Combining Loop-Tree decomposition with Accelerated Cartesian Expansion

A low frequency stable plane wave addition theorem

Calderon preconditioners have recently been demonstrated to be very successful in stabilizing the electric field integral equation (EFIE) for perfect electric conductors at lower frequencies. Previous authors have shown that, by using a dense matrix preconditioner based on the Calderon identities, the low frequency instability is removed while still maintaining the inherent accuracy of the EFIE. It was also demonstrated that the spectral properties of the Caldero-n preconditioner are conserved during discretization if the EFIE operator is discretized with Rao-Wilton-Glisson expansion functions and the preconditioner with Buffa-Christiansen expansion functions. In this article we will show how the Calderon multiplicative preconditioner (CMP) can be combined with fast multipole methods to accelerate the numerical solution, leading to an overall complexity of O(N logN) for the entire iterative solution. At low frequencies, where the CMP is most useful, the traditional multilevel fast multipole algorithm (MLFMA) is unstable and we apply the nondirectional stable plane wave MLFMA (NSPWMLFMA) that resolves the low frequency breakdown of the MLFMA. The combined algorithm will be called the CMP-NSPWMLFMA. Applying the CMP-NSPWMLFMA at open surfaces or very low frequencies leads to certain problems, which will be discussed in this article.

The iterative solution of integral equations containing the Green function of the Helmholtz equation as the integration kernel requires repeated matrix-vector products. These products can be accelerated by means of a so-called fast multipole method (FMM). Of the many fast multipole methods in use today, the multilevel fast multipole algorithm (MLFMA) is arguably among the most successful ones. It allows the simulation of electrically large structures that are intractable with direct or unaccelerated iterative solvers. Testimony to the MLFMAs myriad uses is its implementation in various commercial EM software packages such as FEKO and CST Microwave studio. However, the MLFMA has one big drawback: a numerical instability prevents the method from being used on low frequency (LF) interactions, i.e. interactions between sources and observers that are less than approximately one wavelength apart. As a consequence configurations containing significant sub-wavelength geometrical detail cannot be efficiently treated using the MLFMA alone and a hybrid method is necessary. However, the LF methods in use today are generally less efficient due to non-diagonal translation operators (multipole methods) or the need for six radiation patterns (spectral methods). In this contribution a novel algorithm, called the nondirective stable plane wave multilevel fast multipole algorithm (NSPWMLFMA) [1], will be presented that is stable at LF, exhibits diagonal translation operators and requires only one radiation pattern. The method is based on an analytical expression for a translation operator in the z-direction. This translation operator is made numerically stable using a shift of the integration path into the complex plane. It even has a DC-limit. A QR-based method is then used to extend the applicability to all the other translation directions. The algorithm has also been parallelized using open FMM [2]. Finally some numerical results will be shown.

The fast multipole method (FMM) and multilevel fast multipole algorithm (MLFMA) are successfully developed to solve electrically very large sized perfectly electric conducting (PEC) objects. Using the impedance boundary conditions (IBC) the integral equations can be simplified and solved for wave problems involving imperfectly conducting surfaces. Thus, the algorithm is firstly implemented and verified by comparing with the exact analytical solution for a PEC sphere, and then extended for imperfectly conducting (thin material coating) objects and verified with the analytical solution. Subsequently, it has been applied to an aircraft model with both PEC and material coating surface the results and compared.

This paper presents a novel multi-step iterative scheme to solve system of nonlinear equations. Because the cost of calculating the Frechet derivative evaluation and its inversion is significant, in order to achieve its high computational efficiency, we have tried to calculate Frechet derivative and its inverse less per cycle by using the proposed multi-step iterative schemes. The basic iterative scheme has a convergence order of four; therefore, repetition of the second step can achieve higher convergence order. In fact, adding a new step to the base iterative scheme each time increases the convergence order by two units. The multi-step iterative schemes have convergence-order 2m, where m is the number of steps of the multi-step iterative schemes. Also, the computational efficiency of the iterative scheme is compared with other available methods. The numerical results presented confirm the theoretical results. A number of nonlinear system of equations associated with the numerical approximation of ordinary, partial, and fractional differential equations are made up and solved.

A nonlinear reconstruction scheme based on the Distorted Born Iterative Method (DBIM) is presented to solve two-dimensional inverse scattering problems with metallic scatterers. Half-quadratic regularization alleviates the inherently ill-posed problem of nonlinear inverse scattering while simultaneously preserving edges. Reconstruction results are found by minimizing a cost function. A bistatic experimental set-up is studied with angular and frequency diversity. A Multilevel Fast Multipole Algorithm (MLFMA) combined with a conjugate gradient (CG) scheme is introduced to solve the forward as well as the inverse scattering problem involved in the DBIM. The computational complexity per CG iteration using the MLFMA is of order O(N log N) compared to O(N2) in a standard implementation without the MLFMA. Numerical reconstruction results obtained from synthetic scattering data are presented.

A parallel multilevel fast multipole algorithm (MLFMA) based on OpenMP is proposed for shared memory parallel platform. According to requirements of efficient OpenMP parallelization and different numerical characteristics of different parts in MLFMA, the parallelization scheme is carefully designed on the establishment of near-field interaction matrix and on the far-field interaction in matrix vector multiplication in MLFMA. Numerical experiments show that the parallel MLFMA is efficient and has a consistent accuracy with the sequential MLFMA.

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> For electrically large complex electromagnetic (EM) scattering problems, huge memory is often required for most EM solvers, which is too difficult to be handled by a personal computer (PC) even a workstation. Although the multilevel fast multipole algorithm (MLFMA) effectively deals with electrically large problems to some extent, it is still time and memory consuming for very large objects. In order to further reduce the CPU time and the memory requirement, a hybrid algorithm, based on the overlapped domain decomposition method for integral equations (IE-ODDM), MLFMA and block-diagonal, incomplete lower and upper triangular matrices (DILU) preconditioner, is proposed for the analysis of electrically large problems. The dominant memory requirement for plane wave expansions in the three processes of aggregation, translation and disaggregation in the MLFMA is drastically reduced by the first two techniques. The iterative procedure for each overlapped subdomain solved by the MLFMA is effectively sped up by the DILU preconditioner. After integrating these techniques, the proposed hybrid algorithm is more efficient in computing time and memory requirement compared to the conventional MLFMA and is more suitable for analyzing very large EM scattering problems. Enough accurate solution can be obtained within quite a few outer iterations, where an outer iteration means a complete sweep for all the subdomains. Some numerical examples are presented to demonstrate its validity and efficiency. </para>

A multilevel fast multipole algorithm (MLFMA) for analyzing electromagnetic radiation and coupling characteristics of large-scale antenna arrays mounted on the platforms is presented in this paper. Compared with the method of moments (MoM), the MLFMA can be used to calculate larger scale problems with limited resources. First, waveport model of the MLFMA based on the equivalence principle and mode matching theory is established to efficiently and accurately simulate the antenna array. Then, a preconditioning approach for solving the radiation problems with the waveports is designed to improve convergence of the MLFMA. An initial guess construction method is proposed to accelerate the MLFMA computation for the multi-excitation problems, which can reduce the iteration time by at least 50%. Numerical results demonstrate accuracy and efficiency of the proposed method.

We present antenna switch optimizations using an efficient mechanism based on genetic algorithms and the multilevel fast multipole algorithm (MLFMA). Genetic algorithms are used to determine switch states for desired radiation and input characteristics, while cost-function evaluations are performed efficiently via an MLFMA implementation with dynamic error control. MLFMA is integrated into the genetic algorithm by extracting common computations to be performed once per optimization. Iterative convergence rates are further accelerated by using earlier solutions as initial-guess vectors. The efficiency of the developed mechanism is demonstrated on antennas with relatively large numbers of switches.

The multilevel fast multipole algorithm (MLFMA) has a great advantage in analysing the radiation and scattering of complex surfaces with wire attachment. It has a computational complexity of O(NlogN) for a matrix-vector product, which is much lower than the complexity of the traditional full-matrix method in the method of moments (MOM). The low complexity of the MLFMA is especially favorable for fast impedance calculation over a sweep of frequencies for large complex structures. This paper discusses the MLFMA for analyzing surface-wire interconnects. A variational formulation for calculating the input impedance is presented. Measured data and numerical simulation are used to validate the algorithm and the formulation.

It is well known that the conventional diagonalization used in the multilevel fast multipole algorithm (MLFMA) suffers from low-frequency breakdowns, which limit the application of this powerful method to multiscale problems involving large objects with small (but important) details in terms of wavelength. Using multipoles explicitly or resorting to alternative expansions incorporating evanescent waves for subwavelength interactions are common techniques for broadband implementations of MLFMA, while all these techniques need extensive efforts and re-programming. Recently, we presented an approximate diagonalization of the three-dimensional Green's function using scaled Hankel functions and plane waves (O. Ergul and B. Karaosmanoglu, IEEE Antennas Wireless Propag. Lett., 13, 2014, pp. 1054–1056). While its accuracy is limited (e.g., 3–4% in the worst case for the challenging one-box-buffer scheme), the approximate diagonalization provides stable computations of interactions at arbitrarily short distances. As a major advantage, the approximate diagonalization needs only minor modifications in the existing codes, including the parallel programs, to stabilize the conventional implementations for low frequencies.

The application of the multilevel fast multipole algorithm (MLFMA) to higher order moment method discretizations is a continuing open problem. Herein, we present a point-based mixed potential variant of the MLFMA algorithm that exhibits an MLFMA tree of arbitrary height with no restriction related to basis function support size, efficient nearfield precomputation, and that maintains favorable scaling for any mixture of low- and high-order bases. The flexibility of the algorithm is also leveraged to accelerate the precomputation of algebraic preconditioners. We demonstrate the method through application to the generalized method of moments (GMM), a recently introduced moment method discretization capable of combining both low- and high-order bases and geometries in the same simulation; however, the method may be used to accelerate other higher order moment methods as well.

We compare multi-GPU performance of the multilevel fast multipole algorithm (MLFMA) on two different systems: A shared-memory IBM S822LC workstation with four NVIDIA P100 GPUs, and 16 XK nodes (each is employed with a single NVIDIA K20X GPU) of the Blue Waters supercomputer. MLFMA is implemented for solving scattering problems involving two-dimensional inhomogeneous bodies. Results show that the multi-GPU implementation provides 794 and 969 times speedups on the IBM and Blue Waters systems over their corresponding sequential CPU executions, respectively, where the sequential execution on the IBM system is 1.17 times faster than on the Blue Waters System.

A Gaussian point-adaptive grouping scheme for the multilevel fast multipole algorithm (MLFMA) has been proposed in this communication to significantly reduce the heavy memory cost resulted from using the basis functions defined on large patches without sacrificing the accuracy of the numerical solutions. The grouping process in the MLFMA has been considered as a single-objective optimization problem and is solved by using the clustering algorithm for Gaussian quadrature points on each patch. Meanwhile, the constraint of the addition theorem for MLFMA is still satisfied. As a result, the presented scheme is able to acquire an optimal number of multipoles for each basis function. Compared to the conventional octree grouping scheme or other grouping schemes used in the past, the method in this communication is advantageous in terms of the memory efficiency and the solution accuracy. Numerical examples have been given to demonstrate the validity and effectiveness of the proposed scheme.