Tensor Train Accelerated FDTD Method with Logarithmic Cost of Spatial Operations

Quantized tensor train (QTT) decomposition is applied to the finite-difference time-domain (FDTD) method to reduce the cost of spatial operations on structured grids. In a two-dimensional scattered-field TMz formulation, electric and magnetic fields on a uniform Yee grid are tensorized and compressed into low-rank QTT cores, while discrete curl operators and material coefficients are represented in compatible QTT forms. To improve rank stability during long time integration, the dielectric profile is smoothed near material interfaces, reducing high-spatial-frequency content introduced by staircasing and improving practical compressibility. With modest QTT ranks, the resulting update scheme achieves logarithmic scaling in both memory and CPU cost with respect to the number of grid cells. Numerical results demonstrate logarithmic-cost scattered-field FDTD with Mur absorbing boundary conditions and provide a foundation for extending QTT-accelerated FDTD to three-dimensional problems.