We present a massively-parallel solver for large Helmholtz-type inverse scattering problems. The solver employs the distorted Born iterative method for capturing the multiple-scattering phenomena in image reconstructions. This method requires many full-wave forward-scattering solutions in each iteration, constituting the main performance bottleneck with its high computational complexity. As a remedy, we use the multilevel fast multipole algorithm (MLFMA). The solver scales among computing nodes using a two-dimensional parallelization strategy that distributes illuminations in one dimension, and MLFMA sub-trees in the other dimension. Multi-core CPUs and GPUs are used to provide per-node speedup. We demonstrate a 76% efficiency when scaling from 64 GPUs to 4,096 GPUs. The paper provides reconstruction of a 204.8λ×204.8λ image (4M unknowns) executed on 4,096 GPUs in near-real time (almost 2 minutes). To the best of our knowledge, this is the largest full-wave inverse scattering solution to date, in terms of both image size and computational resources.

This is a brief review of the development of computational electromagnetics (CEM) to partially summarize its achievements, issues, and possibilities.

The papers in this special issue are devoted to the topic of large-scale electromagnetic computation methods for modeling and applicatoins.

The computer integrated circuitmn (IC) is inundated with electromagnetic signals. These signals are omnipresent at the device, chip, package, and board level. The length-scales vary from nanometers to centimeters. The physics also changes significantly over these length-scales. It is clear that multi-scale, multi-physics analyses are needed to understand future generation IC's.

The geometric characteristics of low-frequency magnetic dipole fields and their relative insensitivity to the host media have made low-frequency coils useful for underground geolocation. A robust and succinct method is presented for converting dipole field measurements into estimates of receiver position. A noisy half-space model of the Earth is used to model the receiving environment. A perfectly conducting sphere immersed in a homogeneous conducting background is used to study the localized effects of buried metals.

An efficient inversion method is proposed using Cui et al.'s high-order extended Born approximations to reconstruct the conductivity object function of three-dimensional dielectric objects buried in a lossy Earth. High-order solutions of the object function are obtained, which have closed-form relations to the linear inverse-scattering solution. Because such relations can be evaluated quickly using the fast Fourier transform, the high-order solutions have similar simplicity as the linear inversion. When the contrasts of buried objects are large, the high-order solutions are much more accurate due to the approximate consideration of multiple-scattering effects within the objects. Hence, good-resolution images can be obtained for large-contrast objects using the new method by only solving a linear inverse problem. Numerical experiments have shown the validity and efficiency of the proposed method.

The very early time electromagnetic system (VETEM) is an efficient tool for the detection of buried objects in very lossy earth, which allows a deeper penetration depth compared to the ground-penetrating radar. In this paper, the inversion of VETEM data is investigated using three-dimensional (3-D) inverse scattering techniques, where multiple frequencies are applied in the frequency range from 0-5 MHz. For small and moderately sized problems, the Born approximation and/or the Born iterative method have been used with the aid of the singular value decomposition and/or the conjugate gradient method in solving the linearized integral equations. For large-scale problems, a localized 3-D inversion method based on the Born approximation has been proposed for the inversion of VETEM data over a large measurement domain. Ways to process and to calibrate the experimental VETEM data are discussed to capture the real physics of buried objects. Reconstruction examples using synthesized VETEM data and real-world VETEM data are given to test the validity and efficiency of the proposed approach.

It is known that the extended Born approximation (ExBorn) is much faster than the method of moments (MoM) in the study of electromagnetic scattering by three-dimensional (3-D) dielectric objects, while it is much more accurate than the Born approximation at low frequencies. Hence, it is more applicable in the low-frequency numerical simulation tools. However, the conventional ExBorn is still too slow to solve large-scale problems because it requires O(N/sup 2/) computational load, where N is the number of unknowns. In this paper, a fast ExBorn algorithm is proposed for the numerical simulation of 3-D dielectric objects buried in a lossy Earth. When the buried objects are discretized with uniform rectangular mesh and the Green's functions are extended appropriately, the computational load can be reduced to O(N log N) using the cyclic convolution, cyclic correlation, and fast Fourier transform (FFT). Numerical analysis shows that the fast ExBorn provides good approximations if the buried target has a small or moderate contrast. If the contrast is large, however, ExBorn will be less accurate. In this case, a preconditioned conjugate-gradient FFT (CG-FFT) algorithm is developed, where the solution of the fast ExBorn is chosen as the initial guess and the preconditioner. Numerical results are given to test the validity and efficiency of the fast algorithms.

Very early time electromagnetic system (VETEM) is an efficient tool for the detection of buried targets in very lossy medium. In this paper, we will discuss the inversion of VETEM data using a combination of one-dimensional (1D) and three-dimensional (3D) inverse scattering techniques, where the Born iterative method and distorted Born iterative method have been applied. In order to expedite the 3D inversion, an extended Born-type approximation is made in the forward solver, together with FFT acceleration technique. The 1D inversion provides a gross image of the subsurface, but the 3D inversion helps to resolve the subsurface image better.

The electromagnetic field simulation of the target detection by high-resolution radar is performed by incorporating the asymptotic waveform evaluation (AWE) technique and the compression technique of a chirp signal. The electromagnetic field simulation is very helpful for revealing the multi-scattering mechanism of wide-band radar signal between targets or within a target. Our current simulation involves three major steps, in the first step, AWE technique is used to calculate the scattering far field over a wideband at which a radar is operated; in the second step, the frequency response of scattering field is multiplied by the spectrum of the chirp signal transmitted by radar to get the frequency response of radar system; in the third step, the system response is multiplied by the conjugate of the spectrum of the chirp signal and then transformed into time domain by inverse FFT. In this way the target detection by chirp radar can be accurately simulated using electromagnetic field. Numerical examples are presented.

This paper develops a finite-difference time-domain (FDTD) approach suitable for modeling general antenna structures which include cylindrical dipoles. The medium averaging is a key factor of the method since the FDTD grid size is comparable to the dimensions of the antenna. It is essential in the FDTD calculation to account for the effective media seen by different electric field components. The input impedance calculated by the FDTD approach agrees well with the published results for a cylindrical dipole, although only one cell in the transverse direction is assigned as the metallic conductor. This FDTD approach is then used to model and analyze a novel broadband antenna which can be used in a borehole radar system for oil-field imaging while drilling. The antenna is essentially a horizontal cut on a metallic drill collar with a vertical cylindrical dipole placed across the cut and loaded at the ends. Based on the modeling results, we found that a proper resistive loading is the key in order to increase the bandwidth of the antenna and reduce ringing. When properly loaded, the antenna has a fairly flat input impedance ranging from 50 MHz to 600 MHz. The real part of the input impedance varies favorably around 75 /spl Omega/. In addition, the input impedance of the antenna is not very sensitive to the surrounding medium. The simulated antenna radiation of the electric field shows good azimuthal discrimination. For a lossless medium with a permittivity of 20 /spl epsi//sub 0/ the front-back ratio climbs from around 15 dB at 100 MHz to about 45 dB at 600 MHz.

In this paper, we present a method for solving electromagnetic scattering from a composite object at low frequencies using the method of moments (MoM) and loop-tree basis. Many applications involve composite objects which consist of several homogeneous regions. The loop-tree basis used for analyzing scattering from a homogeneous body at low frequencies could not be directly applied to analyze the low-frequency scattering from a composite object. In general, it is very difficult, if not impossible, to find a set of loop-tree basis functions that is valid for the structures on both sides of the interfaces. In this paper, we treat a composite object as a limiting case of multibody problem so that we could setup the MoM equation using the loop-tree basis found on each single body. A process is then developed to eliminate the redundant unknowns. The proposed method makes it possible to analyze low-frequency scattering from an arbitrary composite object. The validity and applications are illustrated with representative numerical examples.

A simple and efficient method for large scale three-dimensional (3D) subsurface imaging of inhomogeneous background is presented. One-dimensional (1D) multifrequency distorted Born iterative method (DBIM) is employed in the inversion. Simulation results utilizing synthetic scattering data are given. Calibration of the very early time electromagnetic (VETEM) experimental waveforms is detailed along with major problems encountered in practice and their solutions. This discussion is followed by the results of a large scale application of the method to the experimental data provided by the VETEM system of the U.S. Geological Survey. The method is shown to have a computational complexity that is promising for on-site inversion.

A method is presented for solving the surface integral equation using the method of moments (MoM) at very low frequencies, which finds applications in geoscience. The nature of the Helmholtz decomposition leads the authors to choose loop-tree basis functions to represent the surface current. Careful analysis of the frequency scaling property of each operator allows them to introduce a frequency normalization scheme to reduce the condition number of the MoM matrix. After frequency normalization, the MoM matrix can be solved using LU decomposition. The poor spectral properties of the matrix, however, makes it ill-suited for an iterative solver. A basis rearrangement is used to improve this property of the MoM matrix. The basis function rearrangement (BFR), which involves inverting the connection matrix, can be viewed as a pre-conditioner. The complexity of BFR is reduced to O(N), allowing this method to be combined with iterative solvers. Both rectilinear and curvilinear patches have been used in the simulations. The use of curvilinear patches reduces the number of unknowns significantly, thereby making the algorithm more efficient. This method is capable of solving Maxwell's equations from quasistatic to electrodynamic frequency range. This capability is of great importance in geophysical applications because the sizes of the simulated objects can range from a small fraction of a wavelength to several wavelengths.

The paper by T. J. Cui and W. C. Chew, IEEE Trans. Geosci. Remote Sensing, vol. 38, p.2033-41 (2000) is corrected. The authors, in that paper, defined the branch cut of K/sub ay/ and k/sub by/ in an equation (7) as Re(k/sub ay/,/sub by/)/spl ges/0 and Im(k/sub ay/,/sub by/)/spl ges/0 to make the Sommerfeld-like integral in another equation(5) satisfy the radiation condition. In the inverse-Fourier expression of yet another equation (24), however, they had to choose the branch cut of k/sub by/ in a different way, where Re(k/sub by/)/spl ges/O and Im(k/sub ay/,/sub by/)/spl les/, so that the integral in this equation can be convergent. Hence, the k/sub by/ in equation (27) must be set in different branch cuts. Although such a procedure gave correct reconstruction results, the mathematical interpretation was confusing. In this correspondence, the authors now modify the mathematical interpretation so that it is clear and easily understood.

An efficient inverse-scattering algorithm is developed to reconstruct both the permittivity and conductivity profiles of two-dimensional (2D) dielectric objects buried in a lossy earth using the distorted Born iterative method (DBIM). In this algorithm, the measurement data are collected on (or over) the air-earth interface for multiple transmitter and receiver locations at single frequency. The nonlinearity due to the multiple scattering of pixels to pixels, and pixels to the air-earth interface has been taken into account in the iterative minimization scheme. At each iteration, a conjugate gradient (CG) method is chosen to solve the linearized problem, which takes the calling number of the forward solver to a minimum. To reduce the CPU time, the forward solver for buried dielectric objects is implemented by the CG method and fast Fourier transform (FFT). Numerous numerical examples are given to show the convergence, stability, and error tolerance of the algorithm.

In this paper, numerical simulations of a new enhanced very early time electromagnetic (VETEM) prototype system are presented, where a horizontal transmitting loop and two horizontal receiving loops are used to detect buried targets, in which three loops share the same axis and the transmitter is located at the center of receivers. In the new VETEM system, the difference of signals from two receivers is taken to eliminate strong direct-signals from the transmitter and background clutter and furthermore to obtain a better SNR for buried targets. Because strong coupling exists between the transmitter and receivers, accurate analysis of the three-loop antenna system is required, for which a loop-tree basis function method has been utilized to overcome the low-frequency breakdown problem. In the analysis of scattering problem from buried targets, a conjugate gradient (CG) method with fast Fourier transform (FFT) is applied to solve the electric field integral equation. However, the convergence of such CG-FFT algorithm is extremely slow at very low frequencies. In order to increase the convergence rate, a frequency-hopping approach has been used. Finally, the primary, coupling, reflected, and scattered magnetic fields are evaluated at receiving loops to calculate the output electric current. Numerous simulation results are given to interpret the new VETEM system. Comparing with other single-transmitter-receiver systems, the new VETEM has better SNR and ability to reduce the clutter.

A previously developed fast inhomogeneous plane wave algorithm is adapted to the analysis of the scattering from the objects above a multilayered medium. Based on the similarity of the inhomogeneous plane wave expansion of the Green's function encountered in the layered medium study to its free space counterpart, the proper steepest descent path can be defined. By using the interpolation and extrapolation techniques, the translation matrix is diagonalized. The contributions of the pole and branch point, resulting from the deformation of the integration path, can be handled easily. Comparing with the previously developed methods in this area, this algorithm is simpler, more efficient, and more general. The multilevel fast inhomogeneous plane wave algorithm has been implemented and the numerical results show that O(N log N) computational complexity of the algorithm is achieved. The algorithm is validated by comparing the results from a conventional method of moments program.

An efficient method for modeling wire antennas of arbitrary shapes and orientations above a lossy ground is presented by using Galerkin method. As opposed to the traditional point matching method, careful treatment is required for the electric-field integral equation (EFIE) and its numerical implementation. Comparing with the point matching method, only three simple Sommerfeld integrals are needed in the new method, and the convergence is much faster. To test the validity of this method, numerous numerical examples are given. Numerical results show that, for equivalent-accuracy solutions, the cell number used in this method is much fewer than that in the point matching method.

A novel diffraction tomographic algorithm has been proposed to detect two-dimensional (2D) dielectric cylinders buried under a lossy Earth. In this algorithm, the air-Earth interface has been taken into account and the exact treatment of Sommerfeld-like integrals has been considered. Using the algorithm, all locations, shapes, and dielectric properties of buried cylinders can be accurately reconstructed under the low-contrast condition. For high-contrast targets, this algorithm can also be used to determine their locations and approximate their dielectric properties. Due to the use of fast Fourier transforms to implement the problem, this algorithm is very fast and quite tolerant to the error of measurement data. Numerical examples are given to show the validity of the algorithm.

We present two novel, fully three-dimensional (3-D) finite-difference time-domain (FDTD) schemes in cylindrical coordinates for transient simulation of electromagnetic wave propagation in complex (inhomogeneous, dispersive, and conductive) and unbounded media. The proposed FDTD schemes incorporate an extension of the perfectly matched layer (PML) absorbing boundary condition (ABC) to three-dimensional (3-D) cylindrical coordinates. Dispersion on the media is modeled by using the piecewise-linear recursive convolution (PLRC) algorithm, accounting for multiterm Lorentz and/or Debye models. Split-field and unsplit (anisotropic medium) formulations of the cylindrical PML-PLRC-FDTD schemes are implemented and compared in the time domain. The comparison includes the late-time stability properties of the update schemes. Numerical simulations of subsurface electromagnetic problems are included. Because the proposed schemes retain the nearest-neighbor property of the ordinary FDTD, they are well suited for implementation on massively parallel computers.

This paper presents a fast method for electromagnetic scattering and radiation problems pertinent to three-dimensional (3D) buried objects. In this approach, a new symmetrical form of the Green's function is derived, which can reduce the number of Sommerfeld integrals involved in the buried objects problem. The integration along steepest descent paths and leading-order approximations are introduced to evaluate these Sommerfeld integrals, which can greatly accelerate the computation. Based on the fast evaluation of Sommerfeld integrals, the radiation of an arbitrarily oriented electric dipole buried in a half space is first analyzed and computed. Then, the scattering by buried dielectric objects and conducting objects is considered using the method of moments (MOM). Numerical results show that the fast method can save tremendous CPU time in radiation and scattering problems involving buried objects.

A fast algorithm for electromagnetic scattering by buried three dimensional (3-D) dielectric objects of large size is presented by using the conjugate gradient (CG) method and fast Fourier transform (FFT). In this algorithm, the Galerkin method is utilized to discretize the electric field integral equations, where rooftop functions are chosen as both basis and testing functions. Different from the 3-D objects in homogeneous space, the resulting matrix equation for the buried objects contains both cyclic convolution and correlation terms, either of which can be solved rapidly by the CG-FFT method. The near-scattered field on the observation plane in the upper space has been expressed by two-dimensional (2-D) discrete Fourier transforms (DFTs), which also can be rapidly computed. Because of the use of FFTs to handle the Toeplitz matrix, the Sommerfeld integrals' evaluation which is time consuming yet essential for the buried object problem, has been reduced to a minimum. The memory required in this algorithm is of order N (the number of unknowns), and the computational complexity is of order N/sub iter/N log N, in which N/sub iter/ is the iteration number, and N/sub iter//spl Lt/N is usually true for a large problem.

A new technique, the steepest descent-fast multipole method (SDFMM), is developed to efficiently analyze scattering from perfectly conducting random rough surfaces. Unlike other prevailing methods, this algorithm has linear computational complexity and memory requirements, making it a suitable candidate for analyzing scattering from large rough surfaces as well as for carrying out Monte Carlo simulations. The method exploits the quasiplanar nature of rough surfaces to efficiently evaluate the dyadic Green's function for multiple source and observation points. This is achieved through a combination of a Sommerfeld steepest descent integral and a multilevel fast multipole-like algorithm based on inhomogeneous plane wave expansions. The fast evaluation of the dyadic Green's function dramatically speeds up the iterative solution of the integral equation for rough surface scattering. Several numerical examples are presented to demonstrate the efficacy and accuracy of the method in analyzing scattering from extremely large finite rough surfaces.

A three-dimensional (3D) time-domain numerical scheme for simulation of ground penetrating radar (GPR) on dispersive and inhomogeneous soils with conductive loss is described. The finite-difference time-domain (FDTD) method is used to discretize the partial differential equations for time stepping of the electromagnetic fields. The soil dispersion is modeled by multiterm Lorentz and/or Debye models and incorporated into the FDTD scheme by using the piecewise-linear recursive convolution (PLRC) technique. The dispersive soil parameters are obtained by fitting the model to reported experimental data. The perfectly matched layer (PML) is extended to match dispersive media and used as an absorbing boundary condition to simulate an open space. Examples are given to verify the numerical solution and demonstrate its applications. The 3D PML-PLRC-FDTD formulation facilitates the parallelization of the code. A version of the code is written for a 32-processor system, and an almost linear speedup is observed.

The inversion of induction tool measurements using the distorted Born iterative method (DBIM) and the conjugate gradient-fast Fourier-Hankel transform (CG-FFHT) is described. The inverse problem is formulated in terms of an integral equation of scattering where the unknown to be sought is the conductivity in the rock formation, when the measurements along a borehole axis are performed. The nonlinear problem is linearized at each stage using the distorted Born approximation. The inhomogeneous medium Green's function in the distorted Born approximation is found by solving a volume integral equation using the CG-FFHT method, which allows a rapid solution to a large problem with reduced computational complexity and memory requirement. In this manner, the inverse problem is solved with a computational complexity proportional to N/sub tl/N log N where N/sub tl/ is the number of transmitter locations used in the data collection and N is the total number of pixels used to model the unknown formation. The memory requirement is of order NN/sub tl/.>

The authors formulate and implement a numerical mode-matching (NMM) method to model electrode-type resistivity tools in invaded thin beds. The authors derive the low-frequency approximation of the Maxwell's equations to obtain the partial differential equation for the potential field. The new NMM program is validated by comparing the numerical results with those obtained from other dc programs. It is found that this new program is much faster than the program using the finite-element method (FEM), and hence is useful for routine interpretation of resistivity logs and for inversion.>

A time-harmonic formulation for the electrical impedance tomography (EIT) inverse problem accounting for electrodynamic effects is derived. This work abandons the standard electrostatic impedance model for a full-wave T-matrix model. The advantage of this method is an accurate physical model that includes finite frequency effects, such as diffusion phenomena, and electrode contact impedance effects. This model offers the potential for increased resolution and larger invertible contrast objects than other methods when used on experimental data, because it may represent a more realistic physical model. Also, an accurate gradient matrix is used in the Newton iterative method so the image reconstruction converges in a few iterations. These advantages are realized with no increase in the computational complexity of this algorithm, compared to the static finite element model. A calibration technique is suggested for measurement systems, to test the validity of a theoretical model that includes electrode contact impedance effects.

An iterative algorithm is developed to solve the nonlinear inverse scattering problem for two-dimensional lossless dielectric inhomogeneities using time-domain scattering data. The method is based on performing Born-type iterations on a volume integral equation and, hence, successively calculating higher-order approximations to the unknown object profile. Both the full-angle and the limited-angle problems are considered. Solutions are obtained for cases where the first-order Born approximation is severely violated. Wideband time-domain scattered field measurements make it possible to use sparse data sets and thus reduce experimental complexity and computation time. Several examples are given to show the ability of this method to invert arbitrarily shaped permittivity profiles using few transmitters and receivers. The high-resolution capability of the algorithm is also demonstrated.>

A symmetrical form of the solution for an electrical source in a multibed well-logging environment is derived. The method uses local reflection and transmission operators of a single-bed boundary and a general recursive algorithm to derive generalized reflection and transmission operators. Using this method, the computation time scales linearly as N, where N is the number of beds in the environment. A computer program was developed to implement the solution. The program is robust and generates accurate results from 20 kHz to 25 MHz.>

A large dielectric measurement cell was designed and calibrated to operate over a 1-MHz to 3-GHz frequency range. The cell is much larger than commercially available cells and is useful for measuring samples with large-grain heterogeneities like concrete or rock. The cell works on a coaxial transmission line principle. Once the S parameters of the two end sections are calibrated, the S parameters of the sample region can be found. Then the complex permittivity of the sample can be deembedded. Because of the broad frequency range, three calibration methods are used for the low-, intermediate-, and high-frequency ranges. The accuracy of the cell is tested by measuring materials of a known dielectric constant. Error analysis is also performed to determine the uncertainty of the measurements.>