Convergence of Eigenvector Continuation

Abstract

Eigenvector continuation is a computational method that finds the extremal eigenvalues and eigenvectors of a Hamiltonian matrix with one or more control parameters. It does this by projection onto a subspace of eigenvectors corresponding to selected training values of the control parameters. The method has proven to be very efficient and accurate for interpolating and extrapolating eigenvectors. However, almost nothing is known about how the method converges, and its rapid convergence properties have remained mysterious. In this letter we present the first study of the convergence of eigenvector continuation. In order to perform the mathematical analysis, we introduce a new variant of eigenvector continuation that we call vector continuation. We first prove that eigenvector continuation and vector continuation have identical convergence properties and then analyze the convergence of vector continuation. Our analysis shows that, in general, eigenvector continuation converges more rapidly than perturbation theory. The faster convergence is achieved by eliminating a phenomenon that we call differential folding, the interference between non-orthogonal vectors appearing at different orders in perturbation theory. From our analysis we can predict how eigenvector continuation converges both inside and outside the radius of convergence of perturbation theory. While eigenvector continuation is a non-perturbative method, we show that its rate of convergence can be deduced from power series expansions of the eigenvectors. Our results also yield new insights into the nature of divergences in perturbation theory.

arXiv comment5 pages and 4 figures (main text), 4 pages and 8 figures (supplemental), new analysis of the multi-parameter case, new application to BCS-BEC crossover and the unitary limit

References in corpus (24)

• Observation of a Strongly-Interacting Degenerate Fermi Gas of Atoms
• Revealing the Superfluid Lambda Transition in the Universal Thermodynamics of a Unitary Fermi Gas
• Accurate nuclear radii and binding energies from a chiral interaction
• Measurement of positive and negative scattering lengths in a Fermi gas of atoms
• Radio-Frequency Spectroscopy of Ultracold Fermions
• The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects
• Novel chiral Hamiltonian and observables in light and medium-mass nuclei
• Light-nuclei spectra from chiral dynamics
• Eigenvector continuation with subspace learning
• Radii and binding energies in oxygen isotopes: a puzzle for nuclear forces
• Global sensitivity analysis of bulk properties of an atomic nucleus
• Nuclear binding near a quantum phase transition
• Eigenvector Continuation as an Efficient and Accurate Emulator for Uncertainty Quantification
• Essential elements for nuclear binding
• Few- and many-nucleon systems with semilocal coordinate-space regularized chiral nucleon-nucleon forces
• Atomic nuclei from quantum Monte Carlo calculations with chiral EFT interactions
• Improved many-body expansions from eigenvector continuation
• New Ideas in Constraining Nuclear Forces
• High-precision numerical solution of the Fermi polaron problem and large-order behavior of its diagrammatic series
• Prediction of the higher-order terms based on Borel resummation with conformal mapping
• Precision benchmark calculations for four particles at unitarity
• Borel Resummation of the Perturbative Free Energy of Hot Yang-Mills Theory
• Variational Perturbation Theory for Summing Divergent Non-Borel-Summable Tunneling Amplitudes
• Summing Non-Borel Tunnelling Amplitudes by Variational Perturbation Theory

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• Nuclear Forces for Precision Nuclear Physics -- a collection of perspectives
• Training and Projecting: A Reduced Basis Method Emulator for Many-Body Physics
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• Accelerated quantum Monte Carlo with mitigated error on noisy quantum computer
• Self-learning Emulators and Eigenvector Continuation
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• Excited states from eigenvector continuation: the anharmonic oscillator
• Emulating \emph{ab initio} computations of infinite nucleonic matter
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• Nuclear-matter saturation and symmetry energy within $\Delta$--full chiral effective field theory
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• Emulating generator coordinate method with extended eigenvector continuation: Lipkin-Meshkov-Glick model
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• IMSRG-Net: A machine learning-based solver for In-Medium Similarity Renormalization Group
• Non-Hermitian quantum mechanics approach for extracting and emulating continuum physics based on bound-state-like calculations: Detailed description
• Fundamental limitations of the eigenvalue continuation approach
• Deformation, halo, and bubble structure: A paradigm shift of exotic phenomena in light to medium mass nuclei
• Efficient emulation of nuclear ground states with neural-network variational Monte Carlo and eigenvector continuation
• Chiral Effective Field Theory after Thirty Years: Nuclear Lattice Simulations