Fernando L. Teixeira

Haksu Moon, Fernando L. Teixeira, Yuri A. Omelchenko

We describe a charge-conserving scatter-gather algorithm for particle-in-cell simulations on unstructured grids. Charge conservation is obtained from first principles, i.e., without the need for any post-processing or correction steps. This algorithm recovers, at a fundamental level, the scatter-gather algorithms presented recently by Campos-Pinto et al. [1] (to first-order) and by Squire et al. [2], but it is derived here in a streamlined fashion from a geometric viewpoint. Some ingredients reflecting this viewpoint are (1) the use of (discrete) differential forms of various degrees to represent fields, currents, and charged particles and provide localization rules for the degrees of freedom thereof on the various grid elements (nodes, edges, facets), (2) use of Whitney forms as basic interpolants from discrete differential forms to continuum space, and (3) use of a Galerkin formula for the discrete Hodge star operators (i.e., "mass matrices" incorporating the metric datum of the grid) applicable to generally irregular, unstructured grids. The expressions obtained for the scatter charges and scatter currents are very concise and do not involve numerical quadrature rules. Appropriate fractional areas within each grid element are identified that represent scatter charges and scatter currents within the element, and a simple geometric representation for the (exact) charge conservation mechanism is obtained by such identification. The field update is based on the coupled first-order Maxwell's curl equations to avoid spurious modes with secular growth (otherwise present in formulations that discretize the second-order wave equation). Examples are provided to verify preservation of discrete Gauss' law for all times.

Ananda Chakrabarti, Haitham H. Saleh, Indranil Nayak et al.

We present parameter-interpolated dynamic mode decomposition (piDMD), a parametric reduced-order modeling framework that embeds known parameter-affine structure directly into the DMD regression step. Unlike existing parametric DMD methods which interpolate modes, eigenvalues, or reduced operators and can be fragile with sparse training data or multi-dimensional parameter spaces, piDMD learns a single parameter-affine Koopman surrogate reduced order model (ROM) across multiple training parameter samples and predicts at unseen parameter values without retraining. We validate piDMD on fluid flow past a cylinder, electron beam oscillations in transverse magnetic fields, and virtual cathode oscillations -- the latter two being simulated using an electromagnetic particle-in-cell (EMPIC) method. Across all benchmarks, piDMD achieves accurate long-horizon predictions and improved robustness over state-of-the-art interpolation-based parametric DMD baselines, with less training samples and with multi-dimensional parameter spaces.

Lianlin Li, Long Gang Wang, Fernando L. Teixeira

The solution of nonlinear electromagnetic (EM) inverse scattering problems is typically hindered by several challenges such as ill-posedness, strong nonlinearity, and high computational costs. Recently, deep learning has been demonstrated to be a promising tool in addressing these challenges. In particular, it is possible to establish a connection between a deep convolutional neural network (CNN) and iterative solution methods of nonlinear EM inverse scattering. This has led to the development of an efficient CNN-based solution to nonlinear EM inverse problems, termed DeepNIS. It has been shown that DeepNIS can outperform conventional nonlinear inverse scattering methods in terms of both image quality and computational time. In this work, we quantitatively evaluate the performance of DeepNIS as a function of the number of layers using structure similarity measure (SSIM) and mean-square error (MSE) metrics. In addition, we probe the dynamic evolution behavior of DeepNIS by examining its near-isometry property. It is shown that after a proper training stage the proposed CNN is near optimal in terms of the stability and generalization ability.

Lianlin Li, Long Gang Wang, Fernando L. Teixeira et al.

Nonlinear electromagnetic (EM) inverse scattering is a quantitative and super-resolution imaging technique, in which more realistic interactions between the internal structure of scene and EM wavefield are taken into account in the imaging procedure, in contrast to conventional tomography. However, it poses important challenges arising from its intrinsic strong nonlinearity, ill-posedness, and expensive computation costs. To tackle these difficulties, we, for the first time to our best knowledge, exploit a connection between the deep neural network (DNN) architecture and the iterative method of nonlinear EM inverse scattering. This enables the development of a novel DNN-based methodology for nonlinear EM inverse problems (termed here DeepNIS). The proposed DeepNIS consists of a cascade of multi-layer complexvalued residual convolutional neural network (CNN) modules. We numerically and experimentally demonstrate that the DeepNIS outperforms remarkably conventional nonlinear inverse scattering methods in terms of both the image quality and computational time. We show that DeepNIS can learn a general model approximating the underlying EM inverse scattering system. It is expected that the DeepNIS will serve as powerful tool in treating highly nonlinear EM inverse scattering problems over different frequency bands, involving large-scale and high-contrast objects, which are extremely hard and impractical to solve using conventional inverse scattering methods.

Dong-Yeop Na, Ben-Hur V. Borges, Fernando L. Teixeira

We present a finite-element time-domain (FETD) Maxwell solver for the analysis of body-of-revolution (BOR) geometries based on discrete exterior calculus (DEC) of differential forms and transformation optics (TO) concepts. We explore TO principles to map the original 3-D BOR problem to a 2-D one in the meridian plane based on a Cartesian coordinate system where the cylindrical metric is fully embedded into the constitutive properties of an effective inhomogeneous and anisotropic medium that fills the domain. The proposed solver uses a TE/TM field decomposition and an appropriate set of DEC-based basis functions on an irregular grid discretizing the meridian plane. A symplectic time discretization based on a leap-frog scheme is applied to obtain the full-discrete marching-on-time algorithm. We validate the algorithm by comparing the numerical results against analytical solutions for resonant fields in cylindrical cavities and against pseudo-analytical solutions for fields radiated by cylindrically symmetric antennas in layered media. We also illustrate the application of the algorithm for a particle-in-cell (PIC) simulation of beam-wave interactions inside a high-power backward-wave oscillator.