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The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem

Patrick Henning

TL;DR

The paper analyzes the convergence of generalized inverse iterations for the Gross–Pitaevskii eigenvector problem, establishing explicit local linear rates tied to the first spectral gap of the linearized GP operator and refining these via a weighted eigenvalue problem. It shows the basic inverse iteration converges locally with a rate bounded by $\lambda_1/\lambda_2$ and improves to $|\mu_1|$ through a weighted formulation; it extends the results to GFDN and to damped variants, and explains why spectral shifts can degrade convergence due to a blow-up in the weighting. A damping-based scheme is shown to globally converge with provable rates, and a discrete Sobolev gradient-flow perspective provides complementary convergence insights. Numerical experiments in 1D corroborate the theoretical rates, illustrating that the sharp weighted-rate $|\mu_1|$ accurately predicts asymptotic convergence even in small-gap regimes, and confirming the limited usefulness of spectral shifts for the GPE.

Abstract

In this paper we consider the generalized inverse iteration for computing ground states of the Gross-Pitaevskii eigenvector problem (GPE). For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross-Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao, Q. Du, SIAM J. Sci. Comput., 25 (2004)] or the damped inverse iteration suggested in [P. Henning, D. Peterseim, SIAM J. Numer. Anal., 53 (2020)]. Our analysis also reveals why the inverse iteration for the GPE does not react favourably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.

The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem

TL;DR

The paper analyzes the convergence of generalized inverse iterations for the Gross–Pitaevskii eigenvector problem, establishing explicit local linear rates tied to the first spectral gap of the linearized GP operator and refining these via a weighted eigenvalue problem. It shows the basic inverse iteration converges locally with a rate bounded by and improves to through a weighted formulation; it extends the results to GFDN and to damped variants, and explains why spectral shifts can degrade convergence due to a blow-up in the weighting. A damping-based scheme is shown to globally converge with provable rates, and a discrete Sobolev gradient-flow perspective provides complementary convergence insights. Numerical experiments in 1D corroborate the theoretical rates, illustrating that the sharp weighted-rate accurately predicts asymptotic convergence even in small-gap regimes, and confirming the limited usefulness of spectral shifts for the GPE.

Abstract

In this paper we consider the generalized inverse iteration for computing ground states of the Gross-Pitaevskii eigenvector problem (GPE). For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross-Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao, Q. Du, SIAM J. Sci. Comput., 25 (2004)] or the damped inverse iteration suggested in [P. Henning, D. Peterseim, SIAM J. Numer. Anal., 53 (2020)]. Our analysis also reveals why the inverse iteration for the GPE does not react favourably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.
Paper Structure (13 sections, 12 theorems, 109 equations, 2 figures)

This paper contains 13 sections, 12 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Analytical setting and preliminaries
  3. Basic inverse iteration and convergence rates
  4. Ostrowski theorem and calculation of Fréchet derivatives
  5. Proof of Theorem \ref{['thm_conv_basic_inverse_iteration']}
  6. Convergence rates for GFDN iterations
  7. Convergence rates for iterations with shift
  8. Convergence rates for iterations with damping
  9. Discrete projected Sobolev gradient flow and its convergence
  10. Proof of Theorem \ref{['thm_conv_inverse_iteration_damping']}
  11. Numerical experiments
  12. Inverse iteration applied to GPE with moderate spectral gap
  13. Inverse iteration applied to GPE with small spectral gap

Key Result

Proposition 2.1

Assume A1-A3, then there exist exactly two $L^2$-normalized minimizers $u$ of $E$ with which are $|u|$ and $-|u|$. Furthermore, it holds $u \in H^2(\Omega) \cap C^{0,\alpha}(\overline{\Omega})$ for some $0<\alpha < 1$.

Figures (2)

  • Figure 1: Contraction rates for model problem 1. We compare the actual contraction rate $r(n)$ at iteration $n$ with the upper bound $\tfrac{\lambda_1}{\lambda_2}$ that depends on the first spectral gap and the sharper bound $|\mu_1|$ that is obtained through the weighted eigenvalue problem \ref{['weighted-evp']}.
  • Figure 2: Contraction rates for model problem 2. The numerical contraction rates $r(n)$ at iteration $n$ are computed as in \ref{['contraction-rate-H1']}. The upper bounds for the rates are given by $\tfrac{\lambda_1}{\lambda_2}$ according to Theorem \ref{['thm_conv_basic_inverse_iteration']} and by $|\mu_1|$ according to Remark \ref{['remark-sharper-rates']}.

Theorems & Definitions (25)

  • Proposition 2.1: Minimizers of the Gross--Pitaevskii energy
  • Proposition 2.2: Gross--Pitaevskii eigenvector problem
  • Proposition 2.3
  • Definition 3.1: Basic inverse iteration for the GPE
  • Theorem 3.2: Local convergence of the basic inverse iterations
  • Remark 3.3: Sharper characterization of the convergence rates
  • Proposition 3.1: Ostrowski theorem
  • Lemma 3.4
  • proof
  • Lemma 3.6
  • ...and 15 more