M. G. Araújo

MLFMA-FFT PARALLEL ALGORITHM FOR THE SO- LUTION OF LARGE-SCALE PROBLEMS IN ELECTRO- MAGNETICS (INVITED PAPER)

MLFMA-FFT PARALLEL ALGORITHM FOR THE SO-LUTION OF LARGE-SCALE PROBLEMS IN ELECTRO-MAGNETICS (INVITED PAPER)J. M. TaboadaDepartment Tecnolog¶‡as de los Computadores y de lasComunicaciones, Escuela Polit¶ecnicaUniversidad de ExtremaduraC¶aceres 10071, SpainM. G. Araujo¶ and J. M. B¶ertoloDepartment Teor¶‡a do Sinal e Comunicaci¶ons, E.T.S.E.Telecomunicaci¶onUniversidade de VigoVigo (Pontevedra) 36310, SpainL. LandesaDepartment Tecnolog¶‡as de los Computadores y de lasComunicaciones, Escuela Polit¶ecnicaUniversidad de ExtremaduraC¶aceres 10071, SpainF. Obelleiro and J. L. RodriguezDepartment Teor¶‡a do Sinal e Comunicaci¶ons, E.T.S.E.Telecomunicaci¶onUniversidade de VigoVigo (Pontevedra) 36310, SpainAbstract|An e–cient hybrid MPI/OpenMP parallel implementationof an innovative approach that combines the Fast Fourier Transform(FFT) and Multilevel Fast Multipole Algorithm (MLFMA) has beensuccessfully used to solve an electromagnetic problem involving 620millions of unknowns. The MLFMA-FFT method can deal withextremely large problems due to its high scalability and its reducedcomputational complexity. The former is provided by the use of the

Method-of-moments formulation for the analysis of plasmonic nano-optical antennas

We present a surface integral equation (SIE) to model the electromagnetic behavior of metallic objects at optical frequencies. The electric and magnetic current combined field integral equation considering both tangential and normal equations is applied. The SIE is solved by using a method-of-moments (MoM) formulation. The SIE-MoM approach is applied only on the material boundary surfaces and interfaces, avoiding the cumbersome volumetric discretization of the objects and the surrounding space required in differential-equation formulations. Some canonical examples have been analyzed, and the results have been compared with analytical reference solutions in order to prove the accuracy of the proposed method. Finally, two plasmonic Yagi-Uda nanoantennas have been analyzed, illustrating the applicability of the method to the solution of real plasmonic problems.

Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers

The performance of most widespread surface integral equation (SIE) formulations with the method of moments (MoM) are studied in the context of plasmonic materials. Although not yet widespread in optics, SIE-MoM approaches bring important advantages for the rigorous analysis of penetrable plasmonic bodies. Criteria such as accuracy in near and far field calculations, iterative convergence and reliability are addressed to assess the suitability of these formulations in the field of plasmonics.

Solution of large-scale plasmonic problems with the multilevel fast multipole algorithm.

A surface integral equation together with the multilevel fast multipole algorithm is successfully applied to fast and accurate resolution of plasmonic problems involving a large number of unknowns. The absorption, scattering, and extinction efficiencies of several plasmonic gold spheres of increasing size are efficiently obtained solving the electric and magnetic current combined-field integral equation. The numerical predictions are compared with reference analytic results to demonstrate the accuracy, suitability, and capabilities of this approach when dealing with large-scale plasmonic problems.

SUPERCOMPUTER AWARE APPROACH FOR THE SOLUTION OF CHALLENGING ELECTROMAGNETIC PROBLEMS

Abstract|It is a proven fact that The Fast Fourier Transform(FFT) extension of the conventional Fast Multipole Method (FMM)reduces the matrix vector product (MVP) complexity and preservesthe propensity for parallel scaling of the single level FMM. In thispaper, an e–cient parallel strategy of a nested variation of the FMM-FFT algorithm that reduces the memory requirements is presented.The solution provided by this parallel implementation for a challengingproblem with more than 0.5 billion unknowns has constituted the worldrecord in computational electromagnetics (CEM) at the beginning of2009.1. INTRODUCTIONRecent years have seen an increasing efiort in the development of fastand e–cient electromagnetic solutions with a reduced computationalcost regarding the conventional Method of Moments. Among others,the Fast Multipole Method (FMM) [1] and its multilevel version, theMLFMA [2,3] have constituted one of the most important advances inthat context.This development of fast electromagnetic solvers has gone handin hand with the constant advances in computer technology. Dueto this simultaneous growth, overcoming the limits in the scalabilityof the available codes became a priority in order to take advantageof the large amount of computational resources and capabilities thatare available in modern High Performance Computer (HPC) systems.For this reason, works focused on the parallelization improvement ofthe Multilevel Fast Multipole Algorithm (MLFMA) [4{13] have gainedinterest in last years.Besides, the FMM-Fast Fourier Transform (FMM-FFT) deservesbe taken into account as an alternative to beneflt from massivelyparallel distributed computers. This variation of the single-level FMMwas flrst proposed in [14] as an acceleration technique applied to almostplanar surfaces. Later on, a parallelized implementation was applied togeneral three-dimensional geometries [15]. The method uses the FFTto speedup the translation stage resulting in a dramatic reduction ofthe matrix-vector product (MVP) time requirement with respect tothe FMM. Although in general the FMM-FFT is not algorithmically ase–cient as the MLFMA, it has the advantage of preserving the naturalparallel scaling propensity of the single-level FMM in the spectral (

MLFMA-FFT Parallel Algorithm for the Solution of Extremely Large Problems in Electromagnetics

An efficient parallel implementation of the multilevel fast multipole algorithm-fast Fourier transform (MLFMA-FFT) has been successfully used to solve an electromagnetic problem involving one billion of unknowns, which indeed becomes the largest problem solved with the surface integral-equation approach up to now. In this paper, we present a deep review of this challenging execution, focusing on the details of the parallel implementation step by step, with the aim of describing the different stages of the parallel algorithm and analyzing its overall parallel performance.

On the Use of the Singular Value Decomposition in the Fast Multipole Method

The authors present a new matrix compression algorithm to improve the efficiency of the fast multipole method (FMM). The method is based on the application of the singular value decomposition (SVD) to the plane wave FMM aggregation matrices. These matrices are low-ranked, which is exploited to provide alternative sets of orthonormal singular basis functions, obtained as linear combinations of the original basis. By choosing only the most relevant singular functions, a much more compact representation is obtained to accurately handle the interactions between the FMM groups. The new formulation provides a reduction close to one order of magnitude both in computational cost and memory requirements, with a moderate impact on the accuracy of the solution.

COMPARISON OF SURFACE INTEGRAL EQUATIONS FOR LEFT-HANDED MATERIALS

A wide analysis of left-handed material (LHM) spheres with difierent constitutive parameters has been carried out employ- ing difierent integral-equation formulations based on the Method of Moments. The study is focused on the accuracy assessment of for- mulations combining normal equations (combined normal formula- tion, CNF), tangential equations (combined tangential formulation, CTF, and Poggio-Miller-Chang-Harrington-Wu-Tsai formulation, PM- CHWT) and both of them (electric and magnetic current combined fleld integral equation, JMCFIE) when dealing with LHMs. Relevant and informative features as the condition number, the eigenvalues dis- tribution and the iterative response are analyzed. The obtained results show up the suitability of the JMCFIE for this kind of analysis in con- trast with the unreliable behavior of the other approaches.

MLFMA-MoM for Solving the Scattering of Densely Packed Plasmonic Nanoparticle Assemblies

In this paper, we present a judicious combination of two renowned surface integral equation (SIE)-based techniques, namely, the multilevel fast multipole algorithm (MLFMA) and the method of moments (MoM), which synergize into a hybrid method that allows to address the analysis of large densely packed particle assemblies in an efficient and accurate way. This hybridization takes advantage of the repetition pattern inherent to these kinds of structures. Basically, the repeated self-coupling problems are squarely solved throughout the factorization of their MoM impedance matrix, whereas the cross-couplings through the surrounding medium are expedited via the MLFMA in the framework of a global iterative scheme. Some results are presented here to demonstrate the suitability of the proposed hybrid method to address large-scale nanoparticle arrays in the framework of nanoplasmonic biosensing applications.

Improving condition number and convergence of the surface integral-equation method of moments for penetrable bodies

Most of the surface integral equation (SIE) formulations for composite conductor and/or penetrable objects suffer from balancing problems mainly because of the very different scales of the equivalent electric and magnetic currents. Consequently, the impedance matrix usually has high- or ill-condition number due to the imbalance between the different blocks. Using an efficient left and right preconditioner the elements of the impedance matrix are balanced. The proposed approach improves the matrix balance without modifying the underlying SIE formulation, which can be selected solely in terms of accuracy. The numerical complexity of this preconditioner is O(N) with N the number of unknowns, and it can be easily included on any existing SIE code implementation.