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

Abstract

In this work, we propose a method for the compression of the coupling matrix in volume-surface integral equation (VSIE) formulations. VSIE methods are used for electromagnetic analysis in magnetic resonance imaging (MRI) applications, for which the coupling matrix models the interactions between the coil and the body. We showed that these effects can be represented as independent interactions between remote elements in 3D tensor formats, and subsequently decomposed with the Tucker model. Our method can work in tandem with the adaptive cross approximation technique to provide fast solutions of VSIE problems. We demonstrated that our compression approaches can enable the use of VSIE matrices of prohibitive memory requirements, by allowing the effective use of modern graphical processing units (GPUs) to accelerate the arising matrix-vector products. This is critical to enable numerical MRI simulations at clinical voxel resolutions in a feasible computation time. In this paper, we demonstrate that the VSIE matrix-vector products needed to calculate the electromagnetic field produced by an MRI coil inside a numerical body model with 1 mm 3 voxel resolution, could be performed in ∼ 33 seconds in a GPU, after compressing the associated coupling matrix from ∼ 80 TB to ∼ 43 MB.

A Quantized Tensor Train Method for High-Frequency Scattering Problems Involving Heterogeneous Dielectric Layers

“…We see that the QTT method remains globally efficient even if results are a little disappointing compared to [7] or [6] where rough surfaces or irregular lattices were treated spectacularly. Actually, it appears that applying tensor methods in the context of high-frequency regime is still challenging, and our results probably call to experiment other tensor formats, more suited to these special situations (as perhaps the Tucker format successfully used in [8]).…”

“…Initially designed to break the "Curse of the dimension" (exponential growth of the storage with the dimension of an array), tensor methods, especially the Quantized Tensor Train (QTT) decomposition [4], are now able to tackle some difficult problems of computational physics such as the solution of very big linear systems [5]. Despite this promising surge, it is worth noting that the QTT format is still not widely used in the field of boundary integral equations, for instance, we can cite [6]- [8]. The technique we investigate is a contribution in this field and consists of treating the FEM part of our linear system classically and compressing the BEM parts with a QTT algorithm.…”