A Fast Volume Integral Equation Solver With Linear Basis Functions for the Accurate Computation of EM Fields in MRI

PIFON-EPT: MR-Based Electrical Property Tomography Using Physics-Informed Fourier Networks

“…The simulations were performed with the volume [52] and the volumesurface integral equation [53], [54] methods. The volume equations were solved using higher-order polynomials [55] as basis functions to ensure accuracy in the B + 1 distributions. All experiments were performed on a server running Ubuntu 20.04.3 LTS operating system, with an Intel(R) Xeon(R) Silver 4216 CPU at 2.10GHz, 64 cores, 2 threads per core, and an NVIDIA RTX 3090 GPU with 24 GB of memory.…”

Computational methods for the estimation of ideal current patterns in realistic human models

“…$$$ {c}_e=\mathrm{i}\omega {\epsilon}_0 $$$, where $$$ \mathrm{i} $$$ is the imaginary unit, $$$ \omega $$$ is the angular frequency, and $$$ {\epsilon}_0 $$$ is the permittivity of vacuum. $$$ \mathbf{G}\in {\mathbb{C}}^{qn\times qn} $$$ is the Grammian, $$$ \mathbf{N}\in {\mathbb{C}}^{qn\times qn} $$$ is the discretized version of the dyadic Green's function operator that maps volumetric electric currents to electric fields, 33,34 and $$$ {\mathbf{e}}_{\mathrm{inc}}\in {\mathbb{C}}^{qn\times 1} $$$ is the excitation or incident electric field from an external source. The electric field $$$ \mathbf{e}\in {\mathbb{C}}^{qn\times 1} $$$ and magnetic field $$$ \mathbf{h}\in {\mathbb{C}}^{qn\times 1} $$$ in the sample can be computed as follows: …”

“…$$$ \mathbf{G}\in {\mathbb{C}}^{qn\times qn} $$$ is the Grammian, $$$ \mathbf{N}\in {\mathbb{C}}^{qn\times qn} $$$ is the discretized version of the dyadic Green's function operator that maps volumetric electric currents to electric fields, 33,34 and $$$ {\mathbf{e}}_{\mathrm{inc}}\in {\mathbb{C}}^{qn\times 1} $$$ is the excitation or incident electric field from an external source. The electric field $$$ \mathbf{e}\in {\mathbb{C}}^{qn\times 1} $$$ and magnetic field $$$ \mathbf{h}\in {\mathbb{C}}^{qn\times 1} $$$ in the sample can be computed as follows: $$$$ {\displaystyle \begin{array}{ll}\mathbf{e}& ={\mathbf{G}}^{-1}\left(\frac{1}{c_e}\left(\mathbf{N}-\mathbf{I}\right){\mathbf{j}}_{\mathrm{b}}+{\mathbf{e}}_{\mathrm{inc}}\right),\\ {}\mathbf{h}& ={\mathbf{G}}^{-1}\left(\mathbf{K}{\mathbf{j}}_{\mathrm{b}}+{\mathbf{h}}_{\mathrm{inc}}\right),\end{array}} $$$$ where …”

Compression of volume-surface integral equation matrices via Tucker decomposition for magnetic resonance applications

“…As a result, only the defining columns of the BTTB matrix need to be stored and the matrix-vector product can be accelerated using the FFT, as in [12], [31]- [37]. The unknown polarization currents (J p ∈ C qnv×p ) can be discretized with polynomial basis functions, either piecewise constant [12] (PWC, 3 unknowns per voxel) or piecewise linear [14] (PWL, 12 unknowns per voxel), and a single-voxel support.…”

“…MARIE combines surface and volume integral equations (SIE,VIE), employing a triangular tessellation for the RF coils' conductors and a uniform voxelized grid discretization for the body models. RWG basis functions [13] and polynomial basis functions [12], [14] are used to compute the unknowns of the surface and volume IE, respectively. Matrix-vector products are accelerated using the fast Fourier transform (FFT).…”