Tensor‐Train Compression of a Volume Integral Eddy Current Problem in Discontinuous Galerkin

Abstract

ABSTRACT The paper presents a low‐rank tensor compression strategy based on the tensor‐train (TT) format for the integral operator arising in eddy current problems. Unlike classical matrix compression techniques such as the fast multipole method (FMM) or hierarchical matrices (‐matrices), which typically compress only far‐field interactions, the proposed approach compresses the entire operator, leading to significantly reduced memory requirements. The method relies on a tensorization step that transforms the discretized integral operator into a higher‐order tensor represented in TT format. A major challenge of tensor‐based compression for integral operators is the limited flexibility of tensorization, as it imposes strong constraints on the number of degrees of freedom. These constraints are particularly restrictive for conforming discretizations, where degrees of freedom are associated with mesh facets, as commonly encountered in eddy current formulations. To overcome this limitation, a discontinuous Galerkin discretization is employed, allowing the degrees of freedom to be defined at the cell level and enabling tensorization on more general meshes. The TT representation further allows the definition of an efficient matrix–vector product, which can be combined with an iterative solver to compute the solution of the resulting linear system. Numerical results for a thin plate, a torus with square cross‐section, and the COMPUMAG Team 7 benchmark demonstrate a significant reduction in memory consumption compared to an FMM‐based conforming discretization. While the solver performance is slightly lower, more advanced TT algebraic tools offer promising perspectives for further improvements.

The authors' abstract, as published at the source. International Journal for Numerical Methods in Engineering, 2026 · DOI ↗