Tensor Completion algorithms, fast direct volume integral equation solvers, and other optimal complexity solvers
An approach to analyze induced E-fields is to formulate the scenario as a boundary or volume integral equation. This approach can yield considerable advantages such as reduction of the number of unknowns N, operator with a bounded condition number, and easier handling of geometric complexity. However, the solution of integral operators requires storing a dense matrix requiring O(N2) memory. Fast multipole methods (FMM) and Fast Fourier transform (FFT) methods can be used to accelerate integral equation solvers and permit the efficient and accurate analysis of electromagnetic scattering from complex and large-scale structures. We have developed numerous novel techniques for improving the compression of integral techniques including: (i) a tensor-train based compression scheme that enables the storage of a Toeplitz matrix in O(logN) and O(NlogN) memory for VIE with the Laplace and Helmholtz kernels, respectively; (ii) the auxiliary dipole method for evaluating the optimum coil placement during TMS; (iii) a hierarchical off-diagonal butterfly (HOD-BF) approximate inverse with O(Nlog2N) setup and O(N1.5logN) inversion computational cost for the preconditioner; (iv) Tucker decomposition-based compression scheme for the translation operators of FMM-FFT that oftentimes resulted in 90% reduction storage requirements.
[J7] A. C. Yucel, L. J. Gomez, and E. Micielssen, "Compression of Translation Operator Tensors in FMM-FFT Accelerated SIE Solvers via Tucker Decomposition," IEEE Antennas and Wireless Propagation Letters, vol. 16, pp. 2667-2670, 2017.
[J12] C, Zhuotong, L. J. Gomez, S. Zheng, A. C. Yucel, Z. Zhang, V. Okhmatovski, "Sparsity Aware Pre-Corrected Tensor Train Algorithm For Fast Solution of 2D Scattering Problems and Current Flow Modelling on Unstructured Meshes," Transactions on Microwave Theory and Techniques, vol. 67, no. 12, 2019.
[J16] S. B. Sayed, Y. Liu, L. J. Gomez, A. Yucel, "A Butterfly-Accelerated Volume Integral Equation Solver for Broad Permittivity and Large-Scale Electromagnetic Analysis," Transactions on Antennas and Propagation, vol. 70, no. 5, pp. 3549 - 3559, 2022.