Fast Preconditioned Krylov Methods for Boundary Integral Equations in Electromagnetic Scattering

Carpentieri, Bruno

doi:10.5772/32848

Cited by 2 publications

(2 citation statements)

References 80 publications

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“…However, large size parameters were not investigated and the improvement for small size parameters is extremely modest, if anything at all. More general preconditioning strategies exist, such as incomplete-LU, block-Jacobi [20], and the inverse fast multipole method [21], but these are potentially expensive, are not effective for high frequency problems, or are complicated to implement. The distinct advantages of circulant preconditioning are that it is well suited to the Toeplitz structure of DDA, is comparatively straightforward to implement, and it is inexpensive.…”

Section: Introductionmentioning

confidence: 99%

Preconditioning the discrete dipole approximation

Groth¹,

Polimeridis²,

White³

2019

Preprint

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The discrete dipole approximation (DDA) is a popular numerical method for calculating the scattering properties of atmospheric ice crystals. The standard DDA formulation involves the uniform discretization of the underlying volume integral equation, leading to a linear system with a block-Toeplitz Toeplitz-block matrix. This structure permits a matrix-vector product to be performed with O(n log n) complexity via the fast-Fourier transform (FFT). Thus, in principle, the system can be solved rapidly using an iterative method. However, it is well known that the convergence of iterative methods becomes increasing slow as the optical size and refractive index of the scattering obstacle are increased. In this paper, we present a preconditioning strategy based on the multi-level circulant preconditioner of Chan and Olkin [Numer. Algorithms 6, 89 (1994)] and assess its performance for improving this rate of convergence. In particular, we approximate the system matrix by a block-circulant circulant-block matrix which can be inverted rapidly using the FFT. We present numerical results for scattering by hexagonal ice prisms demonstrating that this serves as an effective preconditioning strategy, reducing simulation times by orders of magnitude in many cases. A Matlab implementation of this work is freely available online.

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Section: Introductionmentioning

confidence: 99%

Preconditioning the discrete dipole approximation

Groth¹,

Polimeridis²,

White³

2019

Preprint

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“…We briefly review recent developments of preconditioners in the context of FMM-accelerated BEM (FMBEM) or methods of moment (MoM) for oscillatory problems, involving the Helmholtz equation as well as the Maxwell and Navier-Cauchy equations. A more comprehensive review (especially for the Maxwell equations) is found in [16]. General preconditioning techniques [17] (e.g., point-Jacobi, ILU, etc.)…”

Section: Introductionmentioning

confidence: 99%

Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method

Takahashi

Coulier

Darve

2017

Journal of Computational Physics

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We investigate an efficient preconditioning of iterative methods (such as GMRES) for solving dense linear systems Ax = b that follow from a boundary element method (BEM) for the 3D Helmholtz equation, focusing on the low-frequency regime. While matrix-vector products in GMRES can be accelerated through the low-frequency fast multipole method (LFFMM), the BEM often remains computationally expensive due to the large number of GMRES iterations. We propose the application of the inverse fast multipole method (IFMM) as a preconditioner to accelerate the convergence of GMRES. The IFMM is in essence an approximate direct solver that uses a multilevel hierarchical decomposition and low-rank approximations. The proposed IFMM-based preconditioning has a tunable parameter ε that balances the cost to construct a preconditioner M, which is an approximation of A −1 , and the cost to perform the iterative process by means of M. Namely, using a small (respectively, large) value of ε takes a long (respectively, short) time to construct M, while the number of iterations can be small (respectively, large). A comprehensive set of numerical examples involving various boundary value problems with complicated geometries and mixed boundary conditions is presented to validate the efficiency of the proposed method. We show that the IFMM preconditioner (with a nearly optimal ε of 10 −2) clearly outperforms some common preconditioners for the BEM, achieving 1.2-10.8 times speed-up of the computations, in particular when the scale of the underlying scatterer is about five wavelengths or more. In addition, the IFMM preconditioner is capable of solving complicated problems (in a reasonable amount of time) that BD preconditioner can not.

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Fast Preconditioned Krylov Methods for Boundary Integral Equations in Electromagnetic Scattering

Cited by 2 publications

References 80 publications

Preconditioning the discrete dipole approximation

Preconditioning the discrete dipole approximation

Application of the inverse fast multipole method as a preconditioner in a 3D Helmholtz boundary element method

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