Yuriy A. Gryazin

Imaging the diffusion coefficient in a parabolic inverse problem in optical tomography

The elliptic systems method (ESM), previously developed by the second and third authors, is extended to the reconstruction of the diffusion coefficient of an inverse problem for the parabolic equation in the n-dimensional case (n = 2,3). This inverse problem has applications to optical imaging of small abnormalities hidden in a random media, such as biological tissues, foggy atmospheres, murky water, etc. Results of numerical experiments are presented in the two-dimensional case, for realistic ranges of the parameters.

Numerical Solution of a Subsurface Imaging Inverse Problem

In this paper a new solution method for an inverse problem for the two-dimensional (2D) Helmholtz equation is developed. The underlying application area which motivated this work is the imaging of land mines using ground penetrating radar, formulated as an inverse problem for the Helmholtz equation. A second generation version of the elliptic systems method is developed to numerically solve this problem. Uniqueness and convergence theorems and a variety of numerical results are presented.

Two numerical methods for an inverse problem for the 2-D Helmholtz equation

Two solution methods for the inverse problem for the 2-D 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 solutions of these systems and an efficient preconditioner has been developed for this class of problems. Underlying target applications include the imaging of land mines, unexploded ordinance, and pollutant plumes in environmental cleanup sites, each formulated as an inverse problem for a 2-D Helmholtz equation. The images represent the electromagnetic properties of the respective underground regions. Extensive numerical results are presented.

Novel inverse methods in land mine imaging

The imaging of buried land mines continues to present significant signal-processing challenges in the development of inverse methods for the detection of plastic mines buried in soil. To address this difficult problem, mathematical advances in the development of the elliptic systems method are used to generate images of the buried land mines. The-proposed approach adapts earlier methods, successfully applied in laser tomography of breast tumors using the diffusion equation, to the present problem of land mine imaging using the Helmholtz equation. The images generated by the new method represent electromagnetic properties of underground regions, providing effective differentiation of plastic land mines from surrounding soil. Experimental results are presented to demonstrate the new method.

Comparison of 2D and 1D approaches to forward problem in mine detection

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.

Tomographic Images of Land Mines by the Elliptic Systems Method Using GPR: Efficient Solution of the Forward Problem

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.

SCREAMER V4.0 — A powerful circuit analysis code

Screamer is a fast, highly optimized circuit-analysis code that was originally developed at Sandia National Laboratories in 1985. Screamer is written in Fortran 77 (with some extensions) and is highly optimized for speed. Screamer V3 solved electrical circuits having a limited, in-line circuit topology with restricted branches. This restricted circuit topology allowed for a very efficient matrix solver. We will describe the mathematical basis of Screamer and show how the topology leads to a very sparse matrix. We will describe the evolution of Screamer V4.0 and show how the development of powerful (> 0.1 TFlops) PCs with large amounts of memory (> 8 GB) enables major extensions to Screamers circuit topology without a significant loss in speed. Screamer incorporates many physics-based models such as dynamic loads, gas switching, water switching, oil switching, magnetic switching, and vacuum transmission lines, which are important to the high-voltage, pulsed-power community. Additional circuit models or modifications to existing models can be readily implemented in Screamer. Screamer runs on the Macintosh OS (9 & 10), LINUX, and Windows 7 & 8.

GPR imaging of land mines by solution of an inverse problem

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 Maxwells 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.

Screamer: A optimized pulsed-power circuit-analysis tool

Screamer was developed to solve a wide range circuits with a focus on pulsed-power systems. Screamer is a highly optimized code written in Fortran 77. We will describe the mathematical foundations of Screamer and show how Screamer uses a wide range of pulsed power circuit elements. Screamer incorporates many physics-based models such as lossy transmission lines, dynamic loads, gas switching, water switching, oil switching, magnetic switching, and magnetically insulated transmission lines, which are important to the high-voltage, pulsed-power community. Additional circuit models or modifications to existing models can be readily implemented in Screamer. We show an example of Screamer modeling a gas switch. Screamer is openly available to the community without restrictions. Screamer runs on the Macintosh, LINUX, and Windows platforms.

Application of a modified FFT approach to the subsurface imaging problem

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.