Eigenvector Continuation and Projection-Based Emulators

TL;DR

The paper surveys eigenvector continuation (EC), a model-driven reduced-basis emulator for parametric eigenvalue problems, highlighting its offline-online workflow, convergence properties, and broad applicability from few-body to many-body quantum systems. By constructing a low-dimensional subspace from eigenvector snapshots and projecting the full problem, EC achieves substantial speed-ups while preserving accuracy, and is extended to scattering, finite-volume resonances, and quantum Monte Carlo contexts. The work surveys theoretical foundations (affine parameter dependence, variational/Galerkin formulations, and convergence bounds) and diverse nuclear-structure applications (No-Core Shell Model emulators, subspace-projected CC, and shell-model extensions), as well as extensions to scattering, resonances, and QMC. The findings demonstrate EC’s potential to enable rapid parameter studies, uncertainty quantification, and large-scale statistical analyses, with broad relevance to atomic/molecular physics and quantum chemistry, while also identifying remaining challenges such as non-Hermitian target-state identification and handling non-affine dependencies.

Abstract

Eigenvector continuation is a computational method for parametric eigenvalue problems that uses subspace projection with a basis derived from eigenvector snapshots from different parameter sets. It is part of a broader class of subspace-projection techniques called reduced-basis methods. In this colloquium article, we present the development, theory, and applications of eigenvector continuation and projection-based emulators. We introduce the basic concepts, discuss the underlying theory and convergence properties, and present recent applications for quantum systems and future prospects.

Figures (17)

• Figure 1: Schematic classification of model order reduction emulators into data-driven methods, including Gaussian processes, artificial neural networks, and dynamic mode decomposition; model-driven methods, including reduced-basis methods (RBMs); and hybrid methods. Eigenvector continuation (EC) approaches are a subset of RBM.
• Figure 2: Ground state energy $E$ of the Bose-Hubbard model divided by $t$ versus $U/t$. The exact ground-state energies are shown with open circles, while the EC results are shown for variational subspace dimensions varying from $2$ to $4$. In order to highlight the avoided level crossing, the exact excited state energies are also shown as open squares. Adapted from Frame:2017fah.
• Figure 3: While the power series expansion at $\theta=0$ converges only for $|\theta|<|z|$, we can choose a secondary point $w$ with $|w|<|z|$. The power series expansion at $\theta = w$ converges in the shaded region shown and can be re-expressed as a double series around $\theta=0$. Frame:2017fah.
• Figure 4: Reduced-basis model workflow for a matrix eigenvalue problem. a) High-fidelity calculations of snapshots, each of large size $N_h$, are b) projected in the offline stage to a reduced-basis matrix of small size $n_b\times n_b$. In the c) online stage, the emulator only uses size $n_b$ operations. Adapted from a figure in Drischler:2022ipa.
• Figure 5: Normalized singular values from $n_b=50$ snapshots of various functions that enter a nuclear physics energy density functional. The snapshots correspond to different parameter sets to be used in a Galerkin formulation of the energy density functional, which will be solved many times with different sets for Bayesian parameter estimation. The rapid decrease with principal component number $k$ indicates that a small basis size will be accurate, leading in this case to speed-ups of several thousand compared to the original solver. Adapted from a figure in Giuliani:2022yna.