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Newton-based alternating methods for the ground state of a class of multi-component Bose-Einstein condensates

Pengfei Huang, Qingzhi Yang

Abstract

The computation of the ground states of special multi-component Bose-Einstein condensates (BECs) can be formulated as an energy functional minimization problem with spherical constraints. It leads to a nonconvex quartic-quadratic optimization problem after suitable discretizations. First, we generalize the Newton-based methods for single-component BECs to the alternating minimization scheme for multi-component BECs. Second, the global convergent alternating Newton-Noda iteration (ANNI) is proposed. In particular, we prove the positivity preserving property of ANNI under mild conditions. Finally, our analysis is applied to a class of more general "multi-block" optimization problems with spherical constraints. Numerical experiments are performed to evaluate the performance of proposed methods for different multi-component BECs, including pseudo spin-1/2, anti-ferromagnetic spin-1 and spin-2 BECs. These results support our theory and demonstrate the efficiency of our algorithms.

Newton-based alternating methods for the ground state of a class of multi-component Bose-Einstein condensates

Abstract

The computation of the ground states of special multi-component Bose-Einstein condensates (BECs) can be formulated as an energy functional minimization problem with spherical constraints. It leads to a nonconvex quartic-quadratic optimization problem after suitable discretizations. First, we generalize the Newton-based methods for single-component BECs to the alternating minimization scheme for multi-component BECs. Second, the global convergent alternating Newton-Noda iteration (ANNI) is proposed. In particular, we prove the positivity preserving property of ANNI under mild conditions. Finally, our analysis is applied to a class of more general "multi-block" optimization problems with spherical constraints. Numerical experiments are performed to evaluate the performance of proposed methods for different multi-component BECs, including pseudo spin-1/2, anti-ferromagnetic spin-1 and spin-2 BECs. These results support our theory and demonstrate the efficiency of our algorithms.
Paper Structure (18 sections, 15 theorems, 77 equations, 4 figures, 7 tables, 2 algorithms)

This paper contains 18 sections, 15 theorems, 77 equations, 4 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. The alternating minimization method
  4. Alternating Newton-Noda iteration
  5. Properties of $\mathbf{u}_k$ and $\mathbf{v}_k$
  6. Properties of $\theta_k$
  7. Convergence of alternating Newton-Noda iteration
  8. Extensions
  9. The strategy for choosing $\lambda_k$ and $\mu_k$
  10. Multi-block problems
  11. Numerical experiments
  12. Implementation details
  13. Preconditioner
  14. Choice of $\lambda_k$ and $\mu_k$
  15. Application in pseudo spin-1/2 BECs
  16. ...and 3 more sections

Key Result

Theorem 2.3

(varga1962iterative) \newlabelthm:M-matrix0 Let $A=sI-C$, where $C\in\mathbb{R}^{n\times n}$ is nonnegative. The following are equivalent: Futhermore, if $A$ is a real, symmetric and nonsingular irreducible matrix, then $A^{-1}>0$ if and only if $A$ is positive definite.

Figures (4)

  • Figure 1: nrmG and objective value versus the number of iterations with different $\lambda_k$.
  • Figure 2: The convergence behaviour of ANNI. The left is for the 2D case of Example \ref{['exm:spin-1']} with $U=[-16,16]^2,~\beta_0=0.3,~\beta_1=0.1,~M=0.9$. In this case, ANNI shows quadratically convergence. The right is for the 3D case of Example \ref{['exm:spin-2']} with $U=[-16,16]^3,~\beta_0=243,~\beta_1=12.1,~\beta_2=-13,~M=0.5$. In this case, ANNI converges rather slowly.
  • Figure 3: $\hat{\lambda}_k$, $\hat{\mu}_k$, $\bar{\lambda}_k$, $\bar{\mu}_k$ versus the number of iterations for Example \ref{['exm:1']} with $\beta=100$, $\alpha=0.9$.
  • Figure 4: Wave functions of the ground state of a spin-1/2 BEC for Example \ref{['exm:1']} with $\beta=100$, i.e., $\phi_1(x)$ (dash line) and $\phi_2(x)$ (solid line). The left is for $\alpha=0.2$, and the right is for $\alpha=0.9$.

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Proof 1
  • Theorem 3.2
  • Proof 2
  • Remark 3.3
  • Lemma 4.1
  • ...and 28 more