compression. The efficiency of the method is demonstrated on a common benchmark problem - a 6 times 6 bus crossing. The numerical cost of the algorithm is shown, both theoretically and numerically, to scale with N log3 N and the storage space with N log2 N.

Sparsified Adaptive Cross Approximation Algorithm for Accelerated Method of Moments Computations

This paper presents a modification of the adaptive cross approximation (ACA) algorithm for accelerated solution of the Method of Moments linear system for electrically large radiation and scattering problems. As with ACA, subblocks of the impedance matrix that represent the interaction between well separated subdomains are substituted by “compressed” approximations allowing for reduced storage and accelerated iterative solution. The modified algorithm approximates the original subblocks with products of sparse matrices, constructed with the aid of the ACA algorithm and of a sub-sampling of the original basis functions belonging to either subdomain. Because of the sampling, an additional error is introduced with respect to ACA, but this error is controllable. Just like ordinary ACA, sparsified ACA is kernel-independent and needs no problem-specific information, except for the topology of the basis functions. As a numerical example, RCS computations of the NASA almond are presented, showing an important gain in efficiency. Furthermore, the numerical experiment reveals a computational complexity close to N logN for sparsified ACA for a target electrical size of up to 50 wavelengths.

Accelerated Direct Solution of the Method-of-Moments Linear System

This paper addresses the direct (noniterative) solution of the method-of-moments (MoM) linear system, accelerated through block-wise compression of the MoM impedance matrix. Efficient matrix block compression is achieved using the adaptive cross-approximation (ACA) algorithm and the truncated singular value decomposition (SVD) postcompression. Subsequently, a matrix decomposition is applied that preserves the compression and allows for fast solution by backsubstitution. Although not as fast as some iterative methods for very large problems, accelerated direct solution has several desirable features, including: few problem-dependent parameters; fixed time solution avoiding convergence problems; and high efficiency for multiple excitation problems [e.g., monostatic radar cross section (RCS)]. Emphasis in this paper is on the multiscale compressed block decomposition (MS-CBD) algorithm, introduced by Heldring , which is numerically compared to alternative fast direct methods. A new concise proof is given for the N2 computational complexity of the MS-CBD. Some numerical results are presented, in particular, a monostatic RCS computation involving 1 043 577 unknowns and 1000 incident field directions, and an application of the MS-CBD to the volume integral equation (VIE) for inhomogeneous dielectrics.

Multilevel MDA-CBI for fast direct solution of large scattering and radiation problems

This paper presents the multilevel MDA-CBI method for accelerated direct solution of scattering and radiation problems with the method of moments. The method has all the advantages of a direct solution (no convergence issues, very little marginal cost per excitation vector). For a medium size problem of about 70.000 unknowns it performs some 25 % better than the one level algorithm, but this advantage is expected to grow for larger problems since the computational complexity is considerably better, with a theoretically attainable value of N5/3log2N.

Multiscale CBD for fast direct solution of MoM linear system

This paper presents the Multiscale CBD method for accelerated direct solution of scattering and radiation problems with the Method of Moments. The method has all the advantages of a direct solution and it is essentially parameter-free (there is only a parameter tau controlling accuracy against efficiency). It has an improved efficiency and storage requirements with respect to the ordinary CBD. Two significant computations have been presented, an open pipe and the NASA almond.