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. In this work, we present a novel, adaptive solution to this high computational overhead 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 and control theory. 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. On the basis of 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, for the systems studied, 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.

The Chronus Quantum software package

“…In addition to providing solvers which solve Equation (20b) in the full dimension of H, ChronusQ implements a series of model-order reduction (MOR) techniques 26,104 to improve the efficiency of evaluating Equation (20b) for multiple frequencies in a spectral region of interest. The MOR technique constructs a rational Krylov subspace which interpolates Equation (20a) in the frequency domain.…”

“…Calculations involving thousands of CPU cores and thousands of basis functions are regularly performed with ChronusQ on some of the world's largest supercomputers. 26,27 ChronusQ is compatible with any message passing implementation (MPI-2) compliant MPI, such as MPICH, OpenMPI, and Intel(R) MPI. To allow for the use of ChronusQ on work stations which may not possess an MPI implementation, message passing parallelism is an optional feature which may be enabled or disabled by the build system.…”

“…After several years of development, ChronusQ has become a full-fledged, standalone, ab initio quantum chemistry software package, which has proven itself useful both as a development platform to rapidly prototype electronic structure methods, as well as a robust production code capable of tackling complex science problems. 18,26,[33][34][35] In this article, we briefly review the design, features, and current developments of the ChronusQ software package.…”

Approximate Green’s Function Coupled Cluster Method Employing Effective Dimension Reduction

“…, m. The sampling of the selected frequencies in the regime follows the adaptive refinement strategy described in Ref. [73]. Basically, one can start with a uniformly sampled frequencies in the regime to construct a preliminary level reduced order model.…”

GFCCLib: Scalable and Efficient Coupled-Cluster Green's Function Library for Accurately Tackling Many Body Electronic Structure Problems

“…In the first loop, ω's are sampled following the adaptive midpoint refinement strategy described in Ref. 66 In the second loop, the GFCCSD singles and doubles equations need to be solved for all (ω,p) pairs in a given level. Here we are focused on Lines 4−15 of Algorithm ??…”