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 achieve logarithmic complexity of spatial operations. For a two–dimensional TM$_z$ scattered–field formulation on a uniform Yee grid with $N_x = N_y = 2^d$ cells, the scattered electric and magnetic field samples are tensorized into $2d$–dimensional QTT tensors and approximated by products of small cores with low QTT ranks. The discrete curl operators are recast as QTT matrix–product operators, while spatially varying material coefficients and equivalent volume currents are represented by diagonal QTT tensors. The resulting TT–FDTD updates for all field components at a given time step can be carried out in $O(r^2 d) = O(\log N)$ CPU time and memory, N = N_x N_y, provided that the QTT ranks $r$ remain bounded. The present work demonstrates such logarithmic–cost scattered–field FDTD for structured grids with Mur's absorbing boundary conditions, laying the foundation for QTT–accelerated 3D FDTD.