Han Guo

On MLMDA/Butterfly Compressibility of Inverse Integral Operators

The multilevel matrix decomposition algorithm (MLMDA) is shown to permit effective compression of inverse integral operators pertinent to the analysis of scattering from electrically large structures. Observed compression ratios exceed those realized by low-rank (LR) compression methods, leading to substantial memory savings and a faster application of the inverse operator, and suggesting a new application for schemes traditionally used for compressing forward integral operators.

Analyzing Large-Scale Arrays Using Tangential Equivalence Principle Algorithm With Characteristic Basis Functions

In this paper, the tangential equivalence principle algorithm (T-EPA) combined with characteristic basis functions (CBFs) is presented to analyze the electromagnetic scattering of large-scale antenna arrays. The T-EPA is a kind of domain decomposition scheme for the electromagnetic scattering and radiation problems based on integral equation (IE). CBFs are macrobasis functions which are constructed by conventional local basis functions. By utilizing CBFs together with the T-EPA, the scattering analysis of large-scale arrays will be much more efficient with decreased unknowns compared with the original T-EPA. Further, the multilevel fast multipole algorithm (MLFMA) is applied to accelerate the matrix-vector multiplication in the T-EPA. Numerical results are shown to demonstrate the accuracy and efficiency of the proposed technique.

Hierarchical Matrices Method and Its Application in Electromagnetic Integral Equations

Hierarchical (-) matrices method is a general mathematical framework providing a highly compact representation and efficient numerical arithmetic. When applied in integral-equation- (IE-) based computational electromagnetics, -matrices can be regarded as a fast algorithm; therefore, both the CPU time and memory requirement are reduced significantly. Its kernel independent feature also makes it suitable for any kind of integral equation. To solve -matrices system, Krylov iteration methods can be employed with appropriate preconditioners, and direct solvers based on the hierarchical structure of -matrices are also available along with high efficiency and accuracy, which is a unique advantage compared to other fast algorithms. In this paper, a novel sparse approximate inverse (SAI) preconditioner in multilevel fashion is proposed to accelerate the convergence rate of Krylov iterations for solving -matrices system in electromagnetic applications, and a group of parallel fast direct solvers are developed for dealing with multiple right-hand-side cases. Finally, numerical experiments are given to demonstrate the advantages of the proposed multilevel preconditioner compared to conventional “single level” preconditioners and the practicability of the fast direct solvers for arbitrary complex structures.

Fast Simulation of Array Structures Using T-EPA With Hierarchical LU Decomposition

In this letter, a novel technique that combines the tangential equivalence algorithm (T-EPA) with hierarchical (H-) LU decomposition is proposed to solve the electromagnetic problems of large arrays. The T-EPA is a kind of domain decomposition method based on integral equation. The characteristic basis functions (CBFs) have been used on equivalence surface to reduce the number of unknowns. Multilevel fast multipole algorithm (MLFMA) has also been used to accelerate the matrix-vector multiplication. However, the inversion of dense matrix required in T-EPA is very time-consuming. Here, an efficient H-LU decomposition based on H-matrix framework is proposed to reduce the computational complexity of the inversion of dense matrix. Numerical results are shown to demonstrate the accuracy and efficiency of the proposed method.

An MPI-OpenMP Hybrid Parallel -LU Direct Solver for Electromagnetic Integral Equations

In this paper we propose a high performance parallel strategy/technique to implement the fast direct solver based on hierarchical matrices method. Our goal is to directly solve electromagnetic integral equations involving electric-large and geometrical-complex targets, which are traditionally difficult to be solved by iterative methods. The parallel method of our direct solver features both OpenMP shared memory programming and MPl message passing for running on a computer cluster. With modifications to the core direct-solving algorithm of hierarchical LU factorization, the new fast solver is scalable for parallelized implementation despite of its sequential nature. The numerical experiments demonstrate the accuracy and efficiency of the proposed parallel direct solver for analyzing electromagnetic scattering problems of complex 3D objects with nearly 4 million unknowns.

A Hybrid Solvers Enhanced Integral Equation Domain Decomposition Method for Modeling of Electromagnetic Radiation

The hybrid solvers based on integral equation domain decomposition method (HS-DDM) are developed for modeling of electromagnetic radiation. Based on the philosophy of “divide and conquer,” the IE-DDM divides the original multiscale problem into many closed nonoverlapping subdomains. For adjacent subdomains, the Robin transmission conditions ensure the continuity of currents, so the meshes of different subdomains can be allowed to be nonconformal. It also allows different fast solvers to be used in different subdomains based on the property of different subdomains to reduce the time and memory consumption. Here, the multilevel fast multipole algorithm (MLFMA) and hierarchical (H-) matrices method are combined in the framework of IE-DDM to enhance the capability of IE-DDM and realize efficient solution of multiscale electromagnetic radiating problems. The MLFMA is used to capture propagating wave physics in large, smooth regions, while H-matrices are used to capture evanescent wave physics in small regions which are discretized with dense meshes. Numerical results demonstrate the validity of the HS-DDM.

Investigation on the butterfly reconstruction methods for MLMDA-based direct integral equation solver

Summary form only given: Fast direct methods for solving integral equations are rapidly gaining traction as viable alternatives to iterative schemes. Indeed, iterative methods are often expensive when used on problems involving many excitations; moreover, they converge slowly when applied to ill-conditioned problem. We recently demonstrated direct EFIE and CFIE solvers that leverage the multilevel matrix decomposition algorithm (MLMDA) (sometimes also termed “butterfly methods”) to achieve O (N log2 N) storage and CPU complexity when applied to both 2D and 3D surface scattering phenomena (H. Guo, J. Hu and E. Michielssen, AP-S/URSI IEEE 2013, Orlando, Florida, USA). These (observed) memory and CPU complexities stand in stark contrast to those of low-rank based solvers, which scale as O (Nα log N) (α=1.3~2.0) and O (Nα logβN) (α=2.0~3.0, β≥1.0), respectively. The above-referenced direct solver constructs a hierarchical block LU factorized EFIE or CFIE impedance matrix by recursively operating on butterflycompressed blocks of partial LU factors. In this process, the solver continuously constructs new butterfly-compressed representations of sums and products of butterfly-compressed matrices. The computational complexity of the direct solver strongly hinges on the efficiency of the “butterfly algebra” for computing these representations. Techniques for implementing this algebra generally fall into one of two categories: direct, deterministic methods and iterative, randomized schemes. Direct schemes arrive at a compressed representation of the product or sum of two butterfly-compressed operators by computing judiciously selected components of the resulting operator. Randomized schemes, on the other hand, only rely on information gathered by multiplying the sum or product of the two butterfly-compressed operators by random vectors to arrive at a compressed representation of the resulting operator. These randomized schemes can be regarded as far-reaching generalizations of randomized schemes for computing low-rank approximations of linear operators (E. Liberty, F. Woolfe, P. G. Martinsson, V. Rokhlin, and M. Tygert, PNAS, vol. 104, no. 51, 20167-20172, 2007). The computational efficiency and convergence rate of the randomized schemes depends on many implementation choices. We compare deterministic and randomized schemes on their efficiency and accuracy, as well as their ease of integration into direct solvers.

Multilevel sparse approximate inverse preconditioning for solving dynamic integral equation by H-matrix method

A novel sparse approximate inverse (SAI) preconditioner in multilevel fashion is proposed to accelerate the convergence rate of Krylov iterations for solving 3D electromagnetic scattering by integral equation. This multilevel formatted preconditioning is derived from the hierarchical data structure of hierarchical (H-) matrix, which overcomes the construction restrict of conventional SAI preconditioner combined with popular fast algorithms like multilevel fast multipole algorithm (MLFMA). Numerical experiments have demonstrated that this proposed preconditioner has a good property, can achieve fast convergence even for very complex structures.

Fast analysis of 3D inhomogeneous dielectric objects using IE-FFT

This paper presents a fast solution of electromagnetic scattering problems using the volume integral equation (VIE) combined with the integral equation fast Fourier transform (IE-FFT). The tetrahedron cells are used to model the geometry of the scatterer and the Schaubert-Wilton-Glisson (SWG) basis function is used to expand the electric flux density. A floating stencil topology is introduced to reduce the memory requirement and computational complexity for the near-interaction. Numerical result shows that the method can reduce both memory requirement and computational complexity while maintains the same accuracy.

An improved Calderón preconditioner for electric field integral equation

In this paper, an improved Calderón preconditioner for electric field integral equation (EFIE) with the adaptive cross approximation (ACA) algorithm is presented. The preconditioning technique based on Calderón formulas is aimed at stabilizing EFIE. A new preconditioning technique by using Buffa-Christiansen (BC) basis has been introduced in literatures recently. Because a dual refined mesh must be constructed by using BC basis, the additional computational cost is obviously risen out, despite of its distinct effect on reducing the solution time. The improved preconditioner in this paper has adopted ACA algorithm to decrease the memory and computational cost during the preconditioning process and using BC basis to generate stabilized EFIE. Numerical results demonstrate that this modified preconditioner makes the MoM EFIE system converging rapidly no matter whatever the discretization density and speeds up the preconditioning process by contrast with original multiplicative Calderón preconditioner.