Memory Footprint Reduction for the FFT-Based Volume Integral Equation Method via Tensor Decompositions

Abstract

We present a method of memory footprint reduction for FFT-based, electromagnetic (EM) volume integral equation (VIE) formulations. The arising Green's function tensors have low multilinear rank, which allows Tucker decomposition to be employed for their compression, thereby greatly reducing the required memory storage for numerical simulations. Consequently, the compressed components are able to fit inside a graphical processing unit (GPU) on which highly parallelized computations can vastly accelerate the iterative solution of the arising linear system. In addition, the element-wise products throughout the iterative solver's process require additional flops, thus, we provide a variety of novel and efficient methods that maintain the linear complexity of the classic element-wise product with an additional multiplicative small constant. We demonstrate the utility of our approach via its application to VIE simulations for the Magnetic Resonance Imaging (MRI) of a human head. For these simulations we report an order of magnitude acceleration over standard techniques.Index terms-Canonical polyadic model, higher order singular value decomposition, Tucker decomposition, volume integral equations.

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

“…Frequency: The work presented in [42] showed that the rank of the integral operators for 3D problems increases linearly with respect to the operating frequency. This was confirmed in [10], for the BTTB defining tensors of the FFT-based VIE systems (discretized integral operators). These tensors were columns of the corresponding Galerkin MoM matrices and modeled the interactions between one voxel's basis function and the whole domain via the N or K operators.…”

“…The aim of this work is to use Tucker decomposition [17] to perform a column-wise compression of the VSIE coupling matrix, in order to limit the associated memory demand and enable the computation of the relevant matrix-vector products in GPUs. Our approach was motivated by previous work [10] on the reduction of the memory footprint of FFT-based VIE Green's function tensors and the acceleration of matrix-vector products in VIE using GPU. Tucker decomposition belongs to a larger family of tensor decompositions and have been used successfully in the past for matrix compression inside IEbased simulations for EM applications.…”

“…Tucker decomposition belongs to a larger family of tensor decompositions and have been used successfully in the past for matrix compression inside IEbased simulations for EM applications. Examples include EM simulations of realistic body models simulations for UHF MRI [10], [18] and capacitance extraction [19]- [21]. Other tensor decompositions could be used [22]- [24], but for the intrinsic 3D nature of the problem at hand, Tucker optimizes operations and memory complexity.…”

FFT-PEEC: A Fast Tool From CAD to Power Electronics Simulations

“…Conductive, dielectric, and magnetic media can be considered by the proposed FFT-PEEC method and both linear and non-linear media can be considered. As proposed in [49], further computational improvements can be obtained with the adoption of a Graphical Processing Unit (GPU) that can speed-up the voxelization process and, mostly, the matrix-vector products via FFT. Moreover, when the considered devices have parts with very different dimensions (e.g.…”

Magnetic-Resonance-Based Electrical Property Mapping Using Global Maxwell Tomography With an 8-Channel Head Coil at 7 Tesla: A Simulation Study

“…Since it was shown that the performance of GMT is not affected by the size of the voxel [20], it will be desirable to use clinical resolutions (∼ 1 mm) for future in-vivo translation. One possible solution that will be explored to make that feasible is to compress the VIE operators using the Tucker decomposition [47], [48], which should allow us to perform the necessary matrix-vector products in GPUs even at high spatial resolutions [49]. Moreover, GMT could be further accelerated to facilitate in-vivo applications by applying a recently proposed preconditioner for the solution of the forward problem [25].…”