The acoustic inverse scattering problem is of critical importance in a number of fields, including medical imaging, sonar, and non-destructive evaluation. The problem of interest can vary from the detection of the shape to the properties of an obstacle. The challenge is that this problem is severely ill-posed and highly nonlinear. Significant efforts have been expended over the years to develop solutions to this problem. However, existing fast data-driven methods primarily focus on the two-dimensional scattering case. This paper explores the potential of using machine learning to accelerate the solution to the three-dimensional (3D) version of the problem. To this end, we develop inverse scattering shape reconstruction network (ISSRNet), a deep learning framework for 3D shape reconstruction using phaseless far-field data. The framework is implemented by (a) using a compact probabilistic shape latent space learned by a 3D variational auto-encoder, and (b) a convolutional neural network trained to extract far-field features due to multiple incident waves and map the acoustic scattering information to this shape representation. We demonstrate ISSRNet's 3D shape reconstruction capabilities on random rock-like particles, and airplane objects from the popular ShapeNet data set. We also evaluate the framework's performance when trained on lower-resolution scattering data and when receiver locations include uncertainty. Our experiments show that the proposed framework is able to capture both global and local details, differentiate between different types of shapes and performs several orders of magnitude faster than its numerical iterative counterparts.

Particle in cell (PIC) methods is a popular means of understanding and exploiting plasma physics. This has given rise to numerous codes, both those are publicly available and others. Given the range of applications of plasma physics, from klystrons to accelerators to pulsed power devices to micro-plasma devices to high precision etching to fusion, there is a widespread need for PIC codes [1]. The literature in developing these codes is long and is summarized in [2]. While we will briefly review the state of the art, we digress to pose the problem.

The most popular method for self-consistent simulation of fields interacting with charged species is using finite difference time domain (FDTD) methods together with Newton's laws of motion to evolve locations and velocities of particles. Despite their popularity, the limitations of FDTD particle-in-cell (EM-FDTDPIC) methods are well-known. To address these, there has been significant interest over the past decade in exploring alternatives. In the past few years, the advances in electromagnetic finite element methods for particle-in-cell (EM-FEMPIC) have advanced by leaps and bounds. The mathematics necessary for implicit FEM methods that are unconditionally stable and charge-conserving are now well understood. Some of these advances are more recent. The next bottleneck necessary to make EM-FEMPIC competitive with the FDTD-based scheme is overcoming computational cost. Our approach to resolving this challenge is to develop two different finite element tearing and integration approaches, and use these to create domain decomposition schemes for EM-FEMPIC. Details of the proposed methodology are presented as well as a number of electrostatic results that demonstrate charge conservation as well as amelioration of costs for a number of problems.

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.

The state-of-the-art in Electromagnetic Finite Element Particle-in-Cell (EM-FEMPIC) has advanced considerably in recent years. By representing fields and sources in terms of consistent spatial basis functions, it is now possible to achieve exact satisfaction of Gauss' Laws. In addition, we have extended these conservation properties to implicit time-marching schemes, thereby allowing for analysis at a time step size determined by the physics being modeled rather than the smallest mesh feature. However, there is still a class of problems for which analysis at the highest frequency of interest is considerably expensive. One example of such a system are devices characterized by a high-frequency but narrowband response. In this work, we propose a novel EM-FEMPIC method that uses techniques derived from envelope tracking to analytically represent the high-frequency component of the fields, thereby allowing for analysis at time discretization defined by bandwidth. We demonstrate through a series of numerical examples that the proposed method retains the accuracy of a traditional EM-FEMPIC scheme, while requiring significantly fewer timesteps.

The state of the art in electromagnetic finite element PIC (EM-FEMPIC) has advanced significantly in recent years, with the state-of-the-art allowing the use of implicit field solvers while satisfying Gauss' Laws to machine precision. As a result, there is growing interest in exploring more robust particle integration schemes - in particular, methods with superior convergence and stability than the boris integrator that conserve energy on asymptotically large timescales. The objective of this work is to construct such a particle integrator using a predictor-corrector method. We show through our numerical results that the proposed method robustly conserves energy both for single particles and within a complete EM-FEMPIC scheme.

The ability to optimize transient systems comprising broadband electromagnetic (EM) devices fed by strongly non-linear circuits is of great interest in a number of design applications. However, existing techniques to optimize these systems are computationally infeasible for large-scale problems due primarily to the cost associated with solving the EM system. In this work, we propose a model order reduction technique based around transient port extraction to efficiently optimize a non-linear EM-circuit system. We show through numerical results that our method agrees with a traditional self-consistent solve, while being less computationally expensive.

The state of art of charge-conserving electromagnetic finite element particle-in-cell has grown by leaps and bounds in the past few years. These advances have primarily been achieved for leap-frog time stepping schemes for Maxwell solvers, in large part, due to the method strictly following the proper space for representing fields, charges, and measuring currents. Unfortunately, leap-frog based solvers (and their other incarnations) are only conditionally stable. Recent advances have made Electromagnetic Finite Element Particle-in-Cell (EM-FEMPIC) methods built around unconditionally stable time stepping schemes were shown to conserve charge. Together with the use of a quasi-Helmholtz decomposition, these methods were both unconditionally stable and satisfied Gauss' Laws to machine precision. However, this architecture was developed for systems with explicit particle integrators where fields and velocities were off by a time step. While completely self-consistent methods exist in the literature, they follow the classic rubric: collect a system of first order differential equations (Maxwell and Newton equations) and use an integrator to solve the combined system. These methods suffer from the same side-effect as earlier--they are conditionally stable. Here we propose a different approach; we pair an unconditionally stable Maxwell solver to an exponential predictor-corrector method for Newton's equations. As we will show via numerical experiments, the proposed method conserves energy within a PIC scheme, has an unconditionally stable EM solve, solves Newton's equations to much higher accuracy than a traditional Boris solver and conserves charge to machine precision. We further demonstrate benefits compared to other polynomial methods to solve Newton's equations, like the well known Boris push.

While finite-difference time-domain methods have long been the foundational basis for particle-in-cell (PIC) codes, there has been increasing momentum in developing a suite of finite element-based PIC methods. The beauty of finite difference-based methods is that it is easily cast within the correct mathematical framework to represent fields, fluxes, currents, and charges. However, more importantly, these methods are cost-effective. In the intervening years, since finite difference methods were developed, the state of the art of field modeling has shifted rather dramatically. Indeed, the most popular and trusted field simulators are based on the finite element method (FEM), thanks, in large part, to the discovery of the correct function spaces for quantities of interest for Maxwell’s equations, but also the flexibility that it brings to modeling geometry with the ability to refine in space and numerical order to better capture the underlying physics. Together with time-stepping schemes that are unconditionally stable, these methods provide the framework necessary to correctly capture the nuance of the physical evolution with high fidelity. The intent of this article is to review advances in electromagnetic finite element PIC (EM-FEMPIC). We will address the progress made in fundamental challenges in such a method for charge conservation to more programmatic ones, such as computational complexity.

The analysis of electromagnetic (EM) scattering in the isogeometric analysis (IGA) framework based on the Loop subdivision has long been restricted to simply connected geometries. The inability to analyze multiply connected objects is a glaring omission. In this article, we address this challenge. IGA provides seamless integration between the geometry and analysis using the same basis set to represent both. In particular, IGA methods using subdivision basis sets exploit the fact that the basis functions used for surface description are smooth (with continuous second derivatives) almost everywhere. On simply connected surfaces, this permits the definition of basis sets that are divergence-free and curl-free. What is missing from this suite is a basis set that is both divergence-free and curl-free, a necessary ingredient for a complete Helmholtz decomposition of currents on multiply connected structures. In this article, we achieve this missing ingredient numerically using random polynomial vector fields. We show that this basis set is analytically divergence-free and curl-free. Furthermore, we show that these bases recover curl-free, divergence-free, and both curl-free and divergence-free fields. Finally, we use this basis set to discretize a well-conditioned integral equation for analyzing perfectly conducting objects and demonstrate excellent agreement with other methods.

We introduce a computational Maxwell-Bloch framework for investigating out-of-equilibrium optical emitters in open systems. To do so, we compute the pulse-induced dynamics of each emitter from fundamental light-matter interactions and self-consistently calculate their radiative coupling, including phase inhomogeneity from propagation effects. This semiclassical framework is applied to open quantum dots systems with different densities and dipolar coupling. We observe signatures of superradiant behavior, such as directionality and faster decay, as well as subradiant emission. We compare and discuss the computed light emission obtained with our method and a master equation approach. Our framework enables quantitative investigations of large optical ensembles in the time domain and could be used to design new systems with enhanced superradiant and subradiant properties.

Optimization of strongly non-linear tightly coupled feeds attached to antennas is a challenging problem from a purely computational perspective. One can imagine that an optimization would (a) need to be in the time domain, and (b) has to be self-consistently coupled with the linear antenna (or electromagnetic) system. These two imply that the cost of optimization is governed by the need to repeatedly evaluate the fully coupled cost function. This paper leverages a recently developed transient port-extraction technique to circumvent this challenge and is agnostic to the optimization scheme. This approach provides a representation of the entire linear electromagnetic system at the port and can readily integrate with any non-linear circuit analysis and optimization methodology. In this paper, we demonstrate optimization of linear and non-linear circuit feed parameters that are tightly coupled to broadband radiating systems.

Many integral equations used to analyze scattering, such as the standard combined field integral equation (CFIE), are not well-conditioned for a wide range of frequencies and multi-scale geometries. There has been significant effort to alleviate this problem. A more recent one is using a set of decoupled potential integral equations (DPIE). These equations have been shown to be robust at low frequencies and immune to topology breakdown. But they mimic the ill-conditioning behavior of CFIE at high frequencies. This paper addresses this deficiency through new Calderón-type identities derived from the Vector Potential Integral Equation (VPIE). We construct novel analytic preconditioners for the vector potential integral equation (VPIE) and scalar potential integral equation (SPIE) constrained to perfect electric conductors (PEC). These new formulations are wide-band well-conditioned and converge rapidly for multi-scale geometries. This is demonstrated though a number of examples that use analytic and piecewise basis sets.

The eigenfunctions of the Laplace-Beltrami operator (LBO), or manifold harmonic basis (MHB), have many applications in mathematical physics, differential geometry, machine learning, and topological data analysis. MHB allows us to associate a frequency spectrum to a function on a manifold, analogous to the Fourier decomposition. This insight can be used to build a framework for analysis. The purpose of this paper is to review and illustrate such possibilities for computational electromagnetics as well as chart a potential path forward. To this end, we introduce three features of MHB: (a) enrichment for analysis of multiply connected domains, (b) local enrichment (L-MHB) and (c) hierarchical MHB (H-MHB) for reuse of data from coarser to fine geometry discretizations. Several results highlighting the efficacy of these methods are presented.

Evaluation of pair potentials is critical in a number of areas of physics. The classical N-body problem has its root in evaluating the Laplace potential, and has spawned tree-algorithms, the fast multipole method (FMM), as well as kernel independent approaches. Over the years, FMM for Laplace potential has had a profound impact on a number of disciplines as it has been possible to develop highly scalable parallel versions of these algorithms. This is in stark contrast to parallel algorithms for oscillatory potentials such as the Helmholtz potential. The principal bottlenecks to scalable parallelism are the computation and communication costs of operations necessary to traverse up, across, and down the tree. In this article, we analyze asymptotic costs for both computation and communication in a parallel implementation, and describe techniques to overcome bottlenecks and achieve high performance evaluation of the Helmholtz potential for different distributions of particles. We demonstrate that the resulting implementation has a load balancing effect that significantly reduces the time-to-solution and enhances the scale of problems that can be treated using full wave physics.

Self-consistent solution to electromagnetic (EM)-circuit systems is of significant interest for a number of applications. This has resulted in exhaustive research on means to couple them. In time domain, this typically involves a tight integration (or coupling) with field and non-linear circuit solvers. This is in stark contrast to coupled analysis of linear/weakly non-linear circuits and EM systems in frequency domain. Here, one typically extracts equivalent port parameters that are then fed into the circuit solver. Such an approach has several advantages: 1) the number of ports is typically smaller than the number of degrees of freedom, resulting in cost savings; 2) is circuit agnostic; and 3) can be integrated with a variety of device models. Port extraction is tantamount to obtaining impulse response of the linear EM system. In time domain, the deconvolution required to effect this is unstable. Recently, a novel approach was developed for time domain integral equations (TDIEs) to overcome this bottleneck. We extend this approach to time domain finite element method, and demonstrate its utility via a number of examples; significantly, we demonstrate that self-consistent solutions obtained using either a fully coupled or port extraction is identical to the desired precision for non-linear circuit systems. This is shown within a nodal network. We also demonstrate integration of port extracted data directly with drift diffusion equation to model device physics.

Active media are materials that consist of quantum systems, such as atoms, impurities, or quantum dots, and are characterized by strong resonant absorption and re-emission of radiation. In this article, we present a general framework for the numerical simulation and analysis of time-dependent radiation-induced phenomena in active media. The formulation used is based on the solution of semiclassical Maxwell–Bloch equations describing the evolution of each quantum element under the effect of external and re-emitted radiation. Within these Maxwell–Bloch equations, the coupling of quantum systems to classical Maxwellian fields poses computational challenges due to the strong nonlinearities involved. In contrast to traditional mesh-based solvers, we adopt an electric field integral-operator approach that can reliably account for near-field effects—including self-radiation—and is scalable to large systems. We then focus on media based on large numbers of quantum dots and demonstrate various physical effects arising from the near-field coupling, including polarization modulations and superradiance.

The eigenfunctions of the Laplace-Beltrami operator have widespread applications in a number of disciplines of engineering, computer vision/graphics, machine learning, etc. These eigenfunctions or manifold harmonics (MHs) provide the means to smoothly interpolate data on a manifold and are highly effective, specifically as it relates to geometry representation and editing; MHs form a natural basis for multi-resolution representation (and editing) of complex surfaces and functions defined therein. In this paper, we seek to develop the framework to exploit the benefits of MHs for shape reconstruction. To this end, a highly compressible, multi-resolution shape reconstruction scheme using MHs is developed. The method relies on subdivision basis sets to construct boundary element isogeometric methods for analysis and surface finite elements to construct MHs. This technique is paired with the volumetric source reconstruction method to determine an initial starting point. The examples presented highlight efficacy of the approach in the presence of noisy data, including a significant reduction in the number of degrees of freedom for complex objects, accuracy of reconstruction, and multi-resolution capabilities.

Recent advances in electromagnetic Finite Element Particle-in-Cell (EM-FEMPIC) simulations have made it possible to construct a simulation scheme that exactly satisfies Gauss’ Laws while using implicit time-marching schemes. In particular, one can now define a set of rules will allow any evolution scheme that obeys them to exactly conserve charge. However, these methods have thus far only been applied and studied for non-relativistic PIC schemes with rudimentary polynomial stencils for particle evolution. Here we seek to advance the state-of-the-art by constructing a relativistic PIC. To do so, we will exploit the advantages offered by our framework that decouples field evolution (via unconditionally stable time stepping methods) and equation of motion for particle evolution (or impressed current). Here we will exploit exponential predictor-corrector schemes to create a self-consistent update scheme, and examine both improved accuracy and conservation of physical quantities (charge and momentum).