Two-step reduction approach based on the scattering-symmetric Lanczos algorithm for TLM-ROM

The non-symmetric properties of the TLM-matrix require the application of general Krylov subspace methods for the purposes of model order reduction (MOR). Application of the Arnoldi algorithm is computational expensive. Furthermore, the classical non-symmetric Lanczos algorithm requires the transpose TLM-matrix in order to form a biorthogonal basis for Krylov subspaces; hence, its algorithmic simplicity is also penalized and its computational complexity is increased. In this paper we propose a novel scattering-symmetric (S-symmetric) Lanczos algorithm, which is faster and consumes less memory in comparison to the conventional non-symmetric Lanczos algorithm, since the S-symmetric Lanczos algorithm generates the biorthogonal basis utilizing a single sequence like the symmetric Lanczos procedure. Along with the details of the proposed S-symmetric Lanczos algorithm, estimates are provided for its computational cost in comparison to the standard TLM time evolution scheme, the general Arnoldi process and the non-symmetric Lanczos process. However, the reduced TLM operator can still be large. Instead of the conventional eigenvalue decomposition (EVD) the second reduction of the TLM system can be applied in order to extract only eigenvalues corresponding to a needed frequency band. The two-step reduction algorithm allows us to decrease the computational effort in TLM-MOR.