KEYWORDS: Algorithm development, Iterative methods, Signal processing, Numerical analysis, Matrices, Land mines, Digital signal processing, Data processing, Computer programming, Scattering
The objective of this paper is to present an efficient parallel implementation of the iterative compact high-order approximation numerical solver for 3D Helmholtz equation on multicore computers. The high-order parallel iterative algorithm is built upon a combination of a Krylov subspace-type method with a direct parallel Fast Fourier transform (FFT) type preconditioner from the authors’ previous work, as shown in Ref. 7. In this paper, we will be presenting the result of our algorithm by computationally simulating data with realistic ranges of parameters in soil and mine-like targets. Our algorithm will also be incorporating second, fourth, and sixth-order compact finite difference schemes. The accuracy and result of the fourth and sixth-order compact approximation will be shown alongside the scalability of our implementation in the parallel programming environment.
This paper introduces a direct parallel partial FFT-type algorithm for the numerical solutions of the two- and three-dimensional Helmholtz equations. The governing equations are discretized by high-order compact finite difference methods. The resulting discretized system is indefinite, making the convergence of most iterative methods deteriorate as frequency increases. In this situation, the parallel direct approaches are a better alternative, especially for the systems with discontinuous and singular right-hand sides. The research focuses on the efficient parallel implementation of the proposed algorithm in shared memory environments (OpenMP). The complexity and scalability of the direct parallel method are investigated on scattering problems with realistic ranges of parameters in soil and mine-like targets.
The goal of this paper is to describe a novel parallel high-resolution 3D numerical method for the solution of high-frequency electromagnetic wave propagation. The sequential numerical method was developed by the first author in 2014. The discussed parallel algorithm will be used later by the authors to computationally simulate data for the solution of the inverse problem of imaging mine-like targets. Thus the solution of the forward problem presented in this paper is a necessary prelude to the future solution of a related inverse problem. In this paper, land mines are modeled as small abnormalities embedded in an otherwise uniform media with an air-ground interface. These abnormalities are characterized by the electrical permittivity and the conductivity, whose values differ from those of the host media. The main challenge in the calculation of the scattered electromagnetic signal in these settings is the requirement of solving the Helmholtz equation for high frequencies. This is excessively time-consuming using standard direct solution techniques. A high-resolution and scalable numerical procedure for the solution of this equation is described in this paper. The kernel of this algorithm is a combination of a second, fourth or sixth order compact finite-difference scheme and a preconditioned Krylov subspace approach. Both fourth and sixth order compact approximations for the Helmholtz equation are considered to reduce approximation and pollution errors, thereby softening the point-per-wavelength constraint. The coefficient matrix of the resulting system is not Hermitian and possesses positive as well as negative eigenvalues. This represents a significant challenge for constructing an efficient iterative solver. In our approach, this system is solved by a combination of Krylov subspace-type method with a direct parallel FFT-type preconditioner. The resulting numerical method allows a natural and efficient implementation on parallel computers. Numerical results for realistic ranges of parameters in soil and mine-like targets confirm the high efficiency of the proposed parallel iterative algorithm.
We develop an efficient iterative approach to the solution of the discrete three-dimensional Helmholtz equation with variable coefficients and PML boundary conditions based on compact fourth and sixth order approximation schemes. The coefficient matrices of the resulting systems are not Hermitian and possess positive as well as negative eigenvalues so represent a significant challenge for constructing an efficient iterative solver. In our approach these systems are solved by a combination of a Krylov subspace-type method with a matching high order approximation preconditioner with coefficients depending only on one spatial variable. In the algorithms considered, the direct solution of high order preconditioning system is based on a combination of the separation of variables technique and Fast Fourier Transform (FFT) type methods. The resulting numerical methods allow for efficient implementation on parallel computers. Numerical results confirm the high efficiency of the proposed iterative algorithms.
The goal of this paper is to describe a novel high-resolution 3D numerical method for the solution of high
frequency electromagnetic wave propagation. This method will be used later by the author to computationally
simulate data for the solution of the inverse problem of imaging mine-like targets. Thus the solution of the forward
problem presented in this paper is a necessary prelude to the future solution of a related inverse problem. In this
paper, land mines are modeled as small abnormalities imbedded in an otherwise uniform media with an air-ground
interface. These abnormalities are characterized by the electrical permittivity and the conductivity, whose values
differ from those of the host media. The main challenge in the calculation of the scattered electromagnetic signal
in these settings is the requirement of solving the Helmholtz equation for high frequencies which is excessively
time consuming using standard direct solution techniques. A high-resolution and rapid numerical procedure for
the solution of this equation is described in this paper. The kernel of this algorithm is a combination of a fourth
order compact finite-difference scheme and a preconditioned Krylov subspace approach. A fourth order compact
approximation for the Helmholtz equation is considered to reduce approximation and pollution errors, thereby
softening the point-per-wavelength constraint. The coefficient matrix of the resulting system is not Hermitian
and possesses as positive as well as negative eigenvalues so represent a significant challenge for constructing
an efficient iterative solver. In our approach this system is solved by a combination of Krylov subspace-type
method with a direct FFT-type preconditioner. The resulting numerical method allows efficient implementation
on parallel computers. Numerical results for realistic ranges of parameters in soil and mine-like targets confirm
the high efficiency of the proposed iterative algorithm.
In this paper, an effective numerical method for the solution of Helmholtz equation with radiation boundary conditions is considered. This approach is based on the combination of the Krylov subspace type of iterative technique and FFT based preconditioner. The main novel element presented in this paper is the use of the modified FFT type preconditioning that allows us to keep the discretized Sommerfeld-like boundary conditions in preconditioning matrices and still have the numerical efficiency similar to the FFT method. The results of numerical experiments are compared to the standard application of GMRES method and FFT type preconditioner obtained by the replacing radiation boundary conditions with Neumann boundary conditions on the preconditioning step. The convergence of proposed algorithm was investigated on two test problems. Numerical results for realistic ranges of parameters in soil and mine-like targets are presented.
KEYWORDS: Numerical analysis, Land mines, Target detection, Iterative methods, Detection and tracking algorithms, Wave propagation, Radio propagation, Mining, Scattering, General packet radio service
In this paper, the multigrid technique was considered for the solution of two dimensional Helmholtz equation with radiation boundary conditions. To achieve h-independent convergence of the iterative method, the original Helmholtz equation was transformed by using so-called “regularizing” plane wave. The recently developed “black-box” type multigrid method was adopted as a solver. This multigrid approach uses matrix dependent prolongation operator. The convergence of proposed algorithm was investigated on several test problems. Numerical results for realistic ranges of parameters in soil and mine-like targets are presented.
KEYWORDS: Inverse problems, Land mines, Algorithm development, Matrices, General packet radio service, Detection and tracking algorithms, Soil science, Mathematics, Electromagnetism, Computing systems
Two novel solution methods for the inverse problem for the 2D Helmholtz equation are developed, tested and compared. The proposed approaches are based on a marching finite-difference scheme which requires the solution of an overdetermined system at each step. The preconditioned conjugate gradient method is used for rapid solution of these systems and an efficient preconditioner has been developed for this class of problems. The underlying target application is the imaging of land mines, which is formulated as an inverse problem for a 2D Helmholtz equation. The images represent the electromagnetic properties of the respective underground regions. Numerical results are presented.
Recently, we have successfully applied the Elliptic Systems Method to inverse problems in laser medical imaging applications. As part of applying this method to mine detection, accurate and fast algorithms are required for solving the forward problem to generate data for the inverse problem. Results for the 2D forward problem using GMRES method are compared with 1D transmission line models. Simulation result for miens and clutter are provided for both methods. The comparison with 1D results suggests that GMRES is an effective approach to modeling the forward problem in mine detection. In addition, the contrast between results for mines and clutter provide useful signal features for initial screening between mines and clutter.
KEYWORDS: Inverse problems, Land mines, General packet radio service, Electronic support measures, Detection and tracking algorithms, Numerical analysis, Antennas, Light wave propagation, Mathematical modeling, Systems modeling
Imaging of land mines using signal of a light-weight GPR is considered as an inverse problem for a Helmholtz-like equation. This equation is derived from Maxwell's system. The inverse problem consists in recovery of electrical permittivity and conductivity of a target(s) using multi- frequency measurements of the back-reflected signal. A novel method of solution is proposed. A crucial advantage of this algorithm over many traditional ones is that it avoids entirely the problem of local minima, since it does not use a least squares cost functional.
KEYWORDS: Land mines, Matrices, General packet radio service, Inverse problems, Mining, Numerical analysis, Tomography, Imaging systems, Radio propagation, Sensors
The ultimate goal of the authors is apply inverse problem methods to image land mines using a electromagnetic GPR signal. Specifically, the intention is to use the recently developed Elliptic Systems Method, which has been successfully applied by these authors to the problem of laser imaging of biological tissues. As the first step, however, one should develop a fast and accurate numerical method for the solution of the forward problem to simulate the data for the inverse problem. The main difficulty of the latter consists of the requirement of solving a Helmholtz- like equation for high frequencies which is very time consuming using standard direct solution techniques. A novel accurate and rapid numerical procedure for the solution of this equation is described in this paper.
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