Title: A Model Order Reduction Algorithm for Estimating the Absorption Spectrum,
Abstract: The ab initio description of the spectral interior of the absorption spectrum
poses both a theoretical and computational challenge for modern electronic
structure theory. Due to the often spectrally dense character of this domain in
the quantum propagator's eigenspectrum for medium-to-large sized systems,
traditional approaches based on the partial diagonalization of the propagator
often encounter oscillatory and stagnating convergence. Electronic structure
methods which solve the molecular response problem through the solution of
spectrally shifted linear systems, such as the complex polarization propagator,
offer an alternative approach which is agnostic to the underlying spectral
density or domain location. This generality comes at a seemingly high
computational cost associated with solving a large linear system for each
spectral shift in some discretization of the spectral domain of interest. We
present a novel, adaptive solution based on model order reduction techniques
via interpolation. Model order reduction reduces the computational complexity
of mathematical models and is ubiquitous in the simulation of dynamical
systems. The efficiency and effectiveness of the proposed algorithm in the ab
initio prediction of X-Ray absorption spectra is demonstrated using a test set
of challenging water clusters which are spectrally dense in the neighborhood of
the oxygen K-edge. Based on a single, user defined tolerance we automatically
determine the order of the reduced models and approximate the absorption
spectrum up to the given tolerance. We also illustrate that the automatically
determined model order increases logarithmically with the problem dimension,
compared to a linear increase of the number of eigenvalues within the energy
window. Furthermore, we observed that the computational cost of the proposed
algorithm only scales quadratically with respect to the problem dimension.