[Paper Review] A Fast Volume Integral Equation Solver with Linear Basis Functions for the Accurate Computation of Electromagnetic Fields in MRI

Ioannis P. Georgakis, Ilias I. Giannakopoulos+2 more|arXiv (Cornell University)|Feb 6, 2019

Microwave Imaging and Scattering Analysis68 references9 citations

TL;DR

This paper presents a fast, stable volume integral equation (VIE) solver using discontinuous piecewise linear (PWL) basis functions for accurate electromagnetic (EM) field computation in high-contrast, inhomogeneous MRI environments. By leveraging Galerkin method of moments with FFT-accelerated matrix-vector products and a novel preconditioner, the solver achieves superior accuracy and convergence—especially at coarse resolutions—outperforming conventional piecewise constant (PWC) schemes with fewer degrees of freedom.

ABSTRACT

A stable volume integral equation (VIE) solver based on polarization/magnetization currents is presented, for the accurate and efficient computation of the electromagnetic scattering from highly inhomogeneous and high contrast objects.We employ the Galerkin Method of Moments to discretize the formulation with discontinuous piecewise linear basis functions on uniform voxelized grids, allowing for the acceleration of the associated matrix-vector products in an iterative solver, with the help of FFT. Numerical results illustrate the superior accuracy and more stable convergence properties of the proposed framework, when compared against standard low order (piecewise constant) discretization schemes and a more conventional VIE formulation based on electric flux densities. Finally, the developed solver is applied to analyze complex geometries, including realistic human body models, typically used in modeling the interactions between electromagnetic waves and biological tissue.

Motivation & Objective

• Address the challenge of accurate and efficient EM field computation in highly inhomogeneous, high-contrast biological tissues typical in high-field MRI.
• Overcome the limitations of low-order (piecewise constant) basis functions in terms of accuracy and convergence stability, especially at coarse mesh resolutions.
• Develop a stable, fast VIE solver suitable for patient-specific SAR calculations and RF coil modeling in ultra-high-field MRI, where mesh refinement is often impractical.
• Enable p-refinement (higher-order basis functions) as an alternative to h-refinement for achieving accuracy without increasing computational cost or degrees of freedom.
• Integrate the solver into existing SIE-based frameworks for hybrid modeling of EM interactions in realistic human body models (RHBMs).

Proposed method

• Formulate the EM scattering problem using equivalent polarization and magnetization currents to enable a stable volume integral equation (VIE) formulation.
• Discretize the VIE using discontinuous piecewise linear (PWL) basis and testing functions defined on uniform voxelized grids, improving field representation over piecewise constant functions.
• Apply the Galerkin method of moments to ensure Galerkin orthogonality and enhance numerical stability and accuracy.
• Accelerate matrix-vector products in iterative solvers (e.g., GMRES) using the Fast Fourier Transform (FFT), enabling efficient computation on large-scale problems.
• Implement a specialized preconditioner to stabilize the iterative solver, especially at fine resolutions where unpreconditioned systems diverge.
• Utilize a uniform voxel grid to enable efficient FFT-based matrix-vector multiplication and maintain compatibility with existing MRI modeling pipelines.

Experimental results

Research questions

• RQ1 Can discontinuous piecewise linear (PWL) basis functions significantly improve the accuracy and convergence stability of VIE solvers in high-contrast, inhomogeneous MRI problems compared to standard piecewise constant (PWC) functions?
• RQ2 To what extent can p-refinement with PWL basis functions achieve accuracy comparable to h-refinement with PWC functions while reducing the number of degrees of freedom?
• RQ3 Does the proposed current-based VIE formulation with PWL functions maintain numerical stability and convergence at extremely fine resolutions (e.g., 1 mm) without divergence?
• RQ4 How effective is the proposed preconditioner in accelerating convergence of the iterative solver for PWL-based VIE systems, especially in challenging scenarios like high-permittivity shimming pads?
• RQ5 Can the PWL-based VIE solver reliably compute EM fields and magnetic energy in realistic human body models (RHBMs) at coarse resolutions, enabling practical patient-specific SAR analysis?

Key findings

• The PWL-based VIE solver achieves higher accuracy than the PWC-based solver even at coarser resolutions: 2 mm PWL magnetic energy is lower than 1 mm PWC, indicating reduced error.
• Despite having only half the degrees of freedom of 1 mm PWC, 2 mm PWL yields lower error in magnetic energy (error_h1), magnetic energy density (error_h2), and magnetic field norm (error_h3) than 1 mm PWC.
• The 5 mm PWL solution achieves comparable or better accuracy than 2 mm PWC, despite the PWC having approximately 3.75 times more degrees of freedom.
• The unpreconditioned PWL solver diverges at the 1 mm resolution, but the proposed preconditioner stabilizes convergence, reducing iterations from 3900 to 700—comparable to the PWC solver’s performance.
• The PWL solver converges stably and efficiently with the preconditioner, enabling practical use in demanding MRI simulations without mesh refinement.
• The proposed framework enables accurate EM field computation in complex RHBMs with high-permittivity shimming pads, demonstrating robustness in high-contrast scenarios.