Global Maxwell Tomography (GMT) is a noninvasive inverse optimization method for the estimation of electrical properties (EP) from magnetic resonance (MR) measurements. GMT uses the volume integral equation (VIE) in the forward problem and assumes that the sample has negligible effect on the coil currents. Consequently, GMT calculates the coil's incident fields with an initial EP distribution and keeps them constant for all optimization iterations. This can lead to erroneous reconstructions. This work introduces a novel version of GMT that replaces VIE with the volume-surface integral equation (VSIE), which recalculates the coil currents at every iteration based on updated EP estimates before computing the associated fields. We simulated an 8-channel transceiver coil array for 7 T brain imaging and reconstructed the EP of a realistic head model using VSIE-based GMT. We built the coil, collected experimental MR measurements, and reconstructed EP of a two-compartment phantom. In simulations, VSIE-based GMT outperformed VIE-based GMT by at least 12% for both EP. In experiments, the relative difference with respect to probe-measured EP values in the inner (outer) compartment was 13% (26%) and 17% (33%) for the permittivity and conductivity, respectively. The use of VSIE over VIE enhances GMT's performance by accounting for the effect of the EP on the coil currents. VSIE-based GMT does not rely on an initial EP estimate, rendering it more suitable for experimental reconstructions compared to the VIE-based GMT.

ABSTRACTFocusing on the homogenization of periodic composite materials, this study investigates computational methods based on volume integral equations. Such formulations are revisited from the standpoint of the preconditioning of the original cell problem by the introduction of a comparison material. This allows for to recovery of simple convergence criteria for iterative steepest‐descent and fixed‐point schemes for composites with general non‐linear behaviour. In the case of linear materials, the preconditioned volume integral formulation coincides with the well‐known Lippmann–Schwinger equation. The spectral properties of the featured linear integral operator, which is bounded and self‐adjoint, are investigated to shed light on the behaviour of conventional computational homogenization methods. The so‐called Lippmann–Schwinger spectrum is analyzed, with its bounds governing the convergence rate of iterative solution methods. The associated eigenvectors, which constitute the eigenstates of the composite material considered, are also described in detail to understand their role in constructing the solution to the cell problem and ultimately in computing the effective properties. Formulated in the continuous setting, this analysis is followed by the investigation of a discrete representation of the integral operator considered. A number of examples on synthetic microstructures are finally considered in the conductivity setting to illustrate the obtained theoretical results and highlight the role of the spectral properties in the operation of computational homogenization methods. This paves the way for the development of reduced models and more efficient computations.

The radiative characteristics of the radiofrequency receive coils dictate the signal-to-noise ratio (SNR) of magnetic resonance images. Despite the crucial importance of RF coils, the practical coil design process has remained a largely empirical one. This work introduces a novel optimization framework for rational coil design, which relies on a fully automated pipeline that combines rapid electromagnetic simulations, shape optimization and coil meshing. The objective function iteratively maximizes SNR performance in a target region of interest with respect to the ultimate intrinsic SNR, which is the theoretically highest SNR independent from any particular coil design. The forward simulation employs a fast electromagnetic solver based on coupled surface and volume integral equations. The coils are represented as B-spline curves with an associated width, and automatically meshed for EM simulation. We implemented a new method to tune and decouple coils at each iteration without manual user intervention. The algorithm optimizes the size and position of a given number of coils with a combination of grid search and a line search. We demonstrated the framework by designing receive arrays of increasing complexity that yield optimal SNR for different target regions inside a numerical head model. SNR simulation time ranged from 15 s for a 3-coil configuration to 32 s for a 12-coil array, constrained to a helmet-like surface, including tuning and decoupling. The optimized 12-coil geometry yielded 9% higher average SNR performance in the brain at 3 T. This work represents the first automated coil optimization framework that uses full-wave electromagnetic simulations and ultimate performance benchmarks. This novel approach enables the systematic design of coils for magnetic resonance imaging with significantly improved SNR performance, potentially transforming coil development from empirical design to physics-driven optimization.

The article proposes a method for reconstruction inhomogeneity parameters based on the results of near-field measurements in medical diagnostics. This is a classical inverse problem arising in various fields of science and technology. At the first stage, the problem of wave propagation inside an object is considered. A rigorous description of the problem is given both as a boundary value problem and as a volume integral equation. Next, using the numerical solution of this equation, the field values outside the body in the near zone are determined. At the second stage, using the obtained near-field values using a two-step algorithm, a search for inhomogeneities occurs. A specially trained neural network filters the values obtained before and after the two-step algorithm, thereby improving the quality of images visualizing inhomogeneities. Graphic illustrations of the original and restored values of inhomogeneities for the objects under consideration are presented. An experiment was conducted demonstrating the features of restoring object parameters using neural networks. The results show the effectiveness of filtering the calculated data by the autoencoder. A software package for determining the parameters of inhomogeneities inside the object is proposed and implemented.

The article proposes a method for reconstruction inhomogeneity parameters based on the results of near-field measurements in medical diagnostics. This is a classical inverse problem arising in various fields of science and technology. At the first stage, the problem of wave propagation inside an object is considered. A rigorous description of the problem is given both as a boundary value problem and as a volume integral equation. Next, using the numerical solution of this equation, the field values outside the body in the near zone are determined. At the second stage, using the obtained near-field values using a two-step algorithm, a search for inhomogeneities occurs. A specially trained neural network filters the values obtained before and after the two-step algorithm, thereby improving the quality of images visualizing inhomogeneities. Graphic illustrations of the original and restored values of inhomogeneities for the objects under consideration are presented. An experiment was conducted demonstrating the features of restoring object parameters using neural networks. The results show the effectiveness of filtering the calculated data by the autoencoder. A software package for determining the parameters of inhomogeneities inside the object is proposed and implemented.

Abstract Magnetic Confinement Fusion (MCF) devices are characterized by complex geometries with both thick and thin components. Integral equation (IE) methods, including Volume Integral Equations (VIE) and Surface Integral Equations (SIE), are particularly advantageous for solving eddy-current problems because they eliminate the need for air/vacuum discretization. This paper explores hierarchical matrices (ℋ-matrices) to reduce the computational complexity of the coupled VIE-SIE formulation, enabling the efficient analysis of large-scale eddy current problems in the conductive structures of MCF devices.

Coupled boundary and volume integral equations for electromagnetic scattering

ABSTRACTVertical fractures are often reported in sedimentary rocks. The detection of these inherent fractures is important before carrying out the carbon dioxide sequestration in these rocks. The detection of the fractures is also crucial for an accurate estimation of the moment tensor from microseismic waveform data. In this study, we use the distorted Born iterative method to perform seismic full waveform inversion for the parameters of vertical fractures in sedimentary formations. The distorted Born iterative method is based on transforming a nonlinear inverse scattering problem into a series of linear inverse problems by using the distorted Born approximation. We work in the frequency domain and use a volume integral equation method to solve the direct scattering problem. A heterogeneous, generally anisotropic medium, is iteratively updated using the matrix‐free formulation of Fréchet derivatives and their adjoint. In the distorted Born iterative method, the heterogeneous medium Green's function is also updated after each iteration, which is not done in the classical Born iterative method. In our implementation, we assume that the fractures are thin, vertical and parallel to each other. The background, in which fractures are embedded, is transversely isotropic with a vertical axis of symmetry. The vertically transversely isotropic background can be inhomogeneous. In an isotropic background, it is common to invert for a single tangential fracture weakness along with a normal fracture weakness. However, in a vertically transversely isotropic background, the horizontal‐tangential and the vertical‐tangential fracture weaknesses vary, and therefore we invert for three fracture weaknesses. In numerical experiments, we employ a cross‐hole seismic configuration and invert synthetic waveform data for fracture weaknesses. The radiation pattern analysis is performed to investigate the cross‐talk among different fracture weaknesses. It is found that the horizontal‐tangential fracture weakness is better resolved than the other fracture weaknesses, which is confirmed through the numerical results.

Focused ultrasound has gained importance in cancer therapy and neuromodulation because it offers a non-invasive treatment that can reach malignant tissue at high precision. Fast and accurate simulation tools are essential to improve safety guidelines and patient-specific treatment planning. The goal is guiding sufficient acoustic energy toward the focus to achieve ablation while sparing healthy tissue in the beam path. We have already achieved realistic simulations with our open-source OptimUS library using the Boundary Element Method (BEM). The BEM is a powerful algorithm to simulate high-frequency acoustics in unbounded domains that scatter at objects with high material contrasts, like soft tissue and bone, and only needs surface meshes at material interfaces. However, the BEM is limited to scatterers with constant density and speed of sound. To investigate the influence of bone heterogeneities on the focus aberrations, we developed a Volume Integral Equation and coupled it with the BEM. To improve computational efficiency and reduce memory footprint, we developed a hierarchical matrix compression technique for the dense system matrices. Our innovations allow for simulating ultrasound scattering at ribcage models for transcostal ultrasound and transmission through skull slabs for transcranial ultrasound at operating frequencies and with material data taken from biomedical images.

For the efficient analysis of electromagnetic scattering from thin, homogenous dielectric objects, a kind of revised pulse vector basis function is proposed for the solution of the volume integral equation. In this revised version of the definition, three pulse vector basis functions are defined in two neighboring tetrahedral elements sharing a common face. Therefore, the number of unknowns can be reduced to about half of the traditionally used one. Numerical results show that for the solution of thin homogenous dielectric object problems, the memory of the proposed solution scheme is less than one-third of the conventional one, while both methods give similar accurate results. In addition, the proposed scheme can be applied for the solution of electromagnetic scattering from the dielectric radome and from thin dielectric arrays with different media parameters for each array element. Furthermore, the proposed scheme iterates as fast as the original one when iterative solvers are used to solve the resulted matrix equations. Therefore, for the solution of thin and homogenous dielectric object problems, the proposed scheme is much more efficient than the original one.

We develop a full-wave electromagnetic (EM) theory for calculating the multipole decomposition in two-dimensional (2-D) structures consisting of isolated, arbitrarily shaped, inhomogeneous, anisotropic cylinders or a collection of such. To derive the multipole decomposition, we first solve the scattering problem by expanding the scattered electric field in divergenceless cylindrical vector wave functions (CVWFs) with unknown expansion coefficients that characterize the multipole response. These expansion coefficients are then expressed via contour integrals of the vectorial components of the scattered electric field evaluated via an electric field volume integral equation (EFVIE). The kernels of the EFVIE are the products of the tensorial 2-D Green's function (GF) expansion and the equivalent 2-D volumetric electric and magnetic current densities. We validate the theory using the commercial finite element solver COMSOL Multiphysics. In the validation, we compute the multipole decomposition of the fields scattered from various 2-D structures and compare the results with alternative formulations. Finally, we demonstrate the applicability of the theory to study an emerging photonics application on oligomer-based highly directional switching using active media. This analysis addresses a critical gap in the current literature, where multipole theories exist primarily for three-dimensional (3-D) particles of isotropic materials. Our work enhances the understanding and utilization of the optical properties of 2-D, inhomogeneous, and anisotropic cylindrical structures, contributing to advancements in photonic and meta-optics technologies.

Application of Randomized Matrix Approximation in 3-D Volume Integral Equation Domain Decomposition Method for Electromagnetic Scattering in Layered Media With Complex Objects

One-Stage $ O(N \log N)$ Algorithm for Generating Nested Rank-Minimized Representation of Electrically Large Volume Integral Equations

Acceleration of Solving Volume Integral Equations through a Physics Driven Neural Network and Its Applications to Random Media Scattering

Fast Well-Conditioned Volume Integral Equation Solver for Analyzing Nonlocal Optical Responses in Quantum Nanostructures

Characteristics of L-Band Microwave Scattering From Layered Rough Soil With a Full-Wave Volume Integral Equation Approach

A new volume integral equation (VIE) approach is introduced to study transcranial magnetic stimulation (TMS) and high-contrast media at low frequencies. This new integral equation offers a simple solution to the high-contrast breakdown observed in low-frequency electric field (E-field) dosimetry of conductive media. Specifically, we employ appropriate approximations that are valid for low frequencies and stabilize the VIE by introducing a basis expansion set that removes solutions associated with high eigenvalues in the equation. The new equation is devoid of high-contrast breakdown and does not require the use of auxiliary surface variables or projectors, providing a straightforward practical solution for the VIE analysis of TMS. Our results indicate that the novel VIE formulation matches boundary element, finite element, and analytical solutions. This new VIE represents a first step towards including anisotropy in integral equation E-field dosimetry for brain stimulation.

Soft x-ray wafer-metrology experiments are characterized by low signal-to-noise ratios and lack phase information, which both cause difficulties with the accurate three-dimensional profiling of small geometrical features of structures on a wafer. To this end, we extend an existing phase-based inverse-scattering method to demonstrate a sub-nanometer and noise-robust reconstruction of the targets by synthetic soft x-ray scatterometry experiments. The targets are modeled as three-dimensional finite dielectric scatterers embedded in a planarly layered medium, where a scatterer's geometry and spatial permittivity distribution are described by a uniform polygonal cross section along its height. Each cross section is continuously parametrized by its vertices and homogeneous permittivity. The combination of this parametrization of the scatterers and the employed Gabor frames ensures that the underlying linear system of the spatial spectral Maxwell solver is continuously differentiable with respect to the parameters for phaseless inverse-scattering problems. In synthetic demonstrations, we demonstrate the accurate and noise-robust reconstruction of the parameters without any regularization term. Most of the vertex parameters are retrieved with an error of less than λ/13 with λ=13.5n m, when the ideal sensor model with shot noise detects at least five photons per sensor pixel. This corresponds to a signal-to-noise ratio of 3.5dB. These vertex parameters are retrieved with an accuracy of λ/90 when the signal-to-noise ratio is increased to 10dB, or approximately 100 photons per pixel. The material parameters are retrieved with errors ranging from 0.05% to 5% for signal-to-noise ratios between 10dB and 3.5dB.

Combination of Methods of Volume and Surface Integral Equations in Problems of Electromagnetic Scattering by Small Thickness Structures

Radially transverse isotropic inclusions in isotropic elastic media: Local fields, neutral inclusions, effective elastic properties