B. Shanker

Jie Li, Xin Fu, Balasubramaniam Shanker

In this work, a new integral equation (IE) based formulation is proposed using vector and scalar potentials for electromagnetic scattering. The new integral equations feature decoupled vector and scalar potentials that satisfy Lorentz gauge. The decoupling of the two potentials allows low-frequency stability. The formulation presented also results in Fredholm integral equations of second kind. The spectral properties of second kind integral operators leads to a well-conditioned system.

Andrew D. Baczewski, Balasubramaniam Shanker

The evaluation of long-range potentials in periodic, many-body systems arises as a necessary step in the numerical modeling of a multitude of interesting physical problems. Direct evaluation of these potentials requires O(N^2) operations and O(N^2) storage, where N is the number of interacting bodies. In this work, we present a method, which requires O(N) operations and O(N) storage, for the evaluation of periodic Helmholtz, Coulomb, and Yukawa potentials with periodicity in 1-, 2-, and 3-dimensions, using the method of Accelerated Cartesian Expansions (ACE). We present all aspects necessary to effect this acceleration within the framework of ACE including the necessary translation operators, and appropriately modifying the hierarchical computational algorithm. We also present several results that validate the efficacy of this method with respect to both error convergence and cost scaling, and derive error bounds for one exemplary potential.

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.

N. V. Nair, A. J. Pray, B. Shanker

The problem of late time instability in time domain integral equations for electromagnetics is longstanding. While several techniques have been suggested for addressing this problem, they either require impractically high degrees of freedom in the basis function or an analytical computation of matrix elements. The authors recently proposed a method that demonstrates stability without requiring either of these. The paper, however, does not present theoretical foundations for the choice of, or a rigorous error bounds on the approximation of, the bilinear form. This paper complements the authors' previous work by presenting a construction of the bilinear form based on the minimization of the energy in the system and a proof for the bounds on the approximation. We present results on the bounds developed and few sample scattering results that demonstrate the stability of the proposed scheme.

Doga Dikbayir, Abdel Alsnayyan, Vishnu Naresh Boddeti et al.

The inverse scattering problem is of critical importance in a number of fields, including medical imaging, sonar, sensing, non-destructive evaluation, and several others. The problem of interest can vary from detecting the shape to the constitutive properties of the obstacle. The challenge in both is that this problem is ill-posed, more so when there is limited information. That said, significant effort has been expended over the years in developing solutions to this problem. Here, we use a different approach, one that is founded on data. Specifically, we develop a deep learning framework for shape reconstruction using limited information with single incident wave, single frequency, and phase-less far-field data. This is done by (a) using a compact probabilistic shape latent space, learned by a 3D variational auto-encoder, and (b) a convolutional neural network trained to map the acoustic scattering information to this shape representation. The proposed framework is evaluated on a synthetic 3D particle dataset, as well as ShapeNet, a popular 3D shape recognition dataset. As demonstrated via a number of results, the proposed method is able to produce accurate reconstructions for large batches of complex scatterer shapes (such as airplanes and automobiles), despite the significant variation present within the data.