On preconditioned Riemannian gradient methods for minimizing the Gross-Pitaevskii energy functional: algorithms, global convergence and optimal local convergence rate

1. Problem Statement

The paper addresses the numerical minimization of the Gross-Pitaevskii (GP) energy functional with a rotation term, which models Bose-Einstein Condensates (BECs) and other quantum systems. The mathematical formulation is a constrained optimization problem:
$$\phi_g = \arg \min_{\phi \in \mathcal{M}} E(\phi)$$
where the energy functional $E(\phi)$ includes kinetic energy, external potential, rotation (angular momentum), and nonlinear interaction terms. The constraint set $\mathcal{M}$ is the unit sphere in the Sobolev space $H^1_0(D)$ under the $L^2$-norm (mass conservation).

Key Challenges:

• Non-convexity: The constraint manifold is non-convex.
• Symmetry and Degeneracy: The energy functional possesses two continuous symmetries:
  1. Phase Shift Invariance: $e^{i\alpha}\phi$ is a minimizer for any $\alpha$.
  2. Rotational Invariance: If the potential is symmetric, $\phi(A_\beta x)$ is a minimizer for any rotation angle $\beta$.
• Consequence: These symmetries imply that minimizers are not isolated points but form a continuous manifold (orbit) $S$. This degeneracy prevents the standard Second-Order Sufficient Condition (SOSC) from holding globally, making the analysis of local convergence rates for iterative solvers significantly more difficult than in non-degenerate cases.

2. Methodology

The authors propose a unified framework for Preconditioned Riemannian Gradient (P-RG) methods.

• Riemannian Geometry: The problem is treated on the infinite-dimensional Riemannian manifold $\mathcal{M}$. The tangent space $T_\phi \mathcal{M}$ consists of functions orthogonal to $\phi$ in $L^2$.
• Preconditioning: Instead of using the standard $L^2$ metric, the authors introduce a variable preconditioner $P_\phi$ (a symmetric, coercive linear operator) to define a new Riemannian metric. The Riemannian gradient is computed as the projection of the preconditioned Euclidean gradient onto the tangent space.
• Morse-Bott Assumption: To handle the degeneracy, the authors assume the GP energy functional is a Morse-Bott functional. This means the Hessian is positive definite on the subspace orthogonal to the symmetry directions (the "symmetry-breaking" subspace $N_\phi \mathcal{M}$), while having a kernel exactly spanned by the symmetry generators ($i\phi$ and $iL_z\phi$).
• Polyak-Łojasiewicz (PL) Inequality: By leveraging the Morse-Bott property, the authors derive a sharp PL inequality around the minimizer manifold. This inequality relates the energy gap to the norm of the Riemannian gradient, enabling convergence rate analysis despite the lack of isolated minimizers.

3. Key Contributions

A. Unified Framework and Global Convergence

• The paper establishes a general framework that encompasses existing methods (e.g., projected Sobolev gradient methods, $L^2$-projected gradient) as specific instances of P-RG with different preconditioners.
• Under mild assumptions on the preconditioner (continuity, coercivity, and invariance), the authors prove global convergence of the algorithm to a critical point satisfying the first-order necessary conditions.

B. Local Convergence Rate Characterization

• Theoretical Breakthrough: The paper rigorously derives the local linear convergence rate for P-RG in the presence of symmetry-induced degeneracy.
• Rate Formula: The convergence rate is shown to depend on the condition number $\kappa = L/\mu$ of the combined operator $P^{-1}_\phi (E''(\phi) - \lambda_g I)$ restricted to the subspace $N_\phi \mathcal{M}$:
  $$\|\phi_n - \phi_g\|_{H^1} \leq C \left( \sqrt{1 - \frac{\mu}{L}} \right)^n$$
  where $\mu$ and $L$ are the lower and upper bounds of the spectrum of the preconditioned Hessian.

C. Optimal Preconditioner Design

• The authors identify a quasi-optimal preconditioner:
  $$P_\phi = E''(\phi) - (\tilde{\lambda}_\phi - \sigma_0)I$$
  where $\tilde{\lambda}_\phi$ is the eigenvalue estimate and $\sigma_0 > 0$ is a small regularization parameter ensuring coercivity.
• Optimal Rate: Using this specific preconditioner, the method achieves the optimal local linear convergence rate known for gradient descent on strongly convex problems:
  $$\rho = \frac{L - \mu}{L + \mu} + \epsilon$$
  This rate is theoretically the best possible for first-order Riemannian gradient methods.

4. Main Results

1. Global Convergence: The sequence of iterates generated by P-RG converges globally to a minimizer (or a point in the minimizer manifold) for any initial guess, provided the step size is within a specific range.
2. Local Linear Convergence: Assuming the Morse-Bott property, the algorithm exhibits local linear convergence. The rate is explicitly characterized by the spectral properties of the preconditioned Hessian on the symmetry-orthogonal subspace.
3. Equivalence of Error Metrics: The paper proves that the energy error ($E(\phi_n) - E(\phi_g)$) and the wavefunction error ($\|\phi_n - \phi_g\|_{H^1}$) are equivalent near the solution, justifying standard stopping criteria based on energy or wavefunction differences.
4. Numerical Verification:
   • Numerical experiments on rapidly rotating BECs confirm the Morse-Bott property (the Hessian has exactly two zero eigenvalues corresponding to symmetries).
   • The proposed quasi-optimal preconditioner ($P_4$) significantly outperforms standard preconditioners ($P_1, P_2, P_3$), reducing the number of iterations from $>10^5$ to $\approx 10^2$ to reach high precision, matching the theoretical optimal rate.

5. Significance

• Theoretical Rigor: This is the first work to rigorously derive the local convergence rate of preconditioned Riemannian gradient methods for the GP functional with two symmetric structures (phase and rotation). Previous works often assumed isolated minimizers or provided only qualitative convergence results.
• Practical Impact: The identification of the quasi-optimal preconditioner provides a concrete recipe for accelerating simulations of rotating BECs, which are computationally expensive due to the need for high resolution and the presence of vortices.
• Generalizability: The framework extends the theory of Riemannian optimization from isolated minimizers to Morse-Bott manifolds, offering a blueprint for analyzing other degenerate optimization problems in physics and engineering.
• Clarification of Symmetry: The paper clarifies that while the problem is degenerate, it retains properties analogous to strongly convex problems when viewed on the quotient manifold, allowing for optimal convergence rates if the preconditioner is designed to respect the geometry of the symmetry.

In summary, the paper bridges the gap between the theoretical analysis of degenerate optimization problems and the practical efficiency of numerical solvers for the Gross-Pitaevskii equation, providing both a unified theoretical framework and a highly efficient algorithmic strategy.