TT-FDTD: Tensor Train Accelerated Three-Dimensional FDTD With Logarithmic Cost of Spatial Operators

Abstract

Quantized tensor-train (QTT) compression is incorporated into a full-vector three-dimensional scattered-field finite-difference time-domain (FDTD) formulation on uniform Yee grids. All six electromagnetic-field components, material-dependent update coefficients, equivalent-current sources, and staggered finite-difference operators are represented in compatible QTT form. Gaussian regularization of voxelized material interfaces is used to reduce the coefficient ranks generated by abrupt dielectric and conductivity transitions. The formulation is evaluated for an anatomically heterogeneous human-head model and a homogeneous dielectric sphere on grids containing up to $512^3$ spatial cells. The reported results show that interface smoothing substantially reduces material-coefficient ranks and that the TT--FDTD solution reproduces the full-grid transient fields with pointwise absolute errors on the order of $10^{-4}$ in the examined slices. Compared with conventional FDTD, the tensor representation greatly reduces storage at fine discretizations, although tensor contractions and recompression introduce additional per-step computational cost. These results demonstrate the feasibility and memory--time tradeoff of QTT-accelerated three-dimensional FDTD for large structured-grid simulations.